# Aesthetic Visual Question Answering of Photographs

Xin Jin<EMAIL_ADDRESS>Beijing Electronic Science and Technology
InstituteBeijingBeijingChina ,  Wu Zhou<EMAIL_ADDRESS>Beijing Electronic
Science and Technology InstituteBeijingBeijingChina ,  Xinghui Zhou∗
<EMAIL_ADDRESS>University of Science and Technology of
ChinaHefeiHefeiChina ,  Shuai Cui<EMAIL_ADDRESS>University of
California, DavisDavisUSA ,  Le Zhang<EMAIL_ADDRESS>Beijing
Electronic Science and Technology InstituteBeijingBeijingChina ,  Jianwen Lv
<EMAIL_ADDRESS>Beijing Electronic Science and Technology
InstituteBeijingBeijingChina and  Shu Zhao<EMAIL_ADDRESS>Beijing
Electronic Science and Technology InstituteBeijingChina

(2018)

###### Abstract.

Aesthetic assessment of images can be categorized into two main forms:
numerical assessment and language assessment. Aesthetics caption of
photographs is the only task of aesthetic language assessment that has been
addressed. In this paper, we propose a new task of aesthetic language
assessment: aesthetic visual question and answering (AVQA) of images. If we
give a question of images aesthetics, model can predict the answer. We use
images from www.flickr.com. The objective QA pairs are generated by the
proposed aesthetic attributes analysis algorithms. Moreover, we introduce
subjective QA pairs that are converted from aesthetic numerical labels and
sentiment analysis from large-scale pre-train models. We build the first
aesthetic visual question answering dataset, AesVQA, that contains 72,168
high-quality images and 324,756 pairs of aesthetic questions. Two methods for
adjusting the data distribution have been proposed and proved to improve the
accuracy of existing models. This is the first work that both addresses the
task of aesthetic VQA and introduces subjectiveness into VQA tasks. The
experimental results reveal that our methods outperform other VQA models on
this new task.

Multimodal learning, Aesthetic computing, Visual question and answering, Semi-
automatic labeling, Transformer

aesthetic mixed dataset with attributes, multitasking, external attribute
features, ECA channel attention

*Corresponding authors

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††journal: JACM††journalvolume: 37††journalnumber:
4††article: 111††publicationmonth: 8

##  1\. Introduction

In the researches of computer vision and natural language processing,
researchers always focus on direct and obvious goals instead of image
aesthetics with different attributes. In the past decade, many researchers
were interested in image aesthetics. However, the popular methods were to
score images and describe attributes of images. Either giving a score ranging
from 0 to 10 to evaluate aesthetics of images or describing a images’
attributes(Jin et al., 2019), such as color, lighting, and composition, in one
sentence are limited and lack description in details.

The state-of-the-art model of visual question answering (VQA) has achieved
multiple purposes on many large-scale datasets. According to the category of
the datasets, the subtask of VQA can be divided into task-based, inference-
based, and text-based, like TextVQA (Singh et al., 2019) dataset. Many famous
models (Li et al., 2020; Chen et al., 2020; Li et al., 2019) have been
developed in these visual question answering tasks. However, they lack the
overall investigation and reasoning of the picture.

Many attribute labels related to image aesthetics can be collected in
photography (for example, image classification and evaluation labels available
in subjectively scored datasets). Although some studies have used image tags
for regression, multi-task evaluation, and image captioning, they ignore the
available information of images in multiple information dimensions.

Figure 1. VQA and Aesthetic VQA. The AVQA focuses on the overall aesthetic
attributes of the image such as shot, lighting and colors.

Some large-scale visual question answering datasets, such as VQA2.0 (Goyal et
al., 2017) and Visual Genome (Krishna et al., 2017), include some questions
related to the overall image; GQA (Hudson and Manning, 2019) and NLVR2 (Suhr
et al., 2017) include some questions about designing image content reasoning.
By a lot of computational cost and testing, we can get the answer related to
the question about overall characteristics of the image. There is no dataset
that focuses on visual question answering based on the overall image.

Most of current VQA researches focus on objective QA pairs. This paper mainly
addresses aesthetic visual question answering (AVQA) of images, as shown in
Figure 1. This task is both aesthetic and subjective; also, it is a
complementary task for the previous QA studies. This problem is important
since subjective questions and answers are very prevalent among human
discourses. In this paper, we present a dataset with aesthetic QA pairs,
aesthetic visual question answering (AesVQA). Images in the new dataset are
all marked with aesthetic labels through series of multiple computer vision
sub-tasks. These labels cover composition, color, subject, lighting, genres,
techniques and emotions of photos.

To obtain reliable labels, we develop some unsupervised and semi-supervised
computer vision methods based on image processing and large restraint model.
At the same time, we propose a threshold function and design a small amount of
manual labelling image adjustment function to get a uniform and credible
dataset distribution.

In particularly, the labels of lights and colors are designed based on the
specific objects in the image. The labels of scene and composition are
designed based on the whole picture. The other labels based on subjective
evaluations of images, especially techniques and emotions, provide extra
dimensional information in question and answering. The task becomes more
challenging by adding some human psychological evaluations. The main
contributions of this paper can be summarized as:

  * •

`New Task`: The new task of image aesthetics question and answer, and it is
the first work that introduces subjective answering to VQA problems.

  * •

`New Labels`: Semi-automatic QA labelling of images gets more information when
answering and methods of dataset distribution adjustment.

  * •

`New Points of VQA`: The traditional visual question answering method based on
objective detection is optimized to half-object-detection-based (such as
lighting and color labels) and half-whole-image-based (such as subject and
composition labels).

##  2\. Related work

Aesthetics and computer vision are inextricably linked, and people are
demanding higher quality of image data. How to process the images into a form
that is more compatible with human preferences has become a critical issue. It
is feasible to use computer technology to study the aesthetics of images(Lu et
al., 2015).

