# Heuristic Vision Pre-Training with Self-Supervised and Supervised Multi-Task
Learning

Zhiming Qian

###### Abstract

To mimic human vision with the way of recognizing the diverse and open world,
foundation vision models are much critical. While recent techniques of self-
supervised learning show the promising potentiality of this mission, we argue
that signals from labelled data are also important for common-sense
recognition, and properly chosen pre-text tasks can facilitate the efficiency
of vision representation learning. To this end, we propose a novel pre-
training framework by adopting both self-supervised and supervised visual pre-
text tasks in a multi-task manner. Specifically, given an image, we take a
heuristic way by considering its intrinsic style properties, inside objects
with their locations and correlations, and how it looks like in 3D space for
basic visual understanding. However, large-scale object bounding boxes and
correlations are usually hard to achieve. Alternatively, we develop a hybrid
method by leveraging both multi-label classification and self-supervised
learning. On the one hand, under the multi-label supervision, the pre-trained
model can explore the detailed information of an image, e.g., image types,
objects, and part of semantic relations. On the other hand, self-supervised
learning tasks, with respect to Masked Image Modeling (MIM) and contrastive
learning, can help the model learn pixel details and patch correlations.
Results show that our pre-trained models can deliver results on par with or
better than state-of-the-art (SOTA) results on multiple visual tasks. For
example, with a vanilla Swin-B backbone, we achieve 85.3% top-1 accuracy on
ImageNet-1K classification, 47.9 box AP on COCO object detection for Mask
R-CNN, and 50.6 mIoU on ADE-20K semantic segmentation when using Upernet. The
performance shows the ability of our vision foundation model to serve general
purpose vision tasks.

##  1 Introduction

To learn the intrinsic universal knowledge of visual world, pre-training
models are motivated to learn fundamental representations to support a broad
range of downstream tasks, similar to what humans would do [YCC+21]. One
milestone for the pre-training issue is the introduction of transfer learning
[PY09], which formalizes a two-stage learning framework: a pre-training stage
to capture knowledge from one or more source tasks, and a fine-tuning stage to
transfer the captured knowledge to target tasks. Owing to the wealth of
knowledge obtained in the pre-training stage, the fine-tuning stage can enable
models to well handle target tasks with limited samples. Specifically,
supervised pre-training with image classification on ImageNet [DDS+09] has
driven the progress in solving many computer vision tasks in the past few
years, such as image classification [DBK+20][LLC+21], object detection
[HGDG17][CV18] and semantic segmentation [KGHD19][XLZ+18]. Recently, study in
self-supervised pre-training shows that it can generalize well for specific
downstream tasks by taking ingenious strategies of many self-supervised
objectives, such as contrastive learning [CKNH20][CXH21][CTM+21][CMM+20] and
Masked Image Modeling (MIM) [BDW21][HCX+21][XZC+21][DBZ+21].

Figure 1: An illustration of a few heuristic insights. For an image, we
understand its visual content by simultaneously perceiving scene properties,
inside objects and their correlations, motivating us to learn visual
representations with the relevant tasks.

To investigate representation between supervised and self-supervised methods,
Grigg et al. [GBRW21] recently find that supervised and self-supervised
methods learn similar intermediate representations through dissimilar means,
but diverge rapidly in the final few layers. The similarity indicates a shared
set of primitives, and the divergence is probably caused by the layers
strongly to the distinct learning objectives. Furthermore, taking both weak
supervision of image labels and self-supervision of each single modality,
multi-model methods [YCC+21][LSG+21] can achieve much competitive results on
the authoritative visual challenge tasks. Besides, the absolute model size for
current vision models is just able to reach about 1-2 billion parameters
[LHL+21], resulting in the fact that the need of large-scale unlabelled data
for self-supervised learning is not urgent [ENIT+21]. Based on these views, we
employ a large-scale multi-label dataset for both supervised and self-
supervised learning, and design a Heuristic Vision Pre-training method with
Multi-Task Learning (HVP-MTL) by combining both supervised multi-label
classification and self-supervised objectives. We believe that the open large-
scale supervised datasets can currently make good generalization performance,
which has been proved with the advanced work [DBK+20] based on JFT-300M
[SSSG17]. Specifically, the Tencent-ML dataset [WCF+19] is employed. Then,
with the purpose of learning fundamental representations, we first set a few
heuristic problems. As seen in Figure 1, given an image, it is natural to ask
some questions for understanding, such as which type of the image is it, what
are the objects and their correlations, where are these objects, and how does
it like in 3D space. To cope with these problems, we propose a novel framework
by taking supervised and self-supervised tasks, including multi-label
classification, reconstruction with masked images, and embedding alignment
with different image views. The relations between the above heuristic problems
and pre-text tasks are illustrated in Figure 1. Our contributions are
summarized as follows:

  * •

We propose a unified framework for multi-task learning by setting a few
heuristic pre-text tasks, with the purpose of learning basic visual
representations. Supervised pre-text tasks can usually achieve sustainable
gain with the increasing of the dataset size, and self-supervised pre-text
tasks are class-agnostic and promising for learning fundamental structures.
Combined both supervised and self-supervised pre-text tasks in a heuristic
way, we can learn more consistent representations with human beings.

  * •

For supervised learning, we adopt multi-label classification, and employ
momentum distillation[LSG+21] for label denoising. To solve the label
imbalance problem, we develop a novel weighted asymmetric loss [RBBZ+21] for
multi-label classification.

  * •

For self-supervised learning, we use Masked Image Modeling (MIM)[XZC+21] for
implicitly infer intrinsic objects with their locations and correlations, and
employ contrastive learning for embedding alignment with different image
views, which can benefit scene understanding in 3D space. To make contrastive
learning more efficient and stable, we take online clustering with SWaV
[CMM+20], and impose a layer truncation to solve the collapse problem of the
exponential computation when using Sinkhorn-Knopp [Cut13].

##  2 Related Work

###  2.1 Multi-task learning

Multi-task learning can bring more insightful interpretation for learning
features, but might suffer from negative transfer due to task conflicts
[NVR+20]. To overcome this, works such as GRAD-CAM [SCD+17] proposes
techniques that provide visual explanations for decisions made by a model to
make them more transparent and explainable. Then, multi-model methods, such as
ALBEF [LSG+21] and Florence [YCC+21], utilize both weak supervision of image
descriptions and self-supervision of each single modality for pre-training,
achieving a great success on downstream visual tasks. In our study, we use
multi-task learning by setting a few heuristic pre-text tasks, with the
purpose of learning shared features among these prompting pre-text tasks and
finding the intrinsic image representation.

###  2.2 Multi-label classification

Multi-label classification are more natural descriptions for images, and can
tell the image types, properties, inside objects, or even the correlations
among objects. For its nature of multiple labels on one image, the co-
occurrence of concepts in a large-scale dataset could be mined as prior
knowledge for subsequent classification. A key characteristic of multi-label
classification is the inherent positive-negative imbalance created when the
overall number of labels is large [WCF+19]. To address this issue, a few work
suggests using a dedicated loss function to statically handle the imbalance,
such as distribution-balanced loss [WHL+20], focal loss [LGG+17], asymmetric
loss [RBBZ+21]. Another key characteristic is label correlation, graph-based
methods [CXH+19] and class-aware maps[CWJG19][YHP+20] are employed to
represent the relationship of labels. While modeling label correlations can
introduce additional gains in multi-label classification, it is also arguable
that it may learn spurious correlations when the label statistics are
insufficient. Rather than using graph, the work in [LZY+21] leads the network
to focus on regions of interest for implicitly capturing label relationships
by introducing a Transformer decoder. However, few work focuses on the
intrinsic noising problem in the multi-class dataset. In our work, the
Transformer decoder is applied with a novel weighted asymmetric loss, and we
employ momentum distillation [LSG+21] for label denoising.

