# HIC-YOLOv5: Improved YOLOv5 For Small Object Detection

Shiyi Tang, Yini Fang, Shu Zhang
Heriot-Watt University, Hong Kong University of Science and Technology, Ocean
University of China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>*This work was
not supported by any organization1Shiyi Tang Author majors in Computer
Science, Ocean University of China & Heriot-Watt University<EMAIL_ADDRESS>
###### Abstract

Small object detection has been a challenging problem in the field of object
detection. There has been some works that proposes improvements for this task,
such as adding several attention blocks or changing the whole structure of
feature fusion networks. However, the computation cost of these models is
large, which makes deploying a real-time object detection system unfeasible,
while leaving room for improvement. To this end, an improved YOLOv5 model:
HIC-YOLOv5 is proposed to address the aforementioned problems. Firstly, an
additional prediction head specific to small objects is added to provide a
higher-resolution feature map for better prediction. Secondly, an involution
block is adopted between the backbone and neck to increase channel information
of the feature map. Moreover, an attention mechanism named CBAM is applied at
the end of the backbone, thus not only decreasing the computation cost
compared with previous works but also emphasizing the important information in
both channel and spatial domain. Our result shows that HIC-YOLOv5 has improved
mAP@[.5:.95] by 6.42$\%$ and<EMAIL_ADDRESS>by 9.38$\%$ on VisDrone-2019-DET dataset.

##  I INTRODUCTION

Object detection algorithm has been widely applied to smart systems of
Unmanned Aerial Vehicles (UAVs), such as pedestrian detection and vehicle
detection. It automates the analysis process of the photos taken by UAVs.
However, the biggest issue of such applications lies in detecting small
objects, as most of the objects in the photos become smaller from a higher
altitude. This fact poses negative effects on the accuracy of object
detection, including target occlusion, low target density, and dramatic
changes in light.

You Only Look Once (YOLO)[15], a one-stage object detection algorithm, is
dominating UAV systems due to its low latency and high accuracy. It takes an
image as input and outputs the information of the objects in one stage. The
lightweight model can achieve real-time object detection in UAV systems.
However, it still has drawbacks in UAV scenarios with a large number of small
objects. To address this issue, there have been previous works to improve the
performance of small object detection. Some works [7][9] improves the whole
structure of the feature fusion network of YOLOv5. Other works[11] add several
attention blocks in the backbone. However, the computation cost is large among
the previous methods and there is still improvement space of the performance.

In this paper, we propose an improved YOLOv5 algorithm: HIC-YOLOv5 (Head,
Involution and CBAM-YOLOv5) for small object detection, with better
performance and less computation cost. We first add an additional prediction
head—Small Object Detection Head (SODH) dedicated to detecting small objects
from feature maps with a higher resolution. The features of tiny and small
objects are more easily extracted when the resolution of the feature map
increases. Secondly, we add a Channel feature fusion with involution (CFFI)
between the backbone and the neck enhance the channel information, thereby
improving overall performance. In this way, the performance is improved with
more information transmitted to the deep network. Finally, we apply a
lightweight Convolutional Block Attention Module (CBAM) at the end of the
backbone, which not only has a lower computation cost than [11] but also
improves the total performance by emphasizing important channel and spatial
features. The experiment result shows that our HIC-YOLOv5 has improved the
performance of YOLOv5 on VisDrone dataset by 6.42$\%$(mAP@[.5:.95]) and
9.38$\%$(mAP@0.5).

Our main contributions can be summarized as follows:

  * •

The additional prediction head is designed especially for small objects. It
detects objects in higher-resolution feature maps, which contain more
information about tiny and small objects.

  * •

An involution block is added as a bridge between the backbone and neck to
increase the channel information of the feature maps.

  * •

CBAM is applied at the end of the backbone, thus more essential channel and
spatial information is extracted while the redundant ones are ignored.

##  II RELATED WORKS

###  II-A Object Detection Primer

The main purpose of object detection is to locate and classify objects in
images, in the form of bounding boxes and confidence scores labeled on the
objects. There are two types of object detection pipelines: two-stage and one-
stage detectors. The two-stage detectors (e.g., R-CNN[1], SPP-net[2], Fast
R-CNN[3] and FPN[4]) first generate region proposals, and then apply object
classification and height and width regression. The one-stage detectors (e.g.,
YOLO series and SSD[16]) use a deep learning model, which directly takes an
image as input and outputs bounding box coordinates and class probabilities.

Among all the YOLO series, YOLOv5 is the most suitable algorithm for real-time
object detection due to its promising performance and excellent computational
efficiency. There have been several versions of YOLOv5, with the same main
structure but a few differences on some small modules. In this paper, we
choose to use YOLOv5-6.0 as our experimental algorithm. The main structure of
YOLOv5-6.0 are shown in Fig. 1.

Figure 1: Structure of YOLOv5-6.0.

The backbone of YOLOv5 firstly extracts features from the input image and
generates different sizes of feature maps. These feature maps are then fused
with the feature maps in the neck. Finally, three different feature maps
generated from the neck are sent to the prediction head. The detailed
information is described as follows:

The backbone includes several Conv, CSPDarkNet53 (C3), and SPPF modules. The
Conv module adopts Conv2d, Batch Normalization, and SiLU activation function.
C3, which is based on the CSPNet[17], is the main module to learn the residual
features. It includes two branches: one branch adopts three Conv modules and
several Bottlenecks, and another branch only uses one Conv module. The two
branches are finally concatenated together and fed into the next modules. SPPF
modules are added at the end of the backbone, which is an improved type of
Spatial Pyramid Pooling (SPP)[2]. It replaces the large-sized pooling kernels
with several cascaded small-sized pooling kernels, aiming to increase the
computation speed while maintaining the original function of integrating
feature maps of different receptive fields to enrich the expression ability of
features.

The neck of YOLOv5 draws on the structure of the Feature Pyramid Network (FPN)
and Path Aggregation Network (PANet). The structure of FPN and PANet in YOLOv5
are shown in Fig. 2. FPN mainly consists of two paths: Bottom-up and Top-down.
Bottom-Up path responds to the backbone of YOLOv5, which gradually decreases
the size of the feature map to increase the semantic information. Top-down
path takes the smallest feature map generated by Bottom-up path as input and
gradually increases the size of the feature map using an upsample, thus
increasing the semantic information of low-level features. Finally, feature
maps with the same size in the two paths are laterally connected together to
increase the semantic representation on multiple scales. PANet adds a Bottom-
up path based on FPN. Therefore, the position information at the low level can
also be transmitted to the deep level, thus enhancing the positioning ability
at multiple scales.

Figure 2: FPN and PANet structure in YOLOv5.

There are totally 3 prediction heads in YOLOv5, which aims to detect three
sizes (80 $\times$ 80, 40 $\times$ 40, 20 $\times$ 20) of objects
respectively: large, medium and small, where the image resolution is 640
$\times$ 640\. The prediction heads divide grids on these three feature maps
according to the dimensions of the feature maps. Then, three groups of anchors
with different aspect ratios for each grid on each feature map are set to
generate candidate bounding boxes. Finally, Non-Maximum Suppression (NMS) is
applied to discard the overlapping bounding boxes and output the final
bounding boxes, which include the locations and sizes of the boxes, and the
confidence scores of the objects.