The earliest image aesthetic quality evaluation was only the two-category
evaluation of scores and good or bad, based on which a considerable number of
scholars have made considerable contributions on how to score and score on
multiple attributes of images. The earliest study of aesthetic image quality
evaluation is this paper(Chang et al., 2017).

Visual question answering is a difficult problem that straddles computer
vision and natural language processing, and its task requires the extraction
of not only image features but also textual partial features. Recent VQA
datasets such as DAQUAR(Ren et al., 2015), Visual7W(Zhu et al., 2016), Visual
Madlibs(Yu et al., 2015), FM-IQA(Gao et al., 2015), VQA(Goyal et al., 2017),
etc. are available. Unlike look-and-talk tasks, simply fusing image and text
features often does not yield the desired features, i.e., answers.

Visual question answering requires the model to process from image and
language to combine and give the answer effectively. So, VQA is both
interdisciplinary and challenging. Also, due to characteristics of image
aesthetics, there is no professional image aesthetics Q & A(Question and
Answer) dataset. Therefore, it is important to propose a common aesthetic Q &
A datasets. By the visual model, we use classical Q& A datasets as reference
to generate our dataset. The classic Q & A datasets are always generated by
machines, such as: VQA(Goyal et al., 2017), TextVQA(Singh et al., 2019), EST-
VQA(Wang et al., 2020) and VizWiz-VQA(Bigham et al., 2010).

One of the difficulties of the image aesthetics task is the disadvantage of
inaccurate subjective evaluations but low number of objective ones. Borrowing
a large amount of data as a foundation, it will be possible to mine a
sufficient number of images and corresponding comments with a high enough
standard, and then further convert the comments into the desired Q&A pairs.
Proxy objectification with statistical features of subjective evaluation is a
common approach in current image aesthetics tasks.

To date, there is no comprehensive aesthetic VQA dataset. Previous research
work proposed the AQUA(Garcia et al., 2020) dataset, which focuses more on
studying the artistic aspects of paintings, while our dataset focuses more on
the aesthetic aspects of photographs. In previous studies, researchers would
apply attention mechanisms to images or text to obtain better results, but it
is difficult to obtain the desired features from the many image features
because of the different feature spaces and the presence of tensor features
with ultra-high dimensionality of images.

Table 1. VQA datasets’ answer words length accounted for the proportion of all answers table. Answers with two words or more are 94.6% in AesVQA dataset, which leads a more difficult VQA task. | The Length of The Answer Words |
---|---|---
Dataset | One Word | Yes/No | Two Words or More | The Number of QA Pairs
VQAv2 (Goyal et al., 2017) | 51.2% | 45.0% | 3.8% | 1,105,904
Visual7W (Zhu et al., 2016) | 51.2% | 21.2% | 27.6% | 139,868
Visual Genome (Krishna et al., 2017) | 52.5% | 21.2% | 26.3% | 1,445,233
AesVQA | 5.4% | 0% | 94.6% | 324,756
Figure 2. Pictures, Questions and Answers in the Aesthetic VQA dataset.

##  3\. Aesthetic Visual Question Answering

To study the answers to questions related to image aesthetics and the overall
content of the image, and to reduce the cost of manual annotation, we proposed
AesVQA, as shown in Figure 2 and Table 1. This dataset is an image aesthetic
question and answer dataset based on an unsupervised model. We will start by
describing how to select the images used in AesVQA. Then, we explain our data
collection pipeline to collect questions and answers.

Figure 3. The distribution of the basic aesthetic labels in the AesVQA
dataset. Figure 4. The distribution of the genres labels in the AesVQA
dataset. Figure 5. The distribution of photos’s subjective labels in the
AesVQA dataset. Including tender, magic, lovely, calm, angry, adventurous and
many other attributes.

###  3.1. Images of AesVQA

We use pictures from the Flickr website as our source pictures, which conforms
to developing a VQA model based on image aesthetics and the overall content of
the image. We are most interested in pictures in categories such as
landscapes, people, still lifes, and animals. Several categories in the
Explore section of the Flickr website meet this condition.

To obtain accurate images, to determine the aesthetic information of the image
and the overall characteristics of the image, we deleted the pictures with
borders and the pictures with a too large aspect ratio (if the aspect ratio of
the picture is greater than 1:2, delete it). We use the OCR model Rosetta on
these images to delete some images that contain watermarks that may be
involved in the infringement.

To automate this process to identify categories that tend to contain images
with text, we select 100 random images for each category (if the maximum
number of images for that category is less than 100, select all images). We
run the state-of-the-art OCR model Rosetta [6] on these images and calculate
the average number of OCR boxes in the category.

Based on the 190,000 Flickr images we crawled, after artificially filtering
out abstract images and unrealistic images from image photography, filtering
these (and a small amount of noisy data from manual annotations) can obtain
72168 images. image. Manual screening requires that the image must be taken
from the real world, not obtained through PS or other software editing tools.

###  3.2. Generation and filtering of AesVQA

In order to generate question and answer pairs, we designed our own model. Our
model based on sequence to sequence method, including three modules: text
content encoder, answer encoder and decoder. In addition, the quality of the
generated AesVQA data is uneven, and there is a big gap with the common VQA
dataset. Therefore, the data needs to be screened again after QA is generated.
We propose an aesthetic question and answer filtering method using LDA(Blei et
al., 2003).

The text content coder is used to process the input comments. At the same
time, it needs to calculate the attention weight of different words and the
weight of specific words in the replication mechanism. The answer encoder is
used to process the selected answer words or phrases, filter the words
according to the attention weight, and encode the words into a format suitable
for the decoder. The decoder is responsible for using the cyclic neural
network to process the input tensor data, and generating questions according
to the weight data between the answer and the comment.