###  2.3 Self-supervised learning

Self-supervised learning has attracted increasing attention over the past few
years, as deep learning networks become more and more data-hungry and it is
impossible to label everything in the world. There are two main categories to
alleviate this issue, w.r.t. contrastive and generative. Contrastive learning
is a discriminative approach that aims at grouping similar samples to be
closer and dissimilar samples to be far from each other. By using a noise
contrastive estimator (NCE) [OLV18] to compare instances instead of
classifying them, dealing with a large number of images simultaneously is
usually required for good performance. In practice, this requires large
batches [CKNH20] or memory banks [CXH21]. In short, contrastive-based methods
heavily depend on the strong data augmentation and effective negatives
sampling. To alleviate this, several variants allow automatic grouping of
instances in the form of clustering [CMM+20]. Here, we take a robust online
clustering method for learning similarity from different views, with the
purpose of pursuing memory efficiency and visual coherence.

The other recent resurgent field is generative self-supervised learning
[BDW21][HCX+21], training an encoder and a decoder under the objective of
reconstruction loss, aiming at recovering the corrupted or masked input, which
has yielded the most successful frameworks in NLP [DCLT18]. Recently, BEiT
[BDW21] proposes a pre-text task of MIM by recovering the original visual
tokens based on the corrupted image patches. Then, MAE [HCX+21] reconstructs
pixels with an asymmetric encoder-decoder architecture by masking a high
proportion of the input image. More recently, PeCo [DBZ+21] refine the visual
codebooks, and SimMIM [XZC+21] further study the influence of patch masking
strategies. In this work, we take the pre-text task of MIM based on
Transformers by directly learning from raw pixels to avoid the information
loss.

##  3 Method

In this section, we first introduce the overview of our framework in Section
3.1. Then, the pre-training objectives are delineated in Section 3.2. In the
end, we describe the pre-training dataset and implementation details in
Section 3.3.

Figure 2: The pipeline of HVP-MTL. A Transformer backbone is first employed to
encode image to a feature map. Then, we take several decoders for tasks of
multi-label classification, MIM, contrastive learning and momentum
distillation. Here, $sg$ represents stop gradient, and EMA is used for update
parameters of the teacher network.

###  3.1 Overall architecture

As illustrated in Figure 2, an image is first transformed into different views
by conducting a few augmentations, such as color jittering, random cropping,
patch masking, random rotation and so on. Then, an image encoder with the Swin
[LLC+21] or ViT [DBK+20] backbone is employed to generate the feature map,
which is usually the output of the last stage of the backbone. Furthermore,
several head decoders with different losses are introduced for heuristic
representation learning, including a Transformer decoder for multi-label
classification, a clustering decoder with prototypes [CMM+20] for contrastive
learning, and a MIM decoder for reconstructive learning. Besides, momentum
distillation is employed for label denoising.

###  3.2 Pre-training objectives

Transformer decoder for multi-label classification. Given an image
$x\in\mathbf{R}^{H_{0}\times W_{0}\times 3}$ as input, we extract its spatial
features $\mathcal{F}_{0}\in\mathbf{R}^{H\times W\times d_{0}}$ using the
backbone, where $H_{0}\times W_{0},H\times W$ represent the height and weight
of the input image and the feature map respectively, and $d_{0}$ denotes the
dimension of features. After that, we add a linear projection layer to project
the features from dimension $d_{0}$ to $d$ to match with the desired query
dimension in the following Transformer decoder, and reshape the projected
features to be $\mathcal{F}\in\mathbf{R}^{H\times W\times d}$. Finally, we use
label embeddings as queries $Q_{0}\in\mathbf{R}^{C\times d}$ and perform
cross-attention to extract category-related features from the spatial features
using the Transformer decoder [LZY+21], where $C$ is the number of categories.
To alleviate the strong imbalance between positive and negative images in each
category when taking multi-label classification, we follow the asymmetric loss
[RBBZ+21], and refine it with a weighted asymmetric loss:

$\begin{cases}\mathcal{L}_{+}^{\text{mcls}}=\eta(1-p)^{\gamma_{+}}\text{log}(p)\\\
\mathcal{L}_{-}^{\text{mcls}}=p^{\gamma_{-}}\text{log}(1-p)\end{cases}$ (1)

where $p$ denotes the posterior probability with respect to a category, $\eta$
is the positive weight, $\gamma_{+}$ and $\gamma_{-}$ are the positive and
negative focusing parameters.

Clustering decoder with prototypes for contrastive learning. Given two image
features $f_{t}$ and $f_{s}$ from two different augmentations of the same
image, we compute their codes $q_{t}$ and $q_{s}$ by matching these features
to a set of $K$ prototypes ${c_{1},...,c_{K}}$. We setup a “swapped”
prediction problem with the following loss function:

$\mathcal{L}^{\text{cl}}(f_{t},f_{s})=\textit{l}(f_{t},q_{s})+\textit{l}(f_{s},q_{t})$
(2)

Then, we define the $\textit{l}(f_{t},q_{s})$ as:

$\textit{l}^{\text{cl}}(f_{t},q_{s})=-\sum_{k}{q_{s}^{(k)}\text{log}p_{t}^{(k)}}$
(3)

where $p_{t}^{(k)}=\text{softmax}({f_{t}}^{\text{T}}c_{k}/\tau)$, $\tau$ is a
temperature parameter. The problem can be optimized by Sinkhorn-Knopp [Cut13].
To avoid the collapse with the exponential operation, we adopt a truncated
strategy by clamping the input tensor with a threshold of $T_{max}$.

MIM decoder for reconstruction. Inspired by the work in SimMIM [XZC+21], and
use a learnable mask token vector to replace each masked patch. Image patches
are the basic processing units of vision Transformers [DBK+20] [LLC+21]. it is
convenient to apply the masking operation on patch-level that a patch is
either fully visible or fully masked. For the model Swin[LLC+21], we consider
equivalent patch sizes of different resolution stages, $4\times 4\to 32\times
32$, and adopt $32\times 32$ by default, which is the patch size of the last
stage. For ViT[DBK+20], we adopt $32\times 32$ as the default masked patch
size. The reconstructive loss is defined as:

$\mathcal{L}^{\text{mim}}=\frac{1}{\Omega(x)}\|y-x\|_{1}$ (4)

where $y\in\mathbf{R}^{H_{0}\times W_{0}\times 3}$ is the reconstruction of
the input image $x$.