###  II-B Previous Works on Small Object Detection

There have been numerous prior efforts aimed at enhancing the detection
performance of small objects. Certain studies have focused on optimizing the
overall architecture of the neck component within YOLOv5. For instance, [9]
replaces the PANet in YOLOv5 with a weighted bidirectional feature pyramid
Mul-BiFPN and [7] introduces a new feature fusion method PB-FPN to the neck of
YOLOv5. However, both methods choose to change the entire structure of neck to
achieve better feature fusion, which result in larger computation cost.
Instead, we introduce a lightweight involution block between the backbone and
the neck, aiming to improve the performance of PANet in the neck of YOLOv5
with less computation cost and higher accuracy. [6] introduces a spatio-
temporal interaction module, which applies recursive gated convolution to make
greater spatial interaction, but causes channel information loss because of 1
$\times$ 1 convolution layers. In our structure, the involution block can
effectively address this problem. Moreover, some works try to apply attention
mechanisms[14][19]. Attention mechanisms have been widely applied in the field
of computer vision, which learns to emphasize essential parts and ignore the
unimportant ones in an image. There are various types of attention mechanisms,
such as channel attention, spatial attention, temporal attention, and branch
attention. There have been previous works that integrate transformer layers
into YOLOv5. [11] adds a transformer layer at the end of the backbone.
However, it requires large computation costs and is difficult to train the
model when the input image size is large. Compared with these methods, we add
a lightweight CBAM block at the end of the backbone, which aims to use less
computation cost and focus more on essential information when extracting
features.

##  III METHODOLOGY

The structure of HIC-YOLOv5 is shown in Fig. 3. Original YOLOv5 consists of 3
sections: backbone for feature extraction, neck for feature fusion, and 3
prediction heads. Based on the default model, we propose three modifications:
1) we add an additional prediction head to detect layers with high-resolution
feature maps for small and tiny objects specifically; 2) an Involution block
is adopted at the beginning of the neck to improve the performance of PANet;
3) we incorporate the Convolutional Block Attention Module (CBAM) into the
backbone network.

Figure 3: Structure of HIC-YOLOv5.

###  III-A Convolutional Block Attention Module(CBAM)

Previous works add CBAM into the Neck block when generating feature pyramids.
However, the parameters and computing cost increase because some feature maps
connected with CBAM have large sizes. Moreover, the model is difficult to
train due to the large amount of parameters. Hence, we adopt CBAM in the
Backbone network with the purpose of highlighting significant features when
extracting features in the backbone, rather than generating feature pyramids
in the neck. Moreover, the feature map size as the input of CBAM is only 20
$\times$ 20 which is 32 times smaller than the 640 $\times$ 640 full image so
that the computing cost will not be large.

CBAM is an effective model based on attention mechanism, which can be
conveniently integrated into CNN architectures. It consists of 2 blocks:
Channel Attention Module and Spatial Attention Module, as shown in Fig. 4. The
two modules respectively generate a channel and a spatial attention map, which
are then multiplied with the input feature map to facilitate adaptive feature
refinement. Therefore, the meaningful features along both channel and spatial
axes are emphasized, while the redundant ones are suppressed. The Channel
Attention Module performs global Max-pooling and Average-pooling for feature
maps on different channels and then executes element-wise summation and
sigmoid activation. The Spatial Attention Module performs a global Max-pooling
and Average-pooling for Values of pixels in the same position on different
feature maps and then concatenates the two feature maps, followed by a Conv2d
operation and sigmoid activation.

Figure 4: Structure of CBAM[14].

###  III-B Channel Feature Fusion with Involution (CFFI)

The neck of YOLOv5 adopts PANet, which introduces the bottom-up path
augmentation structure on the basis of FPN. The corresponding structure of FPN
and bottom-up path augmentation in YOLOv5 is shown in Fig. 2. Particularly,
FPN has great ability to detect small and tiny targets by fusing features of
high and low layers so as to obtain high resolution and strong semantics
features. However, a 1 $\times$ 1 convolution is adopted to reduce the number
of channels at the beginning of the neck in original YOLOv5, where the
calculation efficiency is significantly improved, but the channel information
is also reduced, leading to poor performance of PANet. Inspired by [11], we
add an Involution block between the backbone and the neck. The channel
information is improved and shared, resulting in the reduction of information
loss during the initial phases of FPN. As a result, this improvement
contributes to the enhanced performance of FPN, particularly benefiting the
detection of objects with smaller sizes. Moreover, it is emphasized that
Involution has better adaptation to various visual patterns in terms of
different spatial positions.[13]

The structure of Involution is illustrated in Fig. 5. Involution kernels,
represented as $\mathcal{H}\in\mathbb{R}^{H\times W\times K\times K\times G}$,
are designed to incorporate transformations that exhibit inverse attributes in
both the spatial and channel domains, where $H$ and $W$ represents the height
and width of the feature map, $K$ is the kernel size and $G$ represents the
number of groups, where each group shares the same involution kernel.
Particularly, a specific involution kernel, denoted as
$\mathcal{H}_{i,j,\cdot,\cdot,g}\in\mathbb{R}^{K\times K},g=1,2,...,G$, is
designed for the pixel $\mathbf{X}_{i,j}\in\mathbb{R}^{C}$ (the subscript of C
is omitted for brevity), while being shared across the channels. Finally, the
output feature map of involution $\mathbf{Y}_{i,j,k}$, is obtained as follows:

$\mathbf{Y}_{i,j,k}=\sum_{(u,v)\in\Delta_{K}}\mathcal{H}_{i,j,u+\lfloor
K/2\rfloor,v+\lfloor K/2\rfloor,\lceil kG/C\rceil}\mathbf{X}_{i+u,j+v,k}$ (1)

Therefore, the information contained in the channel dimension of a single
pixel is implicitly dispersed to its spatial vicinity, which is useful to
obtain the enriched receptive field information.

Figure 5: Structure of Involution block[13].

###  III-C Prediction Head

The different resolutions (80 $\times$ 80, 40 $\times$ 40, and 20 $\times$ 20)
of 3 prediction heads in YOLOv5 make a great contribution to the detection
ability in various application scenarios, but also make it difficult to detect
small and tiny objects. The reason why the performance of YOLOv5 on tiny
object detection is poor is that the features of tiny objects which only
contain few pixels are likely to be ignored. Although convolutional blocks
play an important role in extracting the features from feature maps, they also
reduce the resolution of feature maps when the depth of the network increases,
thus the features of the tiny object are difficult to extract. In order to
solve this issue and inspired by [9][12][6], we propose an additional
prediction head—Small Object Detection Head (SODH), which aims to detect
feature maps with larger resolution (160 $\times$ 160). It becomes
increasingly effortless to extract features from small and minuscule objects.

Each prediction head takes the feature extracted and fused by backbone and
neck as input, and finally outputs a vector, which consists of the regression
bounding box (coordinate and size), the confidence of the object’s border and
the class of the object. Before generating the final bounding boxes, we
generate anchors to form the candidate bounding boxes. These anchors are
generated by k-means according to the dataset and are defined in 3 different
scales for the 3 prediction heads, adapting to small, middle and large objects
respectively. Anchors of the additional prediction head are also generated by
k-means.

###  III-D Loss Function

The loss function of HIC-YOLOv5 consists of three sections: objectness,
bounding box and class probability, which can be represented as follows:

$\operatorname{Loss}=\alpha\operatorname{Loss}_{obj}+\beta\text{ Loss
}_{box}+\gamma\operatorname{Loss}_{cls}$ (2)

We use binary cross entropy loss for both objectness and class probability,
and CIoU loss [20] for bounding box regression.

###  III-E Data Augmentation

Data augmentation is an essential technique to enhance the robustness of the
model. In YOLOv5, it includes Mosaic, Copy paste, Random affine, MixUp, HSV
augmentation and Cutout. Except that, we found that many small people and cars
are in the center of a picture Visdrone2019. Therefore we add extra center
cropping to the data augmentation techniques mentioned above..