For the generated question and answer pairs, the manual filtering method is
used to remove the inappropriate question and answer pairs, so as to ensure
that at least one question and answer pair generated by each picture is
reasonable. These question and answer pairs are formed into a small text set,
and the correlation between the text set and multiple topic words obtained
from previous LDA topic calculation is calculated, which is used as the
threshold to evaluate whether a question and answer pair meets the aesthetic
question and answer model.

Calculate the relevance between the remaining sentences and the topics
selected by LDA, and judge whether the relevance of the selected sentences is
greater than the topic threshold. When the relevance value of the selected
sentence is greater than the threshold, it is determined as a question and
answer pair that meets the requirements, otherwise it is determined as a
question and answer pair that does not meet the requirements.

Finally, since the quality of the generated issues is not very high, the
solution is to make manual corrections.

###  3.3. Basic Aesthetic Labels

The first part of the AADB(Kong et al., 2016) dataset we constructed is the
basic image aesthetic tags, including composition, color, subject, and light.
Among them, in order not to reduce the VQA task to a simple classification
task, and to ensure that the specific image attributes are associated with
specific objects, we design light labels and color labels to be related to the
content of the image, while composition labels and subjects labels are related
to the overall image. Here, the whole image is defined as multiple targets or
content obtained by the fusion of background and foreground information. The
biggest feature of this content is that each part of the image is required to
obtain it, rather than part-based objects.

Among them, the symmetry of the composition labels is obtained by the
SSIM(Wang et al., 2004) calculation of the image. When the left and right
parts of the image or the upper and lower parts of the image after the mirror
symmetry, the SSIM is higher than the threshold, and it is judged to be an
asymmetric image. The rule of thirds and the center composition use the faster
R-CNN based on the visual genome to obtain the position of the most
distinctive object in the image. Midas is used for the foreground composition,
and the image is divided into three parts: the upper, middle, and lower parts,
and judge whether there is a significant difference in the depth of field
between the upper part and the lower part.

Color labels depend on the type and color of objects in the image obtained by
faster R-CNN. To enrich the diversity of colors, we have added warm tone and
cold tone, using target detection to cut out the selected objects in the
image, and classify their average colors, and judge based on the correlation
between the average color and other known main colors warm tone or cold tone.
The attributes of the subject are labeled using the CLIP(Radford et al., 2021)
pre-training model. According to the classification results of the subject on
the AVA(Murray et al., 2012), it can be known that CLIP can deal with the
category labeling of multiple subjects. At the same time, to make up for the
errors that may be caused by automatic marking, some manually marked image
attributes are subsequently used to correct the results.

The attribute of Light is judged by judging the grayscale histogram of the
image. When the image is not black and white, but still has more black parts,
it is marked as backlight; if the histogram part of the image is on both
sides, it is marked as split light; If the histogram of the image is evenly
distributed, it is marked as broad light. Similar to the Subject attribute,
this part also uses some artificially marked image attributes to correct the
result.

Figure 6. Question answers which before and after the adjustment of image’s
confidence and the adjustment of the distribution of answers.The AoC means the
adjustment of image’s classify confidence. The AoDA means the adjustment of
distribution of answers. These two tricks will help the model achieving a
higher accuracy.

###  3.4. Distribution in VQA Dataset

The distribution of the dataset is extremely important for the VQA task. Some
work (Teney et al., 2020; Agrawal et al., 2018; Zellers et al., 2018) have
proved it. For the aesthetic question and answer task based on the whole
image, three dimensions of data distribution need to be considered: the
distribution of the whole picture and the specific objects in the image in the
question, the distribution of pictures with obvious objects, and the overall
scenery in the data set, the answer to the question is Distribution under the
influence of subjective factors.

For the impact of the first data distribution, we roughly divide the labels of
the data set into two categories: ”related to specific objects” and ”related
to the image as a whole”, and ensure that the number of images and the number
of questions in the two categories are consistent.

In response to the impact of the second data distribution, we have added two
more general and adaptable categories: genre and skill. These two shooting-
related categories will be further divided into more than 10 sub-categories,
each of which has enough questions and answers. The genre and technique will
be able to balance the effects of pictures with obvious objects and overall
scenery on the model.

In response to the impact of data distribution in Chapter Four, we add a
subjective evaluation. Mark the content of the picture as emotionally related.
At the same time, to prevent excessively strong subjective distribution
deviations, in the labeling of some attributes, we introduced 7000 additional
labeled pictures to constrain the generation of questions and answers.

We show the distributions of the basic aesthetic attribute labels, the genre
labels, the subjective labels and photography labels in Figure 3, Figure 4 and
Figure 5.

###  3.5. Subjective Labels and Photography Component

To overcome the possible impact of data distribution, we divide the attributes
of images and aesthetics into subjective feelings and objective photography
techniques. The subjective description comes from people’s direct perception
of the image, and the label of this part comes from the pre-training label of
CLIP. Including tender, magic, lovely, calm, angry, adventurous, and many
other attributes. The reason for selecting these words is that the results of
the sample survey show that these words can get a more balanced data
distribution.

On the other hand, we provide related attributes of photography, including
genres and techniques, which contain multiple subcategories. Same as
subjective, we use CLIP to mark images. Some properties are shown in Figure 6.
Many attributes of this part have the limitation of vague definition.
Therefore, each category has a confidence level. Only when the confidence
level is high enough, the mark of the picture is considered effective.

CLIP’s image classification model is unsupervised, multiple subcategories can
be used as input, and the model is required to classify them. Specifically,
some attributes such as ”Multiple Exposure” require more professional
photography skills, so the confidence threshold for this category is higher;
such as ”Night Photography”, which can better distinguish various attributes,
so for this the confidence threshold of the class is low.