Momentum distillation for label denoising. As [WCF+19] indicates, the
annotated tags for most images in Open Images [KDA+17] are generated by
machine, while only a few fraction of annotations are verified by humans. The
noisy annotations are unavoidable and they are also included in the Tencent-ML
dataset [WCF+19]. To alleviate this, we propose to learn from pseudo-targets
generated by the momentum model as that in [LSG+21]. The momentum model is a
continuously-evolving teacher which consists of exponential-moving-average
(EMA) versions of the backbone and the Transformer decoder for multi-label
classification. We train the base model such that its predictions match the
ones from the momentum model. Specially, we take the vector of cosine
similarities between image embedding and the corresponding label embeddings
for momentum distillation. Here we define the cosine similarity as:

$\mathcal{S}(g,Q_{0})=Q_{0}\otimes g$ (5)

where $g\in\mathbf{R}^{d}$ is the embedding vector learned from the above
Transformer decoder, and $\otimes$ is the matrix product. Then, the
distillation loss is defined as:

$\begin{split}\mathcal{L}^{\text{mom}}=E_{g,g^{\prime}}(KL(\mathcal{S}(g,Q_{0}),\mathcal{S}(g^{\prime},Q_{0}))+\\\
KL(\mathcal{S}(g^{\prime},Q_{0}),\mathcal{S}(g,Q_{0})))/2\end{split}$ (6)

where $g^{\prime}\in\mathbf{R}^{d}$ is the embedding of momentum distillation,
and $KL(\cdot)$ is the Kullback-Leibler (KL) divergence.

###  3.3 Implementation details for pre-training

Based on the above objectives, the full pre-training loss is as:

$\mathcal{L}=\alpha_{1}(\mathcal{L}_{+}^{\text{mcls}}+\mathcal{L}_{-}^{\text{mcls}})+\alpha_{2}\mathcal{L}^{\text{cl}}+\alpha_{3}\mathcal{L}^{\text{mim}}+\alpha_{4}\mathcal{L}^{\text{mom}}$
(7)

where $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are the weights for multi-
label classification loss, contrastive loss, reconstruction loss and momentum
distillation loss, and are set as 0.001, 0.02, 1.0 and 10.0 in our
implementation, respectively. Besides, the parameters for multi-label
classification, i.e. $\eta$, $\gamma_{+}$ and $\gamma_{-}$, are set as 10, 4,
1, respectively. The truncated threshold $T_{max}$ for Sinkhorn-Knopp is set
as 10, and the momentum parameter for updating the momentum model is set as
0.995. For Swin-B [LLC+21] or ViT-B [DBK+20], we pre-train the model for 30
epochs using a batch size of 1024 on 64 NVIDIA A100 GPUs. We use the AdamW
[LH17] optimizer with a weight decay of 0.05. The learning rate is warmed-up
to 1e-4 in the first 5 epochs, and decayed to 1e-7 following a cosine
schedule. During pre-training, we take random image crops of resolution 224 ×
224 as input, and also apply Randaugment [CZSL20].

##  4 Experimental Results

Generally, computer vision pipelines that employ self-supervised learning
performs two tasks: a pretext task and a downstream task. The pre-training
data with respect to the Tencent-ML dataset [WCF+19] collects about 18 million
images with 11,166 categories from existing well-known datasets, i.e., Open
Images [KDA+17] and ImageNet [DDS+09]. To show the effectiveness of HVP-MTL as
a foundation model, we conduct experiments on ImageNet-1K (IN-1K)
classification [DDS+09], COCO object detection [LMB+14], and ADE20K [ZZP+19]
semantic segmentation, which are the most common downstream tasks in computer
vision. We also provide comprehensive ablation studies on the effects of
scaling backbones and each component of HVP-MTL.

###  4.1 ImageNet-1K Classification

ImageNet-1K was created by selecting a subset of 1.2M images from ImageNet
dataset [DDS+09], that belong to 1000 mutually exclusive classes. For fair
comparison, we follow the training strategy in SimMIM, and train 100 epochs
for all our models with the input size of $224\times 224$. In Table 1 and
Table 2, we compare our proposed HVP-MTL with state-of-the-art (SOTA) pre-
training methods, such as MoCo v3 [CXH21], DINO [CTM+21], MAE [HCX+21], SimMIM
[XZC+21], BEiT [BDW21] and PeCo [DBZ+21], by measuring Top-1 accuracy on
ImageNet-1K classification with the backbones of ViT-B and Swin-B,
respectively. We also compare supervised pre-training models with the datasets
of ImageNet-22K (IN-22K) and JFT-300M [SSSG17]. It shows that our method
achieves the highest Top-1 accuracy with 84.2% for ViT-B and 85.3% for Swin-B,
surpassing the supervised method with IN-1K by 2.4% and 2.0%, respectively. It
is also worth noting that we achieve the same performance with the supervised
method with JFT-300M for ViT-B. However, the later use a much larger dataset,
and train more steps than ours.

| Pre-train
---
Method
| Pre-train
---
dataset
| Pre-train
---
epochs
| Input
---
size
| Top-1
---
accuracy
Supervised | - | - | $224^{2}$ | 81.8
Supervised | IN-22K | 90 | $384^{2}$ | 84.0
Supervised | JFT-300M | 7 | $384^{2}$ | 84.2
MoCo v3 | IN-1K | 300 | $224^{2}$ | 83.2
DINO | IN-1K | 300 | $224^{2}$ | 82.8
MAE | IN-1K | 1600 | $224^{2}$ | 83.6
SimMIM | IN-1K | 800 | $224^{2}$ | 83.8
BEiT | IN-1K | 300 | $224^{2}$ | 82.8
PeCo | IN-1K | 300 | $224^{2}$ | 84.1
HVT-MTL | Tencent-ML | 30 | $224^{2}$ | 84.2
Table 1: Comparison of different pre-training methods on ImageNet-1K classification with the backbone of ViT-B. | Pre-train
---
Method
| Pre-train
---
dataset
| Pre-train
---
epochs
| Input
---
size
| Top-1
---
accuracy
Supervised | - | - | $224^{2}$ | 83.3
Supervised | IN-22K | 90 | $384^{2}$ | 85.2
SimMIM | IN-1K | 800 | $224^{2}$ | 84.0
HVT-MTL | Tencent-ML | 30 | $224^{2}$ | 85.3
Table 2: Comparison of different pre-training methods on ImageNet-1K
classification with the backbone of Swin-B.

###  4.2 COCO Object Detection

Next, we evaluate different pre-training methods with Swin-B on COCO objection
detection [LMB+14] with the Mask R-CNN framework [HGDG17]. Specifically, we
follow the fine-tuning strategy with 1$\times$ schedule, i.e. the 12 training
epoch schedule, on the COCO training set. Table 3 reports the results of
different pre-training methods, such as DINO [CTM+21], PeCo [DBZ+21] and
supervised methods pre-trained on IN-1K and IN-22K. It shows that our proposed
method outperforms all the counterparts. In details, our method outperforms
the method on IN-22K by +1.0 box AP, and surpasses others by large margins.
The promising results validate that large-scale supervised datasets are much
valuable for visual representation, and can deliver useful information by
transferring from classification tasks to object detection tasks.

| Pre-train
---
method
| Pre-train
---
dataset
$AP^{b}$ | $AP_{50}^{b}$ | $AP_{75}^{b}$
Supervised | IN-1K | 43.7 | 66.6 | 47.7
Supervised | IN-22K | 46.9 | 68.8 | 51.6
DINO | IN-1K | 43.2 | 66.2 | 47.6
PeCo | IN-1K | 43.9 | 66.3 | 48.2
HVT-MTL | Tencent-ML | 47.9 | 70.1 | 52.5
Table 3: Comparison of different pre-training methods on COCO object detection
with the backbone of Swin-B.