##  IV EXPERIMENTAL RESULTS

The experiments were conducted on the VisDrone2019 dataset, and the obtained
experimental results demonstrate that our proposed YOLOv5 exhibits excellent
performance in terms of detection accuracy.

###  IV-A Experimental Setting

####  IV-A1 Experimental Equipment

In this experiment, the CPU is 15 vCPU Intel® X Platinum 8358P CPU @ 2.60GHz,
the GPU is NVIDIA A40 with 48 GB of Graphics memory. The algorithm is
implemented by PyTorch, using CUDA 11.6 to operation acceleration.

####  IV-A2 Dataset

The dataset used in this experiment is VisDrone2019, which is a comprehensive
benchmark facilitating the integration of drone technology and visual
perception. VisDrone2019 was collected by the AISKYEYE team at Lab of Machine
Learning and Data Mining, Tianjin University, China. It comprises 288 video
clips consisting of 261,908 frames and 10,209 static images captured by
diverse drone-mounted cameras across different locations separated by
thousands of kilometers in China, environments containing both urban and
rural, objects including pedestrians, vehicles, bicycles etc., and densities
from sparse to crowded scenes. Notably, this dataset was acquired using
multiple drone platforms with varying models under different scenarios as well
as weather and lighting conditions. The dataset is divided into a training
set, a validation set and a testing set, with 6471, 548, 1610 images
respectively.

The detailed information of the dataset is visualized in Fig 6. There are
totally 10 classes in this dataset (pedestrian, people, bicycle, car, van,
truck, tricycle, awning-tricycle, bus, and motor) as shown in 6a.
Specifically, it can be observed from the 6c and I that 75$\%$ objects are
0.001 times smaller than the image size, indicating the large number of small
and tiny objects. Moreover, the label locations 6b indicates that many objects
locate in the center of the pictures, which represents the necessity of
center-crop data augmentation.

TABLE I: Area of objects

mean | std | min | 25% | 50% | 75% | max
---|---|---|---|---|---|---
0.001535 | 0.003852 | 0 | 0.00017 | 0.000462 | 0.001342 | 0.302962

(a) Instances. Information of 10 classes in VisDrone-2019.

(b) Locations. The locations of objects in an image, whose height and width
are assumed to be 1.

(c) Distribution. The distribution of object areas in VisDrone-2019.

Figure 6: Dataset.

###  IV-B Hyperparemter Settings

In order to accelerate the training speed, the input image size is set to be
640 $\times$ 640\. We set the batch size to be 128 and training epoch to be
300. We use early stopping strategy to avoid over-fitting, where the patience
is set to 15. We use Adam as the optimizer, with an initial learning rate of
0.001. Other detailed parameters are listed in Table II. The weights of the
loss function are set to be 0.5 (object), 0.05 (box) and 0.25 (class)
respectively.

There are mainly 5 models in YOLOv5, including YOLOv5n, YOLOv5s, YOLOv5m,
YOLOv5l, YOLOv5x. The depth and width of the models increase in sequence while
other structures stay the same. The larger the model, the more precise the
result. However, in order to accelerate the training speed, we choose to use
YOLOv5s during the experiment, with a depth and width of 0.33 and 0.50
respectively.

Data augmentation applied in YOLOv5 contains Mosaic, Copy paste, Random
affine, MixUp, HSV augmentation and Cutout as listed in Table II. We also
adopt center crop during this experiment. The height and width of center crop
is set to be half of the original image size. It has been observed that center
crop is able to improve the overall performance of the model.

TABLE II: Parameter Settings.

hsv_h | hsv_s | hsv_v | degrees | scale | mosaic | mixup | copy_paste
---|---|---|---|---|---|---|---
0.4 | 0.3 | 0.5 | 0.2 | 0.4 | 1 | 0.2 | 0.1

Some default anchors are predefined for coco data sets. Before the training
starts, annotation details in the dataset will be examined automatically and
the most suitable recall rate for the default anchor will be computed. If the
optimal recall rate equals or exceeds 0.98, it is not necessary to update the
anchor frame. However, YOLOv5 will recalculate the anchors if the optimal
recall rate falls below 0.98. During this experiment, 4 groups of anchors of 4
prediction heads are listed in TableIII. Each group is applied for different
sizes of feature maps. Specifically, there are 3 pairs of anchors in each
group for a single ground truth. Therefore, the overall number of anchors is 4
$\times$ 3=12.

TABLE III: Anchor sizes for prediction heads of HIC-YOLOv5.

Detection head | Anchor frame size
---|---
Tiny | [2.9434,4.0435], [3.8626,8.5592], [6.8534, 5.9391]
Small | [10,13], [16,30], [33,23]
Medium | [30,61], [62,45], [59,119]
Large | [116,90], [156,198], [373,326]

###  IV-C Evaluation criterion

The common criteria used to evaluate the performance of an object detection
algorithm include IoU, Precision, Recall and mAP. The detailed definitions are
listed below.

####  IV-C1 IoU

The Intersection over Union (IoU) is calculated by taking the overlap area
between the predicted region (A) and the actual ground truth (B) and dividing
it by the combined area of the two. The formula can be expressed as

$IoU=\frac{A\bigcap B}{A\bigcup B}$ (3)

The value of IoU ranges from 0 to 1. The larger the value, the more precise
the model. Particularly, a lower numerator value indicates that the prediction
failed to accurately predict the ground truth region. On the other hand, a
higher denominator value indicates a larger predicted region, resulting in a
lower IoU value.

####  IV-C2 Precision

Precision represents the proportion of samples predicted correctly in the set
of samples predicted positively. It can be expressed as

$\text{ Precision }=\frac{\text{ True positives }}{\text{ True positives
}+\text{ False positives }}$ (4)

####  IV-C3 Recall

Recall represents the proportion of samples that are actually positive and
predicted to be correct. It can be expressed as

$\text{ Recall }=\frac{\text{ True positives }}{\text{ True positives }+\text{
False negatives }}$ (5)

####  IV-C4 mAP

The Average Precision (AP) is a measure of the Precision scores at different
thresholds along the Precision-Recall (PR) curve, and is calculated as a
weighted mean. Mean Average Precision (mAP) is the mean values of the AP for
all classes. Specifically<EMAIL_ADDRESS>represents the mAP when IoU is 0.5,
mAP@[.5:.95] is the mean mAP when IoU ranges from 0.5 to 0.95.

###  IV-D Experimental Results

In the conducted experiment, the VisDrone-2019 dataset was utilized to assess
the performance of the improved model. After comparing the experimental
outcomes between YOLOv5s and the improved algorithm, it can be inferred that
our algorithm outperforms YOLOv5s in detecting small targets.

TABLE IV: Comparison of algorithms on VisDrone2019 dataset. Method | Dataset |<EMAIL_ADDRESS>| mAP@[.5:.95]
---|---|---|---
YOLOv5 | Test | 27.57 | 14.43
Shang et al. [5] | Test | 36.4 | 20.1
Liu et al. [6] | Test | 35.3 | 20
Liu et al. [7] | Test | 34.3 | 18.2
Ding et al. [8] | Val | 42.9 | 24.6
HIC-YOLOv5 | Test | 36.95 | 20.85
HIC-YOLOv5 | Val | 44.31 | 25.95

From Table IV, we can see that compared with the YOLOv5s model, the
mAP@[.5:.95] has been improved by 6.42$\%$ and<EMAIL_ADDRESS>has been improved by
9.38$\%$. Precision and Recall has been improved by 10.29$\%$ and 6.97$\%$
respectively. The small object detection head greatly helps to retain the
features of small objects. Additionally, Involution effectively amplifies the
channel information, while the CBAM block selectively emphasizes crucial
features during their extraction within the backbone. The detection effect
between YOLOv5s and HIC-YOLOv5 is shown in Fig 7. It visually indicates that
more small objects can be detected when using the improved method.