###  3.6. Bias of Data

Because of the use of unsupervised models such as CLIP to predict the types of
pictures, how to design the distribution of options in answers for pictures is
the key.

Usually, we will give up pictures with too high an accuracy rate and too low
an accuracy rate, that is, a picture can be judged as B category on the A
attribute, or it is difficult to judge which category it is. We use the design
threshold and the accuracy of a certain type of problem model to estimate how
the threshold of this type of problem should be, including two parameters, the
mean and variance. Only when the mean and the standard deviation of answers’
distribution meet the accuracy of this type of baseline model (LXMERT for
example) higher than 50 %, answers contribution is efficacious.

(1) $\text{accuracy}(V,Q,A,D(\text{avg}),D(\text{std}))>0.5$

where $D(\text{avg})$ means the average of the answers’ distribution, the
$D(\text{std})$ means the standard deviation of the answers’ distribution.
72,168 pictures satisfy the above formula.

Table 2. The AoC in the table means the data with adjustment of image’s class
confidences, and the AoDA in the table means the adjustment of the
distribution of answers for each question. ”M1” means method 1, it is the
LXMERT, the baseline model. ”M2” means method 2, it is the Visual BERT. ”M3”
means method 3, it is the UNITER, the state-of-the art model in VQA. The
training set in the AesVQA database was 58,168 images, and the test set and
validation set were both 7,000 images, which were randomly mixed and assigned.

| M1 | M1&AoC | M1&AoDA | M2 | M2&AoC | M2&AoDA | M3 | M3&AoC | M3&AoDA
---|---|---|---|---|---|---|---|---|---
Composition | 53.4% | 54.6% | 54.8% | 55.6% | 56.3% | 56.5% | 57.6% | 59.2% | 60.3%
Color | 55.3% | 57.6% | 58.8% | 57.2% | 59.5% | 60.2% | 58.2% | 58.5% | 60.5%
Light | 70.6% | 73.5% | 67.3% | 72.5% | 75.8% | 69.0% | 71.5% | 72.5% | 67.8%
Subject | 45.9% | 50.2% | 55.5% | 48.3% | 53.2% | 57.1% | 49.5% | 53.9% | 58.0%
Genres | 48.5% | 56.6% | 54.7% | 50.3% | 58.5% | 56.5% | 52.2% | 61.2% | 60.7%
Techniques | 42.4% | 49.9% | 50.3% | 45.9% | 51.2% | 53.0% | 47.2% | 53.6% | 54.9%
Subject | 51.8% | 52.6% | 53.5% | 53.9% | 54.4% | 55.8% | 55.6% | 57.8% | 59.4%
All | 57.7% | 58.6% | 59.3% | 59.8% | 60.7% | 62.2% | 60.3% | 61.4% | 61.9%

Table 3. By randomly extracting 1000 pictures from each category, the confidence adjustment operation needed to be judged. The following table describes the pre-adjusted and post-adjusted range of each sub-category in the categories of genres and techniques: Genres | Before Adjustment | After Adjustment
---|---|---
Abstract | 0.536 | 0.536
Architectural | 0.912 | 0.615
Astrophotography | 0.990 | 0.653
Conceptual photography | 0.760 | 0.760
Fashion | 0.450 | 0.450
Fine-art photography | 0.784 | 0.635
High-speed photography | 0.856 | 0.654
Landscape | 0.750 | 0.653
Nature | 0.803 | 0.664
Portrait photography | 0.869 | 0.685
Sports | 0.427 | 0.427
Still life | 0.752 | 0.645
Underwater photography | 0.416 | 0.416
Wildlife | 0.947 | 0.703
Techniques | Before Adjustment | After Adjustment
Bokeh | 0.514 | 0.514
Burst mode | 0.272 | 0.452
Contre-jour | 0.634 | 0.634
High-dynamic-range imaging | 0.595 | 0.595
Holography | 0.402 | 0.402
Long-exposure photography | 0.308 | 0.502
Macro photography | 0.869 | 0.652
Multiple exposure | 0.894 | 0.675
Night photography | 0.613 | 0.613
Panning | 0.597 | 0.597
Panoramic photography | 0.749 | 0.678
Stereoscopy | 0.875 | 0.658
Time-lapse photography | 0.605 | 0.605
Vignetting | 0.634 | 0.634
Figure 7. The pictures on the left show that these pictures have more biased
attributes, which can easily cause the model to fall into overfitting; the
pictures on the right have appropriate ”controversial” labels, which can allow
the model to obtain better training results.

###  3.7. Influence from Artificial labels

Complete automatic program marking is not considered to be reliable, and there
may also be cases of inconsistency with the facts in automatic program
marking. Therefore, we selected 7000 pictures for manual marking. When the
result of manual marking does not match the result of machine marking, the
picture will be corrected to the result of manual marking.

Another important significance of manually marked pictures is to determine
whether the distribution of answers of this type conforms to the real
situation. Taking the composition category as an example, the number of
symmetrical pictures is usually much lower than that of the rule of thirds and
the central composition. This feature can be reflected in the sampled pictures
and all pictures. Considering the huge workload of manual marking, we only
added manual markings to the composition, color, scene, and lighting
attributes. Genres and techniques are more complicated and prone to
controversy, so they are not marked.

##  4\. Experiments

In the experiment, we used three models to train and test each category. The
ratio of the training set and validation set is 8:1, and the number of test
sets is the same as the validation set. To test the influence of data
distribution on the model, we constantly adjusted the proportions of each sub-
category during training and obtained more balanced training results.