###  4.3 ADE20K Semantic Segmentation

We further investigate the capability of our method for semantic segmentation
on the ADE20K dataset [ZZP+19] based on the backbone of Swin-B. Here, we
employ Upernet [XLZ+18] as the basic framework. For fair comparison, we follow
the previous work [DBC+21], and train Upernet with 160k iterations by setting
batch size as 16. In Table 4, we report the results in terms of mIoU for
different methods, such as DINO [CTM+21], BEiT [BDW21], PeCo [DBZ+21] and
supervised methods pre-trained on IN-1K and IN-22K. It can be seen that, our
method also achieves the highest performance. Compared to the methods of
purely self-supervised methods, the performance gain is very promising, and
demonstrates the effectiveness of our pre-training method again.

| Pre-train
---
method
| Pre-train
---
dataset
mIoU
Supervised | IN-1K | 48.0
Supervised | IN-22K | 50.31
DINO | IN-1K | 44.2
BEiT | IN-1K | 45.7
PeCo | IN-1K | 46.7
HVT-MTL | Tencent-ML | 50.6
Table 4: Comparison of different pre-training methods on ADE20K semantic
segmentation with the backbone of Swin-B.

###  4.4 Ablation Study

To better understand HVP-MTL, we ablate each key component and evaluate the
performance on ImageNet-1K classification based on ViT-B [DBK+20]. As
explained above, there are four key designs in our methods, i.e., multi-label
classification, MIM, contrastive learning for different image views, and
momentum distillation for label denoising. As seen in Table 6, we observe
relatively large performance drop on ImageNet classification by removing the
multi-label classification or MIM task from our framework, indicating that
learning with MIM and multi-label classification together is very crucial.

Method | Top-1 accuracy
---|---
HVT-MTL | 84.2
w/o Multi-label classification | 83.6
w/o MIM | 83.8
w/o Contrastive learning | 84.1
w/o Momentum distillation | 83.9
Table 5: Ablation study for pre-training using different strategies with ViT-B
on ImageNet-1K classification.

Then, we adopt Swin Transformer of different model sizes for pre-training
experiments, including Swin-T, Swin-S, and Swin-B [LLC+21]. We train 30 epochs
on the Tencent-ML dataset for all the pre-training tasks, and fine-tune with
100 epochs with the input size of $224\times 224$. Table 6 lists the results
of our approach with different model sizes. With our pre-training, all of
models achieve higher accuracy than their supervised counterparts.
Specifically, models with larger size achieve more gains than smaller ones,
showing good scalable charactistics for further improving the performance.

| Pre-train
---
method
| Pre-train
---
dataset
Swin-T | Swin-S | Swin-B
supervised | - | 81.3 | 83.0 | 83.3
HVT-MTL | Tencent-ML | 81.6 | 83.7 | 84.2
Table 6: Ablation study for pre-training with backbones of different sizes on
ImageNet-1K classification.

##  5 Conclusion

This paper proposes HVP-MTL, a new framework for vision representation
learning. HVP-MTL combines self-supervised and supervised visual tasks in a
multi-task manner to cope with a few heuristic problems. We theoretically and
experimentally verify the effectiveness of the proposed multi-task learning
framework. Compared to existing methods, HVP-MTL offers better performance
with the same vision models on multiple downstream tasks. For the future work,
we plan to develop more powerful large models with good scaling performance
for pre-training on large-scale multi-modal datasets, and employ more
downstream tasks, such as depth/flow estimation, tracking, as well as
additional vision and language tasks. In addition, the studies of adversarial
attacks against pre-train models is also an interesting direction.

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# Electronic structure and physical properties of candidate topological
material GdAgGe

D. Ram  Department of Physics, Indian Institute of Technology, Kanpur 208016,
India  J. Singh  Department of Physics, Indian Institute of Technology
Hyderabad, Kandi, Medak 502 285, Telangana, India  M. K. Hooda  Department of
Physics, Indian Institute of Technology, Kanpur 208016, India  O. Pavlosiuk
Institute of Low Temperature and Structure Research, Polish Academy of
Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland  V. Kanchana
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Hyderabad, Kandi, Medak 502 285, Telangana, India  Z. Hossain
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur
208016, India  Institute of Low Temperature and Structure Research, Polish
Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland  D. Kaczorowski
<EMAIL_ADDRESS>Institute of Low Temperature and Structure Research,
Polish Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland

###### Abstract

We grew needle-shaped single crystals of GdAgGe, which crystallizes in a
noncentrosymmetric hexagonal crystal structure with space group
P$\overline{6}$2m (189). The magnetic susceptibility data for H $\perp$ c
reveal two pronounced antiferromagnetic transitions at TN1 = 20 K and TN2 =
14.5 K. The magnetic susceptibility anomalies are less prominent for H
$\parallel$ c. The transition at TN1 is accompanied by a pronounced heat
capacity anomaly confirming the bulk nature of the magnetic transition. Below
TN1, the electrical resistivity data follows a T3/2 dependence. In the
magnetically ordered state, GdAgGe shows positive transverse
magnetoresistance, which increases with decreasing temperature and increasing
field, reaching a value of $\sim$ 27% at 9 T and 10 K. The Hall resistivity
data and electronic band structure calculations suggest that both the hole and
electron charge carriers contribute to the transport properties. The
electronic band structure displays linear band crossings near the Fermi level.
The calculations reveal that GdAgGe has a nodal line with drumhead surface
states coupled with a nonzero Berry phase, making it a nontrivial nodal-line
semimetal.

##  I Introduction

Rare-earth based intermetallic compounds are well known for their complex and
wide range of physical properties such as quantum critical point, field
induced first order to second order phase transition, non-Fermi liquid
behavior, crystal electric field interaction, spin and charge ordering,
valence fluctuation, heavy-fermion (Kondo) behavior, charge density wave,
superconductivity, etc. Paramanik  _et al._ (2014, 2016); Schmiedeshoff  _et
al._ (2011); Das  _et al._ (2014); Bonville  _et al._ (2007); Malick  _et al._
(2022); Nakamura  _et al._ (2015). Recently the interest has burgeoned in the
rare-earth compounds due to the observation of nontrivial topological states.
A special interest is in those materials which show interplay between topology
and magnetism. In these materials, the topological states are protected by
certain crystalline or point or space group symmetry as they have either
broken time-reversal symmetry (TRS) or inversion symmetry (IS) or both Liu
_et al._ (2017). It was reported that broken TRS in combination with strong
spin orbit coupling (SOC) leads to a large anomalous Hall effect (AHE) in
antiferromagnetic Weyl semimetal GdPtBi Suzuki  _et al._ (2016).
Interestingly, besides AHE, another type of Hall effect, i.e. topological Hall
effect (THE) arises in some topological magnetic materials with a noncoplanar
magnetization texture, defined by their nontrivial topology coupled with local
magnetization Puphal  _et al._ (2020). It was theoretically predicted that THE
can occur solely from Berry phase in absence of SOC as opposed to SOC induced
AHE and it is favorable for those systems having low carrier density, and
large spin splitting and exchange interactions Bruno  _et al._ (2004).
However, THE can also occur in higher carrier density ferromagnetic (FM) and
antiferromagnetic (AFM) materials crystallizing in different lattice
structures and is sensitive to several factors such as geometrical
frustrations, spin chirality and thermal fluctuations, antisymmetric
Dzyaloshinskii-Moriya interaction, correlation effects, and magnetic
anisotropy Wang  _et al._ (2019); Neubauer  _et al._ (2009); Kanazawa  _et
al._ (2011); Sürgers  _et al._ (2014); Vistoli  _et al._ (2019); Laha  _et
al._ (2021); Puphal  _et al._ (2020). Several different families of materials
such as VxSb2Te3 Wang  _et al._ (2019), MnSi Neubauer  _et al._ (2009), MnGe
Kanazawa  _et al._ (2011), Mn5Si3 Sürgers  _et al._ (2014), Ca1-xCexMnO3
Vistoli  _et al._ (2019), EuAgAs Laha  _et al._ (2021), EuO Ohuchi  _et al._
(2015), and Pr2Ir2O7 Machida  _et al._ (2007) were reported to display THE.