(a) YOLOv5

(b) HIC-YOLOv5

(c)

Figure 7: Comparison of detection effect between YOLOv5s and HIC-YOLOv5.

We also compared our improved model with other algorithms tested on
VisDrone2019. The results are shown in Table IV. It can be seen that our
proposed model has greater performance compared to other detection models. For
instance, [6] applies recursive gated convolution to make greater spatial
interaction, but it can lead to the loss of channel information due to 1
$\times$ 1 convolution layers. Compared with [6], we adopt involution block to
enhance the channel information of PANet, thus improving the performance of
small object detection. [11] adds a transformer layer at the end of the
backbone, which has the shortcoming of large computation cost. Instead, we
apply a lightweight CBAM block, which decreases the training time and
computational cost. The number of layers, parameters and gradients of [11] and
YOLOv5+CBAM are listed in TABLE V. The mAP is incomparable since [11] uses
another different dataset.

TABLE V: Computational comparison of Wang et al.[11] and YOLOv5+CBAM. Model | Layers | Parameters | Gradients
---|---|---|---
Wang et al. [11] | 297 | 14580167 | 14580167
YOLOv5+cbam | 289 | 8391641 | 8391641

###  IV-E Ablation Study

We conducted several experiments to study the effect of three modifications:
additional prediction head, involution block and CBAM. The results of ablation
study are shown in TableVI. It can be observed that the fourth prediction head
makes best contribution to the performance of the model, which improved<EMAIL_ADDRESS>and mAP@[.5:.95] by 8.31$\%$ and 5.51$\%$ respectively. Instead, the single
block of CBAM and Involution could not improve the model without the help of
fourth prediction head. We speculate that it is because there are so many
small objects in VisDrone2019 dataset that the single block of CBAM and
Involution can not perform well if these small objects can not be detected
first. Based on adding the fourth prediction head, the involution block also
has great improvement on the model, with 0.66$\%$ and 0.57$\%$ increase of
<EMAIL_ADDRESS>and mAP@[.5:.95]. Additionally<EMAIL_ADDRESS>and mAP@[.5:.95] are improved by
0.41$\%$ and 0.34$\%$ respectively.

TABLE VI: Ablation study.

Model | P | R |<EMAIL_ADDRESS>| mAP@[.5:.95]
---|---|---|---|---
Baseline | 37.59 | 31.52 | 27.57 | 14.43
+SODH | 46.18 | 37.89 | 35.88 | 19.94
+cbam | 36.13 | 28.16 | 24.2 | 11.92
+involution | 35.49 | 31.13 | 26.76 | 13.8
+SODH+cbam | 44.5 | 35.35 | 34.53 | 19.03
+SODH+involution | 46.48 | 37.91 | 36.54 | 20.51
+SODH+involution+cbam | 47.88 | 38.49 | 36.95 | 20.85

##  V CONCLUSIONS

In this paper, an improved YOLOv5 algorithm HIC-YOLOv5 has been proposed,
aiming to improve the performance of small and tiny object detection. There
are three main contributions in this paper and the experimental results has
proved the effectiveness of our methodology. Firstly, an additional prediction
head for small objects is added so that the higher-resolution feature maps
could be directly used to detect small targets. Secondly, we adopt an
involution block between the backbone and neck, thus increasing channel
information of the feature map. Furthermore, we also apply an attention
mechanism named CBAM at the end of the backbone to decrease the computation
cost and emphasize the important information in both channel and spatial
domain. Additionally, data augmentation such as center crop is also applied
apart from the original data augmentation methods in YOLOv5. Therefore, the
improved YOLOv5 is able to increase the accuracy of detecting small and tiny
objects.

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# Hysteretic phenomena in a 2DEG in quantum Hall effect regime studied in a
transport experiment

M. V. Budantsev<EMAIL_ADDRESS>A. V. Rzhanov Institute of Semiconductor
Physics of SB RAS, Novosibirsk, Russia  Novosibirsk State University, Russia
D. A. Pokhabov  A. G. Pogosov  E. Yu. Zhdanov  A. V. Rzhanov Institute of
Semiconductor Physics of SB RAS, Novosibirsk, Russia  Novosibirsk State
University, Russia  A. K. Bakarov  A. I. Toropov  A. V. Rzhanov Institute of
Semiconductor Physics of SB RAS, Novosibirsk, Russia

###### Abstract

We investigated experimentally non-equilibrium state of a two-dimensional
electron gas (2DEG) in the quantum Hall effect (QHE) regime, studying the
hysteresis of magnetoresistance of a 2DEG with a constriction. The large
amplitude of the hysteresis enabled us to make the consistent phenomenological
description of the hysteresis. We studied the dependence on the magnetic field
sweep prehistory (minor loop measurements), recovered the anhysteretic curve,
and studied the time dependence of the magnetoresistance. We showed that the
hysteresis of magnetoresistance of a 2DEG in the QHE regime has significant
phenomenological similarities with the hysteresis of magnetization of
ferromagnetic materials, showing multistability, jumps of relaxation, and
having the anhysteretic curve. Nevertheless, we revealed the crucial
difference, manifested itself in an unusual inverted (anti-coercive) behavior
of the magnetoresistance hysteresis. The time relaxation of the hysteresis has
fast and slow regimes, similar to that of non-equilibrium magnetization of a
2DEG in QHE regime pointing to their common origin. We studied the dependence
of the hysteresis loop area on the lithographic width of the constriction and
found the threshold value of width $\sim$1.35 $\mu$m beyond which the
hysteresis is not observed. This points to the edge nature of the non-
equilibrium currents (NECs) and allows us to determine the width of the NECs
area ($\sim$0.5 $\mu$m). We suggest the qualitative picture of the observed
hysteresis, based on non-equilibrium redistribution of the electrons among the
Landau level states and assuming huge imbalance between the population of bulk
and edge electronic states.

###### pacs:

71.10.Ca, 71.70.Di, 73.43.Qt

##  I Introduction

A two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime
demonstrates a number of intriguing phenomena that are not fully explained up
to date. Among them are non-equilibrium phenomena manifested in hysteresis of
sheet electron density, Pudalov1984 ; Dolgopolov1992 ; Dolgopolov1993
magnetization (see for review Ref.[Usher2009, ]) and electrochemical potential
Huels2004 ; Klaffs2004  as a function of magnetic field and gate voltage (for
gated structures). The hysteresis was observed in an ordinary single-layer
2DEG in nonmagnetic materials such as Si MOSFETs and AlGaAs/GaAs
heterostructures at integer and fractional filling factors corresponding to a
longitudinal resistance ($R_{\mathrm{L}}$) vanishing.

It was found that the hysteretic variations of magnetization Usher2009  and
electrochemical potential Huels2004 ; Klaffs2004  exceed equilibrium values by
a factor of 10-60. In contrast the conventional electron transport
measurements in QHE regime with vanishing $R_{\mathrm{L}}$ are not sensitive
to the hysteresis, though, the question of a probable indirect influence of
the non-equilibrium on the transport properties of 2DEG still remains open.

Recently in a number of papers it was demonstrated that creation of a
constriction in a macroscopic bath of 2DEG allows to reveal the hysteretic
behavior in magnetoresistance Budantsev2007 ; Budantsev2009 ; Budantsev22009 ;
Budantsev2011; Pioro-Ladriere2006 ; Smith2011 .