The data distribution without artificial interference comes from all pictures.
Such a data distribution will not be conducive to the training of the neural
network. The operation of adjusting the data set mainly includes: adjusting
the confidence threshold of each sub-category in a large category, and the
distribution of the question and answer results of the pictures in each large
category. The number of pictures is limited by the different types of pictures
that have different requirements in the selection process, so it is impossible
to achieve close results. The results of the comparative experiment with AoC
(Adjustment of Confidence) and AoDA (Adjustment of the Distribution of
Answers) are shown in Table 2. It can be seen that UNITER, based on multiple
datasets has achieved the best results in AesVQA.

###  4.1. Adjustment of Confidence

Taking genres and techniques as examples, we adjusted the confidence of
different categories through training and test results, called it adjustment
of confidence (AoC). For different genres and techniques, it is generally
believed that the judgment criteria are also different, which is reflected in
the labeling process with unsupervised learning as the main method, and the
confidence level needs to be adjusted according to the specific class. As is
shown in Table 3, The adjusted result is artificially limited to the range of
0.3 to 0.7, this range will help the training and prediction of the model.

###  4.2. Adjustment of Distribution of Answers

Limited to automatic labeling, our proposed dataset requires multiple models
to label images. To prevent the occurrence of some over-fitting phenomena, ten
answers are set for each question. For adjusting the distribution of answers
(AoDA), we let questions have no controversial answers and salient answers.

The salient answers mean that it is obvious to know which is the right choice
in all options. The controversial answers mean there are two or more answers
with high confidence. For example, a picture may be classified in ”landscape”
and ”wildlife” at the same time, such a picture needs to be removed, as shown
in Figure 6.

The adjustment of the distribution of answers will help eliminate images in
the data set that lack diversity in labels. These images will affect the
model’s preference and make the model more prone to overfitting. The example
in the Figure 7 illustrates which pictures should be discarded and which
should be kept during adjustment.

##  5\. Conclusions

In this paper, we propose and solve a new task of aesthetic quality
evaluation: VQA of image aesthetics. This paper constructs the VQA dataset for
image aesthetics. We get basic image aesthetics labels, genres labels, and
techniques labels from photography, and subjective emotional labels, and
transfer these labels to question-answers pairs. We used three advanced VQA
models for training and testing and obtained further performance improvement
by adjusting the confidence in image classification and the distribution of
answers in VQA. It can be proved by experiments that exporting the answer to
aesthetic questions and answers is available by training on our datasets. As a
result, display performance model and the advantage of databases, which filled
in the blank in the field of aesthetics in the VQA.

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# Spin-polarization and resonant states in electronic conduction through a
correlated magnetic layer

###### Abstract

The transmission through a magnetic layer of correlated electrons sandwiched
between non-interacting normal-metal leads is studied within model
calculations. We consider the linear regime in the framework of the Meir-
Wingreen formalism, according to which the transmission can be interpreted as
the overlap of the spectral function of the surface layer of the leads with
that of the central region. By analyzing these spectral functions, we show
that a change of the coupling parameter between the leads and the central
region significantly and non-trivially affects the conductance. The role of
band structure effects for the transmission is clarified. For a strong
coupling between the leads and the central layer, high-intensity localized
states are formed outside the overlapping bands, while for weaker coupling
this high-intensity spectral weight is formed within the leads’ continuum band
around the Fermi energy. A local Coulomb interaction in the central region
modifies the high-intensity states, and hence the transmission. For the
present setup, the major effect of the local interaction consists in shifts of
the band structure, since any sharp features are weakened due to the
macroscopic extension of the configuration in the directions perpendicular to
the transport direction.

###### keywords:

electronic transport, transmission, heterostructures, spintronics, model
studies

Andreas Weh* Wilhelm H. Appelt Andreas Östlin Liviu Chioncel Ulrich Eckern

A. Weh, W. H. Appelt, U. Eckern
Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
Email Address<EMAIL_ADDRESS>
A. Östlin, L. Chioncel
Center for Electronic Correlations and Magnetism & Augsburg Center for
Innovative Technologies, Institute of Physics, University of Augsburg, 86135
Augsburg, Germany

##  1 Introduction

The transport properties of inhomogeneous electronic systems, including
charge, spin and heat flow, have been intensely studied for several years [1,
2, 3], in particular, in view of the enormous application potential of such
devices. As a specific example, we wish to mention experimental investigations
of the tunneling magnetoresistance effect in Fe/MgO/Fe magnetic tunnel
junctions, which were created by depositing MgO epitaxially between Fe
electrodes [4]. The quantum transport properties of such structures can be
described employing model Hamiltonians [5], or density functional theory (DFT)
[6].

By both these approaches the spin dependence of tunneling can be explained by
the nature of the Bloch states in the bulk ferromagnetic leads that couple
with evanescent states of the complex band structure of the insulator [7, 8].
Depending on the strength of this coupling, the majority/minority conductance
is expected to be dominant, thus determining the positive/negative spin
polarization of the conductance. The interface bonding between leads and the
insulating region is considered to be responsible for the formation of
interface states [9], which significantly affect the conduction spin
polarization; their existence was experimentally confirmed [10] through spin-
dependent tunneling measurements.

Interfaces between ferromagnetic and non-magnetic metals have also been of
recent interest since they are important for current-perpendicular-to-plane
giant magnetoresistance (CPP-GMR) experiments. Initially the magnetic and the
non-magnetic spacer layers were mainly composed of transition metals like
Co/Ag, as in the pioneering work of Ref. [11]; however, recently more complex
Heusler alloys [12, 13] have also been investigated due to their high spin
polarization.

In the theory of the CPP-GMR [14], spin-asymmetric scattering plays an
important role: the larger the asymmetry, the larger the possible spin
polarization of the current. In order to achieve a high spin asymmetry at the
interface, it was proposed [15] that materials should be employed where the
electronic band structure of the majority spin of the ferromagnetic layer
matches as closely as possible the band structure of the non-magnetic layer.
At the same time, the matching between band structures should be poor for the
minority spin channel.