The rare-earth based equiatomic ternary germanide compounds can be very
exciting from the perspective of topological, magnetic and transport
properties due to their tunable magnetic behavior, which could be manipulated
by number of 4f electrons and external magnetic fields Puphal  _et al._
(2020); Pöttgen  _et al._ (1998); Baran  _et al._ (1998); Morosan  _et al._
(2004); Goddard  _et al._ (2007). Recently, type–I and type-II Weyl semimetal
states were theoretically predicted in some of the rare-earth based aluminium
germanides, where the type of topological state can be tuned by choice of
rare-earth element by breaking IS or TRS Chang  _et al._ (2018). The
theoretical predictions made in ref. Chang  _et al._ (2018) were later
confirmed by the experimental results in PrAlGe Destraz  _et al._ (2020);
Sanchez  _et al._ (2020), and CeAlGe Hodovanets  _et al._ (2018).
Interestingly, both tetragonal compounds were found to show anomalous
transport associated with nontrivial Berry phase for magnetic fields along the
crystallographic c-axis Meng  _et al._ (2019); Destraz  _et al._ (2020).
PrAlGe exhibits large AHE with conductivity value $\sim$ 680 $\Omega^{-1}$cm-1
Meng  _et al._ (2019) and CeAlGe shows THE in field range 0.4-1.5 T at low
temperatures Puphal  _et al._ (2020). PrAlGe was also reported to show
enhanced AHE due to large Berry curvature generated by Weyl nodes, when small
magnetic fields polarize Pr local moments along the c-axis Sanchez  _et al._
(2020). YbAgGe is another interesting compound, which exhibits field induced
quantum criticality, non-Fermi liquid behavior and anisotropic Hall effect
Bud’ko  _et al._ (2004, 2005); Tokiwa  _et al._ (2013). All these interesting
observations motivated us to further explore the equiatomic ternary rare-earth
silver germanide systems in context of their magnetic, thermodynamic,
transport and topological properties.

Here, we describe the results of our study of GdAgGe that crystallizes in
hexagonal crystal structure with space group P$\overline{6}$2m (189) and
orders antiferromagnetically below the Néel temperature, TN $\approx$ 15.6 K
as determined for polycrystalline samples Baran  _et al._ (1998). We grew the
GdAgGe single crystals and investigated their physical properties by means of
magnetic susceptibility, heat capacity, electrical resistivity, Hall effect,
and magnetoresistance measurements as well as electronic band structure
calculations.

##  II Experimental details and methods

Single crystals of GdAgGe were grown using Pb flux. High purity elements Gd
(Alfa Aesar, 99.9%), Ag (Alfa Aesar, 99.99%), Ge (Alfa Aesar, 99.999%), and Pb
(Alfa Aesar, 99.99%) were weighed in a 1:1:1:10 molar ratio and put into
alumina crucible. Then, the crucible was sealed inside a quartz tube under
argon atmosphere. Next, the whole assembly was put into a furnace and
subjected to heat treatment at 1100 ∘C for 10 hours. Subsequently, the furnace
was slowly cooled down to 700 ∘C at a rate 2 ∘C/h and at this temperature flux
was removed by centrifugation. The process resulted in needle-like shiny
single crystals with typical dimensions of 3 $\times$ 0.3 $\times$ 0.4 mm3.

The crystal structure and chemical composition of the grown crystals were
checked by X-ray diffraction (XRD) on a PANalytical X’Pert PRO diffractometer
with Cu-Kα1 radiation, and energy-dispersive X-ray spectroscopy (EDS)
performed in a JEOL JSM-6010LA scanning electron microscope. Transport
measurements were carried out in a Quantum Design Physical Property
Measurement System (PPMS) using standard four-probe method. Heat capacity
measurements were performed using conventional relaxation method in the same
PPMS platform. Magnetization measurements were carried out employing a Quantum
Design Magnetic Property Measurement System (MPMS).

The first-principles calculations were performed based on the density
functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) with
the projector augmented wave (PAW) Blöchl (1994) method as implemented in
Vienna ab initio simulation package (VASP) Kresse and Furthmüller (1996);
Kresse and Joubert (1999). To account for exchange-correlation effects, the
generalised gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE)
Perdew  _et al._ (1996) parametrization was used. The significant correlation
effects of Gd-f states were handled by using a Hubbard U parameter (GGA+U) of
6 eV Petersen  _et al._ (2006); Li  _et al._ (2015). All calculations were
done with a plane wave energy cutoff of 600 eV, and the energy convergence
criterion was chosen to be 10-6 eV. The geometry optimization calculations
were performed with 2 $\times$ 2 $\times$ 2 supercell using dense k-mesh as
per the Monkhorst-Pack technique Monkhorst and Pack (1976). For surface state
calculations, the WANNIER90 package Pizzi  _et al._ (2020) was employed to
create a tight-binding Hamiltonian based on maximally localised Wannier
functions. The iterative Green’s function approach, which is implemented in
the WANNIERTOOLS package, was used to study topological features of the
compound based on the tight-binding model Wu  _et al._ (2018); Sancho  _et
al._ (1985).

##  III Results and discussion

Figure 1: (a) Powder X-ray diffraction pattern of crushed single crystals of
GdAgGe recorded at room temperature. (b) XRD pattern of GdAgGe single crystal.
Inset shows the photograph of needle-shaped single crystals grown along the
$c$-axis.

###  III.1 Crystal structure

Figure 2: The magnetization data of GdAgGe single crystals recorded for fields
parallel and perpendicular to the crystallographic c-axis. Magnetization, M/H
as function of T measured in various fields of H = 0.1 T, 0.5 T and 5 T in the
ZFC (open circles) and FC (filled circles) modes: (a) for H $\perp$ c and (b)
for H $\parallel$ c. The insets of (a) and (b) show the temperature dependence
of the inverse magnetic susceptibility (1/$\chi$) for H $\perp$ c and H
$\parallel$ c in a field of H = 0.1 T, respectively. The green line is fit to
Curie-Weiss law. (c) The isothermal magnetization of GdAgGe for H $\perp$ c
and H $\parallel$ c at T = 1.7 K.

Fig. 1(a) shows the powder XRD pattern of crushed single crystals of GdAgGe
recorded at room temperature. It reveals the single phase growth of GdAgGe
crystals. Further, we carried out Rietveld refinement of the powder XRD data
using the FULLPROF software. The refinement yielded the hexagonal structure
with space group P$\overline{6}$2m (189) and the lattice parameters a = b =
7.170 Å, and c = 4.241 Å, close to those reported in the literature Pöttgen
_et al._ (1998). The single crystal XRD pattern measured on a flat surface of
needle-shaped crystal is shown in Fig. 1(b). The presence of ($l$00) peaks in
the diffraction pattern indicates that a needle axis of the crystal coincides
with the crystallographic c-axis as shown in the inset of Fig. 1(b). The
chemical composition of needle-shaped crystals was found to be very close to
the ideal equiatomic stoichiometry.