It was shown that at low temperatures (lower than 100 mK) the hysteresis
relaxes over a day Pioro-Ladriere2006  which is far beyond the time usually
spent on a conventional measurement (a magnetic field sweep) in the QHE
regime. In terms of usual magnetic hysteresis description Bertotti1998  this
type of hysteresis is considered as time-independent.

The most common explanation of the hysteretic phenomenon is based on non-
equilibrium currents (NECs) slowly relaxing providing $R_{\mathrm{L}}$
vanishing. But there are no unambiguous models in the literature, adequately
describing this phenomenon. The detailed consideration of NECs induction is
proposed in the Ref. [Shikin2002, ], but it concerns only the case of weak
non-equilibrium. Ruhe et al. Ruhe2009  studied the case of time-dependent
hysteresis. The most developed model up to now Matthews2004  is based on the
assumption of redistribution of “frozen” charges over the whole bulk area of
the 2DEG Dyakonov1991 . These charges give rise to the radial (for the disk
geometry) electric field that in turn gives rise to azimuthal drift
magnetization current. The magnetization maximum is determined by the QHE
breakdown current, calculated in the frames of quasi-elastic inter-Landau
level scattering (QUILLS) approach.Eaves1986  The mentioned model correctly
describes experimentally observed linear temperature suppression of the
hysteresis amplitude observed both in magnetization Matthews2004  and
constriction conductance.Budantsev2007 ; Budantsev2009  But the magnetization
maximum calculated in the model is less than experimentally obtained value by
a factor of 4. Moreover, the model predicts bulk NECs distribution, which, to
our opinion, contradicts with experimental data of Klaffs et. al. Klaffs2004

In the present work we study the hysteresis of the magnetoresistance of the
conventional 2DEG with constriction, placed in it. It was found that at
certain critical width of the constriction the hysteresis of the
magnetoresistance vanishes. This observation strongly suggests that NECs are
localized near the edge of the sample.

A phenomenological comparison of the 2DEG magnetoresistance hysteresis in the
QHE regime with the hysteresis of magnetization of ordinary ferromagnetic
materials is performed. Particularly, we discuss the dependence on the
magnetic field sweep prehistory for both cases, studying minor loops,
anhysteretic curves and time relaxations. Mostly we observe a similar
behavior. However significant difference has been found manifested itself in
unusual advancing (anti-coercive) response (2DEG magnetoresistance) to
external parameter (magnetic field) change, which is just opposite to the
behavior of ferromagnetic materials.

We compare our results with the hysteresis of magnetoresistance in systems
with pseudo-spin degree of freedom — so-called “quantum Hall ferromagnets”
Piazza1999 ; DePoortere2000 ; Jungwirth2001 . In some cases (see e.g. Ref.
[Piazza1999, ]) analogical anti-coercive behavior is found. This
phenomenological similarity suggests that there are similar mechanisms
underlying the origin of the hysteresis both in ordinary 2DEG and in QH
ferromagnets.

In addition to the phenomenological description, we provide a qualitative
picture of a spatial distribution of NECs and electrochemical potential in the
sample.

The paper is organized in the following way. In Sec. II we describe the
experimental detail. In Sec. III we present the experimental data of following
measurements: the magnetoresistance of the 2DEG with the constriction in the
QHE regime, minor loop, anhysteretic curve, time relaxation dependence of the
magnetoresistance, and the dependence of the magnetoresistance hysteresis loop
area on the lithographic width of the constriction and discuss them. In Sec.
IV we develop a detailed qualitative picture of spatial distribution of NECs
in a sample, explaining the hysteresis of the magnetoresistance of the
constriction as well as the non-equilibrium magnetization of a 2DEG in the QHE
regime. We discuss inverted (anti-coercive) behavior of the hysteresis and
compare the hysteresis of the magnetoresistance with the hysteresis of the
ferromagnetic materials magnetization. We conclude in Sec. V.

##  II Experimental details

Experimental samples were fabricated from GaAs/AlGaAs heterojunctions with
2DEG of two types (type I and type II) grown in different cycles of molecular
beam epitaxy. The electron mobility $\mu$ and spacer layer thickness $d$ are
$\mu$=0.6$\div$0.8$\times$106 cm2/V$\cdot$s and $d=400$ Å in the type I sample
and $\mu$=0.8$\div$1.0$\times$106 cm2/V$\cdot$s and $d=300$ Å in the type II
sample. The electron densities of the macroscopic 2DEGs of both types are
ns=3$\times$1011 cm-2 at the temperature of 4.2 K. Hall bars of the dimensions
$W$$\times$$L$=50$\times$100 $\mu$m2 have been created on the surface of the
heterostructures by means of photolithography. In the middle part of the Hall
bars constrictions of the effective width 0.6$\div$0.8 $\mu$m and the length 3
$\mu$m have been created by means of electron beam lithography (see the inset
of Fig. 1). The samples of the both type allow to measure longitudinal
magnetoresistance $R_{\mathrm{L}}=U_{35}/I_{12}$ and Hall magnetoresistance of
macroscopic 2DEG $R_{\mathrm{H}}=U_{36}/I_{12}$. The measurements are carried
out in the linear response regime on the alternating current of the magnitude
1$\div$10 nA and the frequency 7 Hz at the temperature of 60 mK. The magnetic
field covering a range of 0$\div$15 T was oriented perpendicular to the 2DEG.

Figure 1: The magnetoresistance of the samples of the type I. At the centers
of the plateaux of macroscopic 2DEG Hall resistance
$R_{\mathrm{H}}~{}=~{}U_{36}/I_{12}$ at odd and even integer filling factors a
giant hysteresis of longitudinal magnetoresistance
$R_{\mathrm{L}}~{}=~{}U_{35}/I_{12}$ is observed. The geometry of the sample
is shown in the inset.

##  III Experimental results

First, we have studied the type I samples. A giant hysteresis of longitudinal
magnetoresistance $R_{\mathrm{L}}$ of the constriction with respect to
magnetic field sweep direction is observed in magnetic fields corresponding to
centers of the plateau of macroscopic 2DEG Hall resistance $R_{\mathrm{H}}$ at
both odd and even integer filling factors (Fig. 1). The effect is independent
on a direction of magnetic field. The most pronounced hysteresis loop is
observed near the filling factor $\nu=1$ in the range of filling factors
between $\nu_{\mathrm{h1}}=0.95$ and $\nu_{\mathrm{h2}}=1.07$ on the
background of macroscopic 2DEG Hall plateau observed in the range of filling
factors between $\nu_{1}=0.85$ and $\nu_{2}=1.22$ (Fig. 1). The hysteresis has
the amplitude 10 k$\Omega$ amounting to 100$\%$ of measuring signal and the
width width of the hysteresis is 1.2 T at filling factor $\nu=1$. Drastic
step-like rise of the magnetoresistance from zero up to 10 $k\Omega$ at up-
sweep of the magnetic field and sharp drop down to zero at down-sweep of the
magnetic field allows us to consider this behavior as the magneto-induced
breakdown of QHE.

The large amplitude of the hysteresis made it possible to carry out a detailed
study. It has been found that the response of the system (magnetoresistance)
outpaces the external excitation (magnetic field change) rather than retards
from it, unlike the case of ferromagnetic magnetization. In mathematical terms
the hysteresis loop is negatively oriented curve, while in ferromagnetic
materials the hysteresis loop is positively oriented. One can see that the
magnetoresistance dramatically changes at the entrance to the area of
hysteresis, that is $|dR/dB|$ tends to a huge value, and it remains almost
unchanged when leaving the area. In the case of retarded behavior the
magnetoresistance would practically not be changed at the entrance to the area
of hysteresis, that is $|dR/dB|$ would be close to zero. In this sense the
observed hysteresis is abnormally inverted having a negative coercivity in
contrast to the ferromagnetic magnetization.