Current theories of spin-dependent tunneling hardly emphasize electronic
correlations, neither in the leads nor in the scattering region. In this
context, we have studied within the framework of DFT the ballistic conduction
through transition metal heterostructures [16], and more recently Heusler
based systems such as Au/NiMnSb/Au [17]. For these system, we modeled
electronic correlations for the Mn atoms via the Hubbard model, which was
solved within dynamical mean-field theory (DMFT) [18, 19, 20]. We obtained a
significant reduction of the spin polarization in the density of states which
is not apparent in the spin polarization of the conduction electron
transmission, and concluded that the interface states hybridized with the
many-body induced states are localized.

In the present work, we revisit our recent studies using a simplified single-
band model Hamiltonian on the one hand, but a more advanced many-body solver
to treat electronic correlations on the other hand. Within this model, we
discuss the nature of the electronic states in the interacting region which
determine the transmission. We show that a modification of the coupling
between leads and the central region, as well as electronic correlations, may
strongly affect the spectral functions and hence the conductance. Our focus is
on a ferromagnetic metal as the central layer, coupled to non-interacting
normal-metal leads. We show, in particular, that electronic interactions may
enhance the spin polarization of the spectral function in the interacting
region, as well as of the transmission through the entire system.

The paper is organized as follows. In the next section, Section 2, we present
the tight-binding model to be employed in this work. Section 3 is devoted to
our results, and Section 4 to the summary and conclusions.

##  2 Generic transport model

The electronic transport through a device can be conveniently addressed by
applying scattering theory, which was pioneered by Landauer [21, 22] and
Büttiker [23, 24], and worked out in detail by Meir and Wingreen [25]. In this
approach (see, e.g., Ref. [26]), one typically considers a mesoscopic system
(say, a molecule or a quantum dot) which is coupled via ideal leads which act
as charge reservoirs which are so large that they can be described by
equilibrium distributions. Hence the corresponding left ($L$) and right ($R$)
leads are characterized by the equilibrium Fermi distribution functions
$f_{L/R}(\omega)$. For the derivation, one assumes that in the infinite past
the various subsystems (microscopic contact, or central region; and the leads)
are separated and in equilibrium, albeit at their respective chemical
potentials and temperatures. The couplings are then turned on adiabatically,
assuming time-reversal invariance [27, 28]. In the following, we apply the
Meir-Wingreen approach [25] to our heterostructure setup, in which electronic
correlations are considered in the scattering region only, i.e., in the
central layer. Note that this layer, and all others, are macroscopic in the
directions perpendicular to the transport direction.

Recently we have described [16, 17] how to take into account local
interactions when computing the transmission of correlated heterostructures.
In comparison to the previous DFT+DMFT approach, in the present tight-binding
Hamiltonian the computation of the transmission becomes even more transparent.
In particular, using the Meir-Wingreen formalism, we can replace the
scattering region Green’s function directly by its interacting counterpart, in
which electronic correlations are taken into account by the local self-energy
$\Sigma(\omega)$. This latter quantity is computed using the recently
developed fork tensor-product states (FTPS) solver [29]; we used this method
recently in order to describe the spectral properties of heterostructures
containing half-metals [30].

As is well known, the Meir-Wingreen approach [25] can be considerably
simplified for non-interacting systems, or when the coupling matrices are
proportional to each other. We emphasize that the latter is fulfilled for our
setup, as long as the central region consists of just one interacting layer.
On the other hand, for two or more interacting layers, the lesser Green’s
function will be needed.

CLRwhite$t_{L}$white$t_{L}$white$t_{-10}$white$t_{01}$white$t_{R}$white$t_{R}$
Figure 1: Schematic graph of the model setup. The layers are labeled with $l$,
such that $l<0$ corresponds to the left, and $l>0$ to the right lead. The
central layer is $l=0$.

Figure 1 shows the geometry of the system: non-interacting leads, left ($L$)
and right ($R$), separated by the central region ($C$). Both leads consist of
a semi-infinite stack of square-lattice planes. The hopping amplitude between
the layers in the left (right) lead is $t_{L}$ ($t_{R}$), and the on-site
energy is $\epsilon_{L\sigma}$ ($\epsilon_{R\sigma}$); the electron dispersion
within the layers is of Bloch type because of two-dimensional translation
invariance. The transverse (i.e., parallel to the layers) wave vector,
$\mathbf{k}_{\parallel}=(k_{x},k_{y})$, is a good quantum number, and the in-
plane dispersion of the electrons is denoted as
$\varepsilon_{\mathbf{k}_{\parallel}}$. The complete Hamiltonian thus can be
written as $\hat{H}=\hat{H}_{L}+\hat{H}_{C}+\hat{H}_{R}$, where $\hat{H}_{C}$
contains the coupling between leads and central region. To be definite, the
Hamiltonians for the semi-infinite left and right leads read:

$\displaystyle\hat{H}_{L}$
$\displaystyle=\sum_{l<0\mathbf{k}_{\parallel}\sigma}(\epsilon_{L\sigma}+\varepsilon_{\mathbf{k}_{\parallel}})\hat{n}_{l\mathbf{k}_{\parallel}\sigma}-\sum_{l<0\mathbf{k}_{\parallel}\sigma}(t_{L}\hat{c}^{\dagger}_{l-1\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{l\mathbf{k}_{\parallel}\sigma}+t_{L}^{*}\hat{c}^{\dagger}_{l\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{l-1\mathbf{k}_{\parallel}\sigma}),$
(1) $\displaystyle\hat{H}_{R}$
$\displaystyle=\sum_{l>0\mathbf{k}_{\parallel}\sigma}(\epsilon_{R\sigma}+\varepsilon_{\mathbf{k}_{\parallel}})\hat{n}_{l\mathbf{k}_{\parallel}\sigma}-\sum_{l>0\mathbf{k}_{\parallel}\sigma}(t_{R}\hat{c}^{\dagger}_{l\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{l+1\mathbf{k}_{\parallel}\sigma}+t_{R}^{*}\hat{c}^{\dagger}_{l+1\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{l\mathbf{k}_{\parallel}\sigma}),$
(2)

where $\sigma=\pm\frac{1}{2}$ labels the spin. In the following, we consider
the leads to be identical, but keep the notation “$L$” which henceforth refers
to “lead”. For the central region, just one layer, we have:

$\hat{H}_{C}=-\sum_{\mathbf{k}_{\parallel}\sigma}(t_{-10}\hat{c}^{\dagger}_{-1\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{0\mathbf{k}_{\parallel}\sigma}+\mathrm{h.c.})+\sum_{\mathbf{k}_{\parallel}\sigma}(\epsilon_{0\sigma}+\varepsilon_{\mathbf{k}_{\parallel}})\hat{n}_{0\mathbf{k}_{\parallel}\sigma}+U_{0}\sum_{i}\hat{n}_{0i\downarrow}\hat{n}_{0i\uparrow}-\sum_{\mathbf{k}_{\parallel}\sigma}(t_{01}\hat{c}^{\dagger}_{0\mathbf{k}_{\parallel}\sigma}\hat{c}^{\vphantom{\dagger}}_{1\mathbf{k}_{\parallel}\sigma}+\mathrm{h.c.})$
(3)

The amplitudes describing the hybridization between the central region and the
leads, $t_{-10}$ and $t_{01}$, are chosen to be real and positive, and equal
to each other. The hopping amplitudes in the transport direction are also
assumed to be real, $t_{L}=t_{L}^{*}$, without loss of generality. In
addition, $\epsilon_{0\sigma}=\epsilon_{0}+\sigma h_{0}$ is the on-site
energy, $h_{0}$ the magnetic splitting, and $U_{0}$ the on-site Hubbard
interaction in the central layer. Since the central region is just a _single_
layer, the effect of the leads is characterized by the _scalar_ level-width
function

$\Gamma^{L}_{\sigma}(\omega,\varepsilon_{\mathbf{k}_{\parallel}})=-2\absolutevalue{t_{01}}^{2}\imaginary
g^{\text{s}}_{L}(\omega^{+}-\epsilon_{L\sigma}-\varepsilon_{\mathbf{k}_{\parallel}}),$
(4)

where $\omega^{+}=\omega+i0^{+}$, with the leads’ surface Green’s function

$g^{\text{s}}_{L}(z)=\frac{z}{2\absolutevalue{t_{L}}^{2}}\left(1-\sqrt{1-{\left(\frac{2\absolutevalue{t_{L}}}{z}\right)}^{2}}\right).$
(5)

Applying the Meir-Wingreen formalism [25] to the present, highly symmetric
heterostructure, we obtain a particularly simple expression for the charge
current perpendicular to the layers:

$J=-\frac{e}{h}N_{\parallel}\sum_{\sigma}\int\mathrm{d}\omega\left[f_{L}(\omega)-f_{R}(\omega)\right]\int\mathrm{d}\varepsilon_{\parallel}\rho_{\parallel}(\varepsilon_{\parallel})\Gamma^{L}_{\sigma}(\omega,\varepsilon_{\parallel})\imaginary
G_{00\sigma}(\omega^{+},\varepsilon_{\parallel}),$ (6)

where $G_{00\sigma}$ is the Green’s function of the central region,
$G_{00\sigma}(z,\varepsilon_{\mathbf{k}_{\parallel}})=\expectationvalue*{\\!\\!\innerproduct*{\hat{c}^{\vphantom{\dagger}}_{0\mathbf{k}_{\parallel}\sigma}}{\hat{c}^{\dagger}_{0\mathbf{k}_{\parallel}\sigma}}\\!}\\!(z)$,
and $N_{\parallel}$ the number of sites within a layer perpendicular to the
transport direction. The layer density of states is given by

$\rho_{\parallel}(\varepsilon_{\parallel})=\frac{1}{N_{\parallel}}\sum_{\mathbf{k}_{\parallel}}\delta(\varepsilon_{\parallel}-\varepsilon_{\mathbf{k}_{\parallel}})=\frac{2}{\pi^{2}D}K(1-{\varepsilon_{\parallel}^{2}}/{D^{2}}),$
(7)

where the last equality is valid for
$\absolutevalue{\varepsilon_{\parallel}}\leq D$. For the square lattice [31],
it can be written in terms of the complete elliptic integral of the first
kind:

$K(m)=\int_{0}^{\pi/2}\mathrm{d}t{[1-m\sin^{2}(t)]}^{-1/2}.$ (8)

From the above quantities, the normalized transmission for the spin channel
$\sigma$ can be computed as follows:

$\displaystyle T_{\sigma}(\omega)$
$\displaystyle=-\int\mathrm{d}\varepsilon_{\parallel}\rho_{\parallel}(\varepsilon_{\parallel})\Gamma^{L}_{\sigma}(\omega,\varepsilon_{\parallel})\imaginary
G_{00\sigma}(\omega^{+},\varepsilon_{\parallel})$ (9)
$\displaystyle=2\absolutevalue{t_{01}}^{2}\int\mathrm{d}\varepsilon_{\parallel}\rho_{\parallel}(\varepsilon_{\parallel})\imaginary
g^{\text{s}}_{L}(\omega^{+}-\varepsilon_{\parallel}-\epsilon_{L\sigma})\imaginary
G_{00\sigma}(\omega^{+},\varepsilon_{\parallel}).$

Within dynamical mean-field theory (DMFT), the local self-energy of the
central region, $\Sigma_{\sigma}(z)$, is included in the central region
Green’s function:

$G_{00\sigma}(z,\varepsilon_{\parallel})=\frac{1}{z-\varepsilon_{\parallel}-\epsilon_{0\sigma}-\Sigma_{\sigma}(z)-2\absolutevalue{t_{01}}^{2}g^{\text{s}}_{L}(z-\epsilon_{L\sigma}-\varepsilon_{\parallel})}.$
(10)

In the next section, we discuss the behavior of the spin dependency of the
spectral functions, and contrast it with the spin-dependent transmissions,
when varying the hopping to/from the central region, as well as the strength
of the local interaction on the central layer.

##  3 Results

We consider the setup as shown in Figure 1. As discussed above, our model
consists of non-magnetic (non-spin-polarized) metallic leads in contact with a
single layer of a ferromagnetic metal. The Hamiltonian describing the leads
(at half-filling) is specified by the on-site energies,
$\epsilon_{L\sigma}=0$, and the electrons’ hopping matrix elements $t_{L}$ in
the direction of transport. The latter are fixed at $t_{L}=0.25D$, where $D$
denotes the parallel half-bandwidth. In addition, the square-lattice parallel
hopping matrix elements, $t_{\parallel}$, are assumed to have the same value.
From now on, $D$ ($=4t_{\parallel}$) will be our energy unit, i.e., formally
$D=1$, and $t_{L}=t_{\parallel}=0.25$.

According to Equation 9, the transmission is determined by the product of the
surface spectral functions of the uncoupled leads and that of the central
region. The surface Green’s function is given above, Equation 5. The
corresponding spectral function

$A_{L}^{s}(\omega-\varepsilon_{\parallel})=-\frac{1}{\pi}\imaginary
g^{\text{s}}(\omega^{+}-\varepsilon_{\parallel})$ (11)

has a semi-circular shape and vanishes at the band edges, $\pm
2\absolutevalue{t_{L}}=\pm 0.5$. The lead spectral function $A_{L}^{s}$ does
not depend on the parameters of the central layer, hence our focus is on the
spectral function of the central region.

First, we present our results for the non-interacting case, i.e., we discuss
how the spectral function of the central region and the transmission depend on
the parameters on-site energy ($\epsilon_{0}$) and coupling ($t_{01}$), cf.
Figures 2, 3 and 4. In the second part of this section, we discuss the
modifications induced by a local interaction ($U_{0}$) within the central
layer: We vary $t_{01}$ for fixed $U_{0}$, see Figure 5, as well as $U_{0}$
for fixed $t_{01}$, see Figure 6.

###  3.1 Non-interacting central layer

In the non-interacting case, the Green’s function depends only on the
difference between frequency and dispersion,
$G_{00\sigma}(\omega,\varepsilon_{\mathbf{k}_{\parallel}})=G_{00\sigma}(\omega-\varepsilon_{\mathbf{k}_{\parallel}})$.
The corresponding spectral function is

$A_{00\sigma}(\omega-\varepsilon_{\parallel})=-\frac{1}{\pi}\imaginary\frac{1}{\omega^{+}-\varepsilon_{\parallel}-\epsilon_{0\sigma}-2\absolutevalue{t_{01}}^{2}g^{s}_{L}(\omega^{+}-\varepsilon_{\parallel})}.$
(12)

In addition, the spin directions are decoupled. We define $\xi_{\sigma}$ as
the (generally complex) root of the denominator of the r.h.s. of Equation 12,
i.e.,

$\xi_{\sigma}-\epsilon_{0\sigma}-2\absolutevalue{t_{01}}^{2}g^{s}_{L}(\xi_{\sigma})=0,$
(13)

which leads to

$(1-2r)\xi^{2}_{\sigma}-2(1-r)\epsilon_{0\sigma}\xi_{\sigma}+\epsilon^{2}_{0\sigma}+4r^{2}\absolutevalue{t_{L}}^{2}=0,$
(14)

where $r=\absolutevalue{t_{01}/t_{L}}^{2}$ characterizes the hopping to the
central region relative to the hopping in the leads. First we consider
energies outside of the lead band,
$\absolutevalue{\omega-\varepsilon_{\parallel}}>2\absolutevalue{t_{L}}=0.5$.
Then the lead Green’s function $g^{s}_{L}$ is real, implying that the spectral
function is a sum over delta functions:

$A_{00\sigma}(\omega-\varepsilon_{\parallel})=\sum_{\xi_{\sigma}}\delta(\omega-\varepsilon_{\parallel}-\xi_{\sigma}).$
(15)

The corresponding $\xi_{\sigma}$-solutions are

$\xi^{\pm}_{\sigma}=\frac{(1-r)\epsilon_{0\sigma}\pm
r\sqrt{\epsilon_{0\sigma}^{2}-4(1-2r)\absolutevalue{t_{L}}^{2}}}{1-2r}.$ (16)

Note that this expression also contains spurious solutions belonging to the
unphysical branch of the square root in $g^{s}_{L}(z)$. On the other hand,
inside the lead band,
$\absolutevalue{\omega-\varepsilon_{\parallel}}<2\absolutevalue{t_{L}}=0.5$,
the denominator has an imaginary part, yielding the spectral function

$A_{00\sigma}=\frac{1}{\pi}\frac{r(\omega-\varepsilon_{\parallel})\sqrt{{\left(\frac{2\absolutevalue{t_{L}}}{\omega-\varepsilon_{\parallel}}\right)}^{2}-1}}{(1-2r){(\omega-\varepsilon_{\parallel})}^{2}-2(1-r)(\omega-\varepsilon_{\parallel})\epsilon_{0\sigma}+\epsilon