###  III.2 Magnetic properties

Figs. 2(a) and 2(b) show the dc magnetization (M) as a function of temperature
(T) for various external fields (H) applied perpendicular and parallel to the
crystallographic $c$-axis. The data was taken in zero-field cooled (ZFC) and
field cooled (FC) modes and it is plotted as M(T)/H curves. For H $\perp$ c,
M/H curve shows two very sharp peaks at TN1 = 20 K and TN2 = 14.5 K, clearly
indicating the AFM ordering in the compound. It is worthy to mention here that
for polycrystalline sample, only one AFM transition was observed at $\sim$
15.6 K Baran  _et al._ (1998). As is evident from Fig. 2(a), an increase in
field from 0.1 to 5 T shifts the peaks towards lower temperatures, which is
consistent with the usual behavior exhibited by an antiferromagnet Kosaka  _et
al._ (2020); Bud’ko  _et al._ (1999). Further, with the change in field
direction from H $\perp$ c to H $\parallel$ c, the observed sharp peaks become
quite broad and less pronounced. The magnetization values observed for H
$\perp$ c and H $\parallel$ c do not differ significantly, suggesting small
magnetic anisotropy. For both configurations, ZFC and FC curves show
bifurcation below TN2. This feature hints at the possibility that the order-
to-order magnetic phase transition in GdAgGe involves small canting of the AFM
moments.

The temperature dependencies of the inverse magnetic susceptibility
($\chi^{-1}$ = M/H) measured in magnetic field $\mu_{0}$H = 0.1 T with H
$\perp$ c and H $\parallel$ c are shown in insets of Figs. 2(a) and 2(b),
respectively. Above 100 K, the data were fitted to the Curie-Weiss equation
$\chi(T)=C/(T-\Theta_{P})$, where $C$ and $\Theta_{P}$ are the Curie constant
and the paramagnetic Curie temperature, respectively. The fit yields the
values $\Theta_{P}$ $\sim$ -56.4 K and -42.0 K for H $\perp$ c and H
$\parallel$ c, respectively. The negative value of $\Theta_{P}$ conforms to
the AFM order in the compound. The frustration parameter f (=$\Theta_{P}$/TN)
estimated for our crystal is $\sim$ 2.82 (3.89) for TN1 (TN2), which lies in
moderate range Ramirez (1994). The f value for GdAgGe crystal is much smaller
than reported for strongly frustrated germanide YbAgGe ($\sim$26) Umeo  _et
al._ , but larger than for polycrystalline GdAgGe ($\sim$1.77) and ErAgGe
($\sim$0.30) compounds Gibson  _et al._ (1996). The effective magnetic moment,
$\mu_{eff}$ estimated from the Curie constant for H $\perp$ c (H $\parallel$
c) is 7.99 (7.42) $\mu_{B}$, which compares well with the theoretical value of
$\sim$ 7.94 $\mu_{B}$ for Gd3+ ion.

Fig. 2(c) shows the isothermal magnetization, M(H) of GdAgGe measured at 1.7 K
for H $\perp$ c and H $\parallel$ c configurations. M increases linearly with
H up to 7 T for both field orientations. This kind of behavior is expected for
AFM systems with metamagnetic transition occurring in stronger fields and
reported before for other Gd based ternary antiferromagnets such as GdPtBi,
GdAuGe and GdAuIn Suzuki  _et al._ (2016); Pöttgen  _et al._ (1998).

###  III.3 Heat capacity and entropy

The temperature dependence of the heat capacity ($C_{p}$) at constant pressure
is shown in Fig. 3. A clear lambda-shaped anomaly near 20 K is observed, which
confirms the AFM ordering at TN1 revealed in our magnetization measurements.
The other AFM transition, clearly visible at TN2 = 14.5 K in the magnetic
susceptibility data, does not manifest itself in $C_{p}$, however its weak
signature is visible in the temperature derivative of the heat capacity (see
the inset to Fig. 3). With application of magnetic field, the anomaly at TN1
is slightly shifted towards lower temperatures with increasing field (not
shown here), as observed in the magnetic susceptibility data. In the
paramagnetic state, $C_{p}$ increases continuously until it gets saturated at
a value of $\sim$ 72.11 J mol-1 K-1 at the room temperature. This value is
close to the limit set by Dulong-Petit law ($C_{p}$ = 3nR = 74.84 J mol-1 K-1,
where n is the number of atoms in the formula units, and R is the universal
gas constant). The temperature variation of $C_{p}$ above TN1 can be well
described by a sum of electronic ($\gamma$T), Debye ($C_{D}$), and Einstein
($C_{E}$) contributions with the expression

$C_{p}(\textit{T})=\gamma\textit{T}+pC_{D}(\textit{T})+(1-p)C_{E}(\textit{T})$
(1)

where $p$ is the weight factor and $\gamma$ is the Sommerfeld coefficient.
$C_{D}$(T) and $C_{E}$(T) are defined as

$C_{D}(\textit{T})=9\textit{n}R\left(\frac{\textit{T}}{\Theta_{D}}\right)^{3}\int_{0}^{\Theta_{D}/\textit{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx$
(2)

and

$C_{E}(\textit{T})=3\textit{n}R\left(\frac{\Theta_{E}}{\textit{T}}\right)^{2}\frac{e^{\Theta_{E}/\textit{T}}}{(e^{\Theta_{E}/\textit{T}}-1)^{2}}$
(3)

where $\Theta_{D}$ and $\Theta_{E}$ are the Debye and Einstein temperatures,
respectively. The obtained values of the fitting parameters are $\gamma$ =
0.47 mJ mol-1 K-2, $\Theta_{D}$ = 164 K, $\Theta_{E}$ = 288 K, and $p$ = 0.66.
The large value of $\Theta_{E}$ suggests the contribution of high frequency
optical modes to the heat capacity of GdAgGe.

Figure 3: Temperature dependence of the heat capacity of GdAgGe at zero field.
The solid red line represents the fit to the experimental data using
Einstein–Debye model. The inset shows the temperature derivative of the heat
capacity.

By subtracting the phonon contribution to $C_{p}$(T), determined from Eq. (1),
the magnetic part of the heat capacity $C_{m}$ was derived and used to
estimate the magnetic entropy according to the formula
$S_{m}=\int\frac{C_{m}}{\textit{T}}d\textit{T}$. The magnitude of $S_{m}$ is
$\sim$ 11.5 J mol-1 K-1 at TN1 = 20 K and saturates above 30 K around 14.5 J
mol-1 K-1, which is about 84 % of the value Rln8 expected for S = 7/2.

###  III.4 Magnetotransport

Figure 4: (a) Temperature dependence of the electrical resistivity of GdAgGe
single crystals in zero magnetic field. The inset shows the expanded view of
the low temperature resistivity data with power-law fit (shown by red line) in
the AFM ordering range. (b) Transverse magnetoresistance (H $\perp$ i) of
GdAgGe single crystal measured at several temperatures.