Figure 2:  The hysteresis of the magnetoresistance of the II type sample at
the filling factor $\nu=1$. (a) Major and minor loops. (b) Anhysteretic curve.
Arrows indicate the magnetic field sweep direction. The sweep-rate is 0.05
T/min. The geometry of the sample and relaxation are shown in the insets. One
can see the fast relaxation regime with $\tau$=2.4 s. The relaxation is
occurred by jumps simultaneously in both structures.

The magnetoresistance of the type II sample also demonstrates the hysteresis
of magnetoresistance (Fig. 2) in the range of filling factors between
$\nu_{\mathrm{h1}}=0.96$ and $\nu_{\mathrm{h2}}=1.06$ on the background of
macroscopic 2DEG Hall plateau observed in the range of filling factors between
$\nu_{1}=0.88$ and $\nu_{2}=1.16$. One can see that the hysteresis of
magnetoresistance of the type II samples is observed at the same range of
filling factors as in the type I samples but it does not demonstrate a giant
amplitude and drastic jumps. This distinction is probably resulted from the
differences in heterostructures parameters and needs to be discussed
separately. Nevertheless, we succeeded to reveal inverted (anti-coercive)
behavior of the magnetoresistance hysteresis of the type II sample studying
the form of minor hysteretic loops. One can see that a change of the magnetic
field sweep direction inside the major hysteretic loop leads to dramatic
change of the magnetoresistance, i.e. the derivative $|dR/dB|$ tends to a huge
value (Fig. 2(a)), while the retarding behavior would correspond to zero
$|dR/dB|$ (as in the case of minor loops of ferromagnetic materials, where
$|dM/dH|=0$). Moreover the analysis of the minor loops allows us to conclude
that there is a large number of sample states inside the major hysteresis
loop, and therefore the sample state is multistable.

It is generally accepted that long-lived NECs are induced on the background of
Hall plateaus where $\sigma_{xx}$ is vanishing. Our measurements shows that
the range of existence of the hysteresis $\Delta\nu_{\mathrm{h}}$ is
approximately 3 times less than the Hall plateau width $\Delta\nu$:
$\Delta\nu_{\mathrm{h}}$/$\Delta\nu$$\approx$1/3. That is the condition
$\sigma_{xx}\approx 0$ is necessary but not sufficient to observe the
hysteresis.

To perform consistent phenomenological comparison of the 2DEG
magnetoresistance hysteresis in QHE regime with the hysteresis of
magnetization of ferromagnetic materials we have measured the anhysteretic
curves (Fig. 2(b)). Each experimental point $R_{i}(B_{i})$ has been obtained
by cycling the magnetic field around $B_{i}$ with a decreasing amplitude. The
form of the anhysteretic curve and its position inside the loop are
phenomenologically similar to that of ferromagnetic materials.

We have studied the time relaxation of the hysteresis, measuring the time
dependence of the magnetoresistances of two conducting channels situated in
the same Hall bar and separated by macroscopic 2DEG reservoir of the length of
250 $\mu$m and the width of 50 $\mu$m (see the top inset of Fig. 2(a)). The
dependencies are shown in the insets of Fig. 2(a) and (b). The curves have
been obtained after stopping the up-sweep of the magnetic field at the value
12.25 T, corresponding to the filling factor $\nu=1$ where the hysteresis has
the maximum amplitude.

It has been found that time relaxation has two phases: the fast initial one
followed by slow phase. After stopping the magnetic field sweep we have
observed the fast exponential relaxation by relatively small value (less than
25$\%$) with relaxation time 2.4 s (see the bottom inset of Fig. 2(a)). Such
relaxation time is too big to be explained by spin-orbital coupling mechanism
of relaxation Muller1992 ; Khaetskii1992  and resembles the nuclear relaxation
time in GaAs which is about 30 s.Dixon1997 ; Devyatov2004  It should be noted
that the non-equilibrium magnetization relaxes in the similar way.Kershaw2007
It also has regime of initial exponential decay followed by a much slower
power-law decay, however the relaxation time in the first regime is longer
($\sim$20 s) than obtained in our work. In Ref. [Matthews2004, ] it has been
suggested that magnetic field change results in QHE breakdown accompanied by
electron transitions between adjacent Landau levels with opposite spin. The
observed fast relaxation can be explained by such inter-Landau levels
scattering and brings the system to a local (not global) minimum of energy
after stopping the magnetic field sweep. According to Ref. [Matthews2004, ]
such scattering results from QUILLS process Eaves1986  and takes place in a
part of the sample where the maximum electric field is reached. In Refs.
[Matthews2004, , Dyakonov1991, ] it is suggested that maximum electric field
is reached in the bulk of a sample. However our results are sensitive to the
sample edge (see Section IV) since the magnetoresistance is defined by the
edge states filling. This allows us to conclude that QHE breakdown takes place
at the sample edge.

Further relaxation is much slower. During the next 40 min the resistances of
both structures relax by a small value ($\sim$20$\%$). Moreover the most
notable changes are occurred by sudden jumps simultaneously in both remote
structures (see the inset of Fig. 2(b)). Taking into account that the
structures are separated by macroscopic 2DEG reservoir of the length of 250
$\mu$m and the width of 50 $\mu$m it can be concluded that these jumps are
caused by relaxation process in the macroscopic 2DEG reservoir. This
observation once again confirms the conclusion that the narrow conductive
channel is a tool for the study of the non-equilibrium phenomena taking place
in a macroscopic 2DEG.Budantsev2007  The observed stepwise relaxation is
phenologically similar to Barkhausen jumps observed in the ferromagnetic
materials and originating from spin domain structure
transformations.Bertotti1998

A formation of a spatial spin polarization in a single quantum wire has been
considered in Ref. [Ihnatsenka2007, ] and it has been shown that this would
result in hysteresis in electron transport measurements. However, all the
calculated characteristics demonstrate the conventional coercive behavior.
Consequently, the spin mechanism is not relevant to our results. Moreover,
earlier it have been shown that the hysteresis of magnetoresistance is
independent on the in-plain component of magnetic field.Budantsev2009  It
allowed us to conclude that the hysteretic effect is not related to the
electron spin.

The measurements of non-equilibrium magnetization of a 2DEG also show stepwise
relaxation.Smith2011  It should be noted that transport measurements give a
valuable complementary information about the non-equilibrium state of a 2DEG.
Fact is that the magnetization is an integral characteristic determined by the
sum of all the magnetic moments, which are caused by all magnetization
currents in a sample, while the magnetoresistance is sensitive to a local
electrochemical potential of edge states. Particularly in this paper we
succeed to observe the correlation between the relaxation of magnetoresistance
of two remote constrictions and have established that NECs are localized near
the edge.

Figure 3: The dependence of the hysteresis loop area on the constriction width
at the filling factors $\nu$=1 and 2 at temperature 0.48 K.

To determine the width of spatial localization of NECs the dependence of
magnetoresistance hysteresis loop area on the lithographic width of the
constriction has been experimentally studied. For this purpose we used the
series of samples fabricated from the same heterostructure. The samples were
the Hall bars with constrictions of different widths. The constrictions of
widths from 0.8 $\mu$m to 1.3 $\mu$m have been fabricated by means of electron
beam lithography in the central part of Hall bars. All the constrictions have
low resistance in zero magnetic field (0.2$\div$0.8 k$\Omega$) and demonstrate
QHE in high magnetic fields. At the temperature of 0.48 K the hysteresis of
magnetoresistance have been observed at the filling factors $\nu$=1, 2 and 4
for all the samples. Fig. 3 shows the hysteresis loop area as a function of
the constriction width at the filling factors $\nu$=1 and 2. Experimental
points are well fitted by linear functions. One can see that the hysteresis
loop area decreases with the constriction width and vanishes at certain
critical width $W_{0}$. Moreover these critical widths coincide for both
filling factors and are about $W_{0}\approx$1.35$\mu$m. Assuming that the
critical lithographic width comprises of two counter-propagating NECs of the
width $W_{\mathrm{NEC}}$ and two depletion regions of the width
$W_{\mathrm{depl}}\approx$0.2$\mu$m:
$W_{0}=2W_{\mathrm{NEC}}+2W_{\mathrm{depl}}$, we obtain
$W_{\mathrm{NEC}}\approx$0.5$\mu$m is comparable with the depletion width. The
edge character of NECs has been also reported in the Ref. [Klaffs2004, ]
devoted to the study of spatial distribution of the non-equilibrium
electrostatic potential on a 2DEG surface, where it has been shown that the
most significant changes of potential take place close to the edge.

The observed hysteresis of magnetoresistance exhibits the following
properties: (i) it has two phases of relaxation, fast and slow; (ii) the slow
relaxation is occurred by jumps resembling the Barkhausen jumps in
ferromagnetics; (iii) it demonstrates inverted anti-coercive behavior; (iv) it
shows multistability and has an anhysteretic curve; (v) it can be observed in
sufficiently narrow channel thus pointing out its edge character.

##  IV Discussion

The existing physical models are able to interpret only separate experiments.
Unfortunately there is no any model which would not contradict all the set of
experimental data on the non-equilibrium state in a 2DEG in the QHE regime.
Analyzing the experimental data obtained in our and other studiesBudantsev2007
; Klaffs2004 ; Matthews2004 ; Usher2009 ; Ruhe2009  we suggest such a
qualitative picture. We propose the picture of spatial distribution of NECs in
a 2DEG and discuss its relation to the experimentally observed non-equilibrium
magnetization of the 2DEG. For simplicity, we consider a 2DEG at the filling
factor near $\nu=1$, when electrons form the incompressible liquid on the
first Landau level and inter Landau level scattering is assumed to be
suppressed.

###  IV.1 Magnetoresistance of constriction

The obtained experimental data on the magnetoresistance can be interpreted in
the frame of the following physical picture.

Down-sweep of the magnetic field gives rise to the vortex electric field that
induces outflow of the electrons from the bulk to the edge.Dolgopolov1992 ;
Laughlin1981  In this case the area of the incompressible liquid increases and
the edge channels are shifted closer to the lithographic borders of the
sample. It means that the counter-propagating edge currents in the
constriction become more distant from each other (Fig. 4(a-c)) than in the
case of equilibrium state and the backscattering is suppressed. In the type II
samples the magnetoresistance is indeed lower than its equilibrium value on
the anhysteretic curve. In the type I samples at filling factor $\nu=1$ the
decrease of magnetic field leads to complete suppression of the backscattering
in the constriction, manifested in the vanishing of the longitudinal
magnetoresistance and the establishment of the QHE regime.

Up-sweep of the magnetic field leads to vortex electric field, which in turn
induces outflow of the electrons from the edge to the bulk. In this case the
area of the incompressible liquid decreases and the edge channels are shifted
away from the lithographic border of the sample. The counter-propagating edge
currents in the constriction come closer together (Fig. 4(d-f)) than in the
case of equilibrium state and the backscattering increases. This results in
magnetoresistance rise as we have observed in both types of the samples. The
magnetoresistance of the type II samples is indeed higher than its equilibrium
value on the anhysteretic curve. In the type I samples at the filling factor
$\nu=1$ the increase of magnetic field switches the constriction from the QHE
regime to the resistive state. In other words, the magnetic field sweep leads
to magnetic field induced QHE breakdown.

Figure 4: Non-equilibrium electron redistribution in the macroscopic 2DEG (a),
(d) and in the constriction (b), (e) and top view of the 2DEG with narrowing
(c), (f) at up- and down-sweep of the magnetic field correspondingly at the
filling factor $\nu$=1. Bulk states in the constriction are occupied (c) or
not occupied (f) with incompressible liquid in different magnetic field sweep
direction. Arrows designate nonequilibrium currents.

###  IV.2 Non-equilibrium magnetization

Let us consider how the proposed picture corresponds to the non-equilibrium
magnetization measurements. The spatial distribution of the NECs have been
discussed in Refs. [Usher2009, , Ruhe2009, , Matthews2004, ]. However, the
details of the mentioned models are not fully consistent with the experimental
data obtained up to date. In particular, the model proposed in Refs.
[Usher2009, , Matthews2004, ] assumes that the non-equilibrium charge is
redistributed over the whole plane of a 2DEG and consequently the electric
field is formed in the bulk. This contradicts experimental data obtained in
the present study and discussed in Ref. [Klaffs2004, ] pointing to the fact
that the non-equilibrium charge redistribution takes place only near the edge.
In Ref. [Ruhe2009, ] the spatial distribution of the magnetization currents at
the 2DEG edge is qualitatively described (Fig.8 in [Ruhe2009, ]), but the
provided picture corresponds rather to equilibrium state because the common
Fermi level for the bulk and the edge is introduced, up to which all the
electronic states are occupied. Besides in this case the observed
magnetization should not exceed the amplitude of the equilibrium dHvA
oscillations while the experimental non-equilibrium magnetization is about
20$\div$60 times larger.

Figure 5: The structure of the non-equilibrium edge currents at down- (a) and
up-sweep (b) of the magnetic field.

The decrease of magnetic field at the filling factor $\nu=1$ shifts the
electrons from the bulk to the edge within the first Landau level. Due to the
fact that the area occupied by the bulk states is much larger than that
occupied by the edge states one can neglect the changes of the electrostatic
potential in the bulk and consider that it remains flat. In this case the most
significant changes are occurred near the edge, where a dipolar strip
decreasing the electrostatic potential in the bulk and increasing it near the
edge is formed. The electrons on the first Landau level populate the edge
states up to the level of non-equilibrium electrochemical potential
$\mu_{down}$ exceeding the potential of the empty bulk states by the value
$\Delta\mu$ (Fig. 5(a)). In the crossed electric and magnetic fields the
electrons drift along the edge forming the magnetization current approximately
equal to $e^{2}/h\cdot\Delta\mu$. This current corresponds to the
experimentally observed non-equilibrium paramagnetic
magnetization.Matthews2004 ; Ruhe2009

The increase of magnetic field at the filling factor $\nu=1$ leads to more
complicated distribution of the magnetization currents. The outflow of the
electrons from the edge to the bulk leads to the formation of the dipolar
strip near the edge that rises the electrostatic potential in the bulk and
forms the potential well near the edge followed by the depletion region. It
means that the electrostatic potential is non-monotonic and has a minimum near
the edge, caused by interplay of two oppositely directed electric fields: the
field of overpopulated bulk and the depletion field (Fig. 5(b)). On either
side of the potential minimum the incompressible electron liquid forms
counter-propagating drifting currents — giant diamagnetic current flowing
closer to the bulk and low paramagnetic current flowing in close proximity to
the edge. The presence of the paramagnetic current is confirmed by the
experimental fact that the sign of the Hall resistance $R_{H}$ does not depend
on the magnetic field sweep direction in a given orientation of the magnetic
field. Hence the direction of the transport current, flowing along the edge of
a sample, coincides with the direction of the paramagnetic current as in case
of equilibrium state.