Fig. 4(a) shows the temperature dependence of the electrical resistivity
$\rho$ of GdAgGe single crystal measured along the c-axis in zero magnetic
field. The resistivity decreases monotonically with decreasing T in a metallic
manner and its magnitude is similar to those reported for GdPdSn and GdPdGe
Talik  _et al._ (2006). Below 30 K, the resistivity decreases more rapidly.
The AFM transition at TN1 manifests itself as a knee in $\rho$(T) and a clear
minimum in the temperature derivative of this curve (not shown here). This
feature can be associated with suppression of spin-disorder scattering in the
magnetically ordered state. Remarkably, $\rho$(T) does not exhibit any clear
anomaly at TN2 = 14.5 K, as displayed in the inset to Fig. 4(a). The measured
resistivity was found to drop suddenly near 7 K. This feature is likely
extrinsic to GdAgGe and originates due to Pb flux residues trapped on the
random sites of crystal surface Matthias  _et al._ (1963), which become
superconducting at this temperature. The extrinsic character of the
superconductivity is confirmed by the absence of any corresponding anomaly in
the heat capacity data. To determine the dominant scattering mechanism in the
AFM range (8 K $\leq$ T $\leq$ 17 K), the low temperature resistivity data
were fitted with the expression $\rho$(T) = $\rho_{0}$ \+ AT${}^{\textit{m}}$,
where $\rho_{0}$ and A represent the residual resistivity and the scattering
coefficient, respectively and m is the exponent representing the scattering
mechanism. Usually, in the AFM regime, where conduction electrons are
scattered by AFM magnons, the exponent, m = 3 is expected Rossiter (1987). In
the present case, we found m = 3/2, predicted by Moriya et al. to occur due to
AFM spin fluctuations Moriya and Takimoto (1995). Similar kind of $\rho$(T)
behavior at low temperatures was reported in the literature for RPdSi (R = Gd,
Tb, Dy) compounds Mukhopadhyay  _et al._ (2021).

The isothermal transverse magnetoresistance (H $\perp$ i) of GdAgGe is shown
in Fig. 4(b), where MR is defined as MR = [$\rho$(H) $-$ $\rho$(0)]/$\rho$(0).
In the ordered state, MR is positive and increases with increasing field,
reflecting the generic feature of an antiferromagnet. The maximum MR value is
$\sim$ 27% at 9 T and 10 K. At the boundary of AFM to paramagnetic phase
transition i.e. at 20 K, MR magnitude sharply decreases to $\sim$ 4% at 9 T,
and becomes nearly zero in weak magnetic fields. With further increase in
temperature (T $>$ TN), MR is very small in small fields and becomes negative
in strong fields, which can be attributed to field-induced alignment of the Gd
magnetic moments in the paramagnetic phase.

Figure 5: Magnetic field dependence of Hall resistivity for GdAgGe at various
temperatures. The red and blue lines represent the two-(Eq. 4) and single-band
model fitting, respectively. The inset shows variation of carrier
concentrations and mobilities as a function of temperature.

Fig. 5 shows the field dependence of Hall resistivity ($\rho_{xy}$) in the
temperature range 15-100 K. It is clearly observed that at low temperatures (T
$<$ 100 K), $\rho_{xy}$ shows the nonlinear field dependence, indicating the
multi-band character of electronic transport. However, the sign of $\rho_{xy}$
is positive below 30 K, which suggests that the holes are the dominating
charge carriers. At somewhat higher temperatures (30 $\leq$ T $<$ 100 K), the
nonlinear behavior of $\rho_{xy}$ persists with the electrons dominating
charge transport. For T = 100 K, the $\rho_{xy}$ becomes linear with complete
dominance of electrons. The slope of linear fit provides the Hall coefficient
$R_{H}$, which is used to estimate electron carrier concentration (n), and the
Hall mobility ($\mu$) by using the relations n = 1/(e$R_{H}$) and $\mu$ =
$R_{H}$/$\rho_{xx}$(H = 0). The estimated value of n and $\mu$ is 2.04
$\times$ 1021 cm-3 and 118 cm2 V-1 s-1, respectively. To further extract
information about the charge carriers and their mobilities at low temperatures
(T $<$ 100 K), we use the semi-classical two-band model, where the Hall
resistivity is given by the expression

$\rho_{xy}=\frac{H}{e}\frac{(n_{h}\mu_{h}^{2}-n_{e}\mu_{e}^{2})+(n_{h}-n_{e})\mu_{e}^{2}\mu_{h}^{2}H^{2}}{{(n_{e}\mu_{e}+n_{h}\mu_{h\
})}^{2}+{(n_{h}-n_{e})}^{2}\mu_{e}^{2}\mu_{h}^{2}H^{2}}$ (4)

The two-band model fit in the temperature range 15-50 K is shown in Fig. 5. It
is clearly evident that the two-band model nicely fits the Hall data above 20
K. However, at 15 K, the quality of fit is rather poor due to the magnetic
ordering effects on the $\rho_{xy}$. From the fitting, we estimate the carrier
density of $\sim$ 1019-1021 cm-3 in the temperature range 15-50 K. The details
of different charge carrier densities and their mobilities are shown in inset
of Fig. 5. The estimated carrier densities in GdAgGe are significantly higher
than that of typical Dirac/Weyl semimetals (n $\sim$ 1017-1019 cm-3), but
comparable to that reported in several nodal-line semimetals Hu  _et al._
(2016); Sankar  _et al._ (2017); Laha  _et al._ (2019); Roy  _et al._ (2021).
We do not observe the signature of THE down to 15 K. This could be due to the
fact that the Dirac node is not precisely located at the Fermi level, and
hence the contribution of Dirac fermions is probably masked by the dominating
conventional charge carriers.

###  III.5 Electronic structure

GdAgGe crystallizes in a hexagonal system with noncentrosymmetric space group
P$\overline{6}$2m (189), which is structurally identical to ZrNiAl, a ternary
ordered form of Fe2P as illustrated in Fig. 6(a). The space group has a
threefold rotational axis (C3z) and a horizontal mirror plane m001, but lacks
inversion symmetry. Our DFT calculations describe the crystal structure with
the lattice parameters a = b = 7.24 Å, c = 4.26 Å, in good agreement to those
determined experimentally. Gd and Ag atoms are positioned at the pyramidal
(3g) and tetrahedral (3f) sites, respectively. From Fig. 6(a), it can be seen
that Gd + Ge and Ag + Ge atoms form the layers that are separated along the
c-axis. According to the experimental results, GdAgGe is an AFM system. Thus,
we used the 2 $\times$ 2 $\times$ 2 supercell to compute the ground state
energy for non-magnetic (NM), FM and AFM configurations to gain a better
understanding of the magnetic characteristics. The possible AFM configurations
are shown in Figs. 6(d) and 6(e). Here, AFM1 configuration contains the AFM
coupling of magnetic moments in- and out of plane, whereas AFM2 (A-type AFM)
configuration exhibit FM coupling in the ab-plane and AFM interactions along
the c-axis. The calculated energy differences among the different magnetic
orderings are tabulated in Table 1. As evident from Table 1, AFM1 has lowest
ground state energy among the calculated structures and it is consistent with
our experimental data. It should be noted that the energy difference between
AFM1 and AFM2 is very small and computed electronic band structures as well as
the topological features for both cases show no significant differences at the
Fermi level [see Figs. 7(a) and 7(b)]. To further check the spin orientations
of moments in AFM1 structure (which explain our experimental data), we have
calculated the ground state energies for [001], [010], [100], [011], [101],
[110] and [111] configurations. The calculated energy differences are given in
Table 2, which shows minimum ground state energy for the [001] configuration.