Edge states are populated up to the non-equilibrium electrochemical potential
level $\mu_{up}$ so that the empty edge states on the first Landau level are
lower than populated bulk states by the value of $\Delta\mu$. The total
magnetization current is dominated by the giant diamagnetic current
$e^{2}/h\cdot\Delta\mu$. To explain the experimentally observed giant non-
equilibrium magnetization of a 2DEG Usher2009  it is necessary that
$\Delta\mu\geq(20\div 60)\hbar\omega_{c}$, where $\omega_{c}$ is a cyclotron
frequency.

Here we do not discuss the finite magnetic field range of the hysteresis loop
as it requires detailed consideration of disorder in nonlinear screening
conditions.Chklovskii1992

###  IV.3 Anti-coercive behavior

There are two possible reasons for anti-coercive behavior: (i) an abrupt
movement of the boundary of the incompressible liquid at small change of the
magnetic field, (ii) topological transitions in quantum Hall liquid. Discuss
them in more detail.

(i) at the filling factor $\nu=1$ the magnetic field change $\Delta B$ causes
the shift of the boundary of the incompressible liquid relative to the
lithographic edge. Besides, the larger the sample the smaller the change of
the magnetic field $\Delta B$ is required to shift this boundary by an amount
of the order of the depletion width. For example for disk-shaped 2DEG of
radius $R$:

$\Delta B\approx\frac{W_{depl}}{2R}\cdot B.$ (1)

(ii) As seen from Figs. 4(c) and 4(f) the shift of the boundary of the
incompressible liquid leads to topological transitions in quantum Hall liquid.
Fig. 4(c) corresponds to the total transmission of the edge channels through
the constriction, while Fig. 4(f) corresponds to the intense backscattering.
Thus, a small change of the magnetic field leads to a drastic change of the
magnetoresistance.

Topological transition has been observed in samples of type I: the vanishing
of the longitudinal magnetoresistance has been observed indicating total
suppression of backscattering in the constriction at down-sweep of the
magnetic field, while up-sweep of the magnetic field causes sharp jump of the
magnetoresistance from zero to 10 k$\Omega$. Abrupt movement of the boundary
of the incompressible electron liquid at small change of the magnetic field is
inevitably realized in the both types samples. This is manifested itself in
the sharp changes of the magnetoresistance in minor loops measurements.

Figure 6: Hysteresis of 2DEG magnetization (large loop) in QHE regime against
the de Haas - van Alphen oscillations (thick line) is inverted (anti-coercive
behavior) that is the response advances the magnetic field sweep. Small loop
demonstrates the retarding (coercive) behavior.

The discovered unusual anti-coercive behavior is in accordance with the law of
energy conservation. Consider the behavior of the non-equilibrium
magnetization of a 2DEG in the QHE regime. In Fig. 6 a typical hysteresis of
the magnetization on the background of dHvA oscillations at the filling factor
$\nu=1$ is shown. Matthews2004  The system response (magnetization) also
advances the external parameter (magnetic field) change as in the case
discussed. However the total work performed by the system $-\oint{Md\,B}$
(shadow area in Fig. 6) is negative, that is in accordance with energy
conservation law. Retarding response behavior would correspond to the positive
total system work (dashed area in Fig. 6) per cycle contrary to the law of
conservation of energy.

###  IV.4 Comparison with ferromagnetics

Let us compare the observed hysteresis of magnetoresistance with the
hysteresis of magnetization of ferromagnetics.

It is well known that ferromagnetism is of spin origin. The exchange
interaction in ferromagnetics causes the spins align. At the same time, the
requirement of a minimum of the total energy of the system leads to its
splitting into domains with opposite spin orientations. A change of the
magnetic field results in spin flip process near the domain walls causing
their “motion”. However, due to a disorder the transformations of domain
structure is retarded with respect to the external magnetic field change. This
is the reason of hysteresis with positive coercivity in ferromagnetics. While
in a 2DEG a small changes of the external magnetic field lead to sharp motion
of the boundary of the incompressible liquid and causes the hysteresis with
negative coercivity.

Earlier it has been found Budantsev2007  that the hysteresis of
magnetoresistance is suppressed by temperature. It has been shown that the
amplitude of hysteresis linearly increases as the temperature decreases and
saturates at certain low temperature ($\sim$400 mK). The non-equilibrium
magnetization of a 2DEG has similar temperature dependence.Matthews2004  The
hysteresis of magnetization of ferromagnetic materials is also suppressed by
temperature but it has different functional dependence described by Weiss
theory of ferromagnetism. Bertotti1998

It should be noted that NECs in the QHE regime are induced by sweep of the
magnetic field and can be considered as eddy currents as they are often
called. However, in ferromagnetics it is enough to reduce the sweep rate to
eliminate the influence of eddy currents. In this sense eddy currents in
ferromagnetics are time-dependent because of finite longitudinal conductivity
$\sigma_{xx}$. While in a 2DEG in the QHE regime when $\sigma_{xx}\approx 0$
the generation of NECs is inescapable at any reasonable sweep rate. According
to our results NECs relaxation lasts for hours after stopping the magnetic
field sweep. In Ref. [Kershaw2007, ] NECs have been shown to circulate for
many hours. In terms of Ref. [Bertotti1998, ] under conventional experimental
conditions NECs can be considered as time-independent.

##  V Conclusions

We have shown that sweep of the magnetic field in the QHE regime results in
drastic changes of electrostatic potential in the bulk with respect to the
edge if the inter Landau levels scattering is suppressed. This leads to giant
magnetization currents flowing along the edge and topological transitions in a
2DEG with a constriction. The latter results in changes of magnetoresistance.

The hysteresis of magnetoresistance of a 2DEG in the QHE regime has
significant phenomenological similarities with the hysteresis of magnetization
of ferromagnetic materials, showing multistability, jumps of relaxation,
temperature suppression and having the anhysteretic curve. At the same time a
fundamental difference expressed in anti-coercive hysteresis behavior of the
magnetoresistance has been found.

The possible causes of the limitations of the inter Landau level scattering as
well as the possible limiting mechanisms of NECs are beyond the scope of this
article and require further investigation. However we have experimentally
shown that NECs are induced in a narrow ($\sim$0.5$\mu$m) area along the edge.

Despite the large number of studies devoted to the non-equilibrium state of a
2DEG induced by the sweep of the magnetic field in the QHE regime there is no
clear understanding of the phenomena at a microscopic level as well as a
systematic phenomenological study of the dependence of the phenomena on
parameters of heterostructures and conditions of epitaxial growth up to date.
Unfortunately the NECs are insensible in a conventional transport measurement
in a linear response regime due to zero longitudinal resistance. In such
measurements the condition of the 2DEG is not clear in advance. Such
uncertainty can bring unexpected difficulties in studies of a 2DEG in the QHE
regime, such as the study of QHE breakdown by the electric current, that has
no full and consistent model up to date (see for a review Ref. [Nachtwei1999,
]). The present study raises the issue of the need to search critical
parameters that define the phenomena and explain the phenomena at a
microscopic level. Therefore, some existing concepts on the phenomena in 2DEG
in the QHE regime can be extended or revised taking into account the found
non-equilibrium.

## Acknowledgments

The reported study was supported by RFBR (research project
No.14-02-31740-mol-a and No.12-02-00532-a), Program of fundamental scientific
research DNIT RAS (project 3.2).

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