Figure 6: (a) The crystal structure of GdAgGe, where Gd and Ag atoms are situated at the pyramidal and tetrahedral sites, respectively. (b) AFM configuration for 1 $\times$ 1 $\times$ 2 supercell of GdAgGe with [001] directed magnetic moments. (c) The irreducible Brillouin zone of the bulk along with the (001) projected surface. (d) AFM1 configuration for 2 $\times$ 2 $\times$ 2 supercell with in- and out of plane AFM interactions. (e) AFM2 configuration for 2 $\times$ 2 $\times$ 2 supercell with FM interactions in ab-pane and AFM interactions along c-axis. Table 1: Calculated energies of different non-magnetic and magnetic configurations (in meV) with the reference energy considered to be 0 meV Configuration | NM | FM | AFM1 | AFM2
---|---|---|---|---
Energy (meV) | 680 | 35.10 | 0 | 1.92
Figure 7: Electronic band structure along M-$\Gamma$-K path for (a) AFM1 and (b) AFM2 configuration. (c) The Fermi surfaces in 3D-Brillouin zone without spin-orbit coupling with 2 $\times$ 2 $\times$ 2 supercell for AFM1 configuration. Table 2: Calculated energies of different spin configurations in AFM1 state with the reference energy considered to be 0 $\mu$eV Configuration | [001] | [010] | [100] | [011] | [101] | [110] | [111]
---|---|---|---|---|---|---|---
Energy ($\mu$eV) | 0.0 | 26.09 | 25.13 | 13.40 | 12.85 | 21.96 | 14.93
Figure 8: (a) Electronic band structure along $\Gamma$-M-K-$\Gamma$-A-L-H-A
path without SOC. (b) Total and projected density of states of GdAgGe. (c) The
character bands Gd-d and Ge-p without SOC. (d) Electronic band structure with
SOC along [001] direction. Figure 9: (a) The illustration of the nodal line,
where a, b, c and d are equally spaced points between M and K. (b) Electronic
band structure along the k paths as indicated in (a). (c) (001) projected
surface states together with (d) energy gap plane, which show the nodal line.

As explained above, the energy difference between AFM1 and AFM2 is very
minimal, and the calculated electronic band structures for both configurations
look alike as shown in Figs. 7(a) and 7(b). Moreover, we have calculated the
Fermi surfaces for AFM1 configuration, which are depicted in Fig. 7(c). We can
see that the electron and hole pockets are present at the Fermi level, which
is in good agreement with the experimentally observed Hall resistivity. The
investigated compound’s electronic band structure displays the band crossings
in kz = 0 plane and it gives us a hint towards the topological features. To
further explore the topological features, we have performed the electronic
bands as well as the surface analysis for lower cell (1 $\times$ 1 $\times$ 2)
of AFM configuration [as shown in Fig. 6(b)] and such type of calculations for
bigger cell is beyond the scope of our work.

In the next step, we have proceeded with 1 $\times$ 1 $\times$ 2 supercell of
AFM configuration. Fig. 8(a) shows the band structure of GdAgGe with the spin-
up (in red color) and the spin-down (in blue color) channel. The examined
compound’s electronic properties demonstrate the semi-metallic nature. Fig.
8(b) displays the total density of states (DOS) along with atomic projected
density of states. The DOS values in both the conduction and valence band
region around the Fermi level are almost same, which reflects the semi-
metallic nature of the compound. Interestingly, the conduction and valence
bands cross each other in a linear manner along the kz = 0 plane (marked in
Fig. 8(a)). These Dirac-like crossings hint toward the presence of nodal line,
which is protected by mirror symmetry located in the xy-plane. To check for
possible nontrivial nature of bands, we calculated the orbital decomposed band
structure (Fig. 8(c)), which infers that Gd-d and Ge-p states are main
contributors to the bands forming nodal line. Close inspections revealed the
Gd-d and Ge-p band inversion together with opposite mirror eigenvalues +1 and
-1, respectively, at the $\Gamma$ point. To analyse these band crossings, we
carefully examined the band structure along equally spaced paths between M and
K, which is shown in Fig. 9(a). It can be seen that the band crossings appear
along $\Gamma$-M/a/b/c/d/K paths, which infers that a $\Gamma$-centered nodal
line should occur in the kz = 0 plane. To demonstrate the topological features
in GdAgGe, we calculated the surface states in the (001) direction using the
WANNIER package. Fig. 9(c) illustrates the nodal line dispersions from bulk
bands and emergent drumhead surface states in the (001) projected surface
state. In addition, the energy gap plane and the Fermi surface corresponding
to the bands crossings also confirm the occurrence of nodal line in GdAgGe,
which is illustrated in Fig. 9(d). The topological invariant, i.e., the Berry
phase (a Z2-type invariant) along a closed path encircling the nodal line,
usually protects the nodal line structure of a topological nodal-line
semimetal Hyart  _et al._ (2018). To confirm this point we have calculated the
Berry phase and found a nonzero quantized Berry phase, which confirms GdAgGe
to be a nontrivial nodal-line semimetal.

The presence of band inversion without SOC and nodal line in kz = 0 plane
further lead us to include the SOC effects here. The electronic band structure
with SOC is shown in Fig. 8(d), which reflects that the band crossings are
nearly unaffected. However, the inclusion of SOC opens a negligible band gap
of 5.7 meV and 5.9 meV along the $\Gamma$-M and K-$\Gamma$ path, respectively.
These gap values are quite smaller than those found in other reported nodal-
line semimetals Laha  _et al._ (2019, 2020); Emmanouilidou  _et al._ (2017).

##  IV Conclusions

We studied the physical properties of GdAgGe single crystals by measuring the
magnetic susceptibility, heat capacity, magnetoresistance, Hall effect, and by
computing the electronic band structure. The magnetic susceptibility
measurements in GdAgGe showed the two successive AFM transitions at TN1 = 20 K
and TN2 = 14.5 K, which are most prominent for fields perpendicular to the
crystallographic c-axis. In the heat capacity data, a clear lambda-shaped
anomaly confirms the bulk character of AFM transition observed at TN1 = 20 K.
The electrical resistivity shows a sharp drop below TN1 and T3/2 dependence in
the AFM state. The transverse magnetoresistance is positive for T $\leq$ TN1,
and becomes negative in the paramagnetic region. It increases with increasing
field and reaches $\sim$ 27% at 9 T and 10 K. The Hall resistivity data show
the multi-band character of the compound, where hole carriers are dominating
at low temperatures and electron carriers at high temperatures. The electronic
band structure calculations reveal the presence of a nodal line with drumhead
surface states in kz = 0 plane, which is protected by the reflection symmetry.

##  V Acknowledgment

We acknowledge IIT Kanpur and the Department of Science and Technology, India,
[Order No. DST/NM/TUE/QM-06/2019 (G)] for financial support. VK acknowledges
IIT Hyderabad for its computational facility. JS was supported through CSIR
scholarship. ZH acknowledges financial support from the Polish National Agency
for Academic Exchange under Ulam fellowship. DK acknowledges financial support
by the National Science Centre (Poland) under research grant
2021/41/B/ST3/01141.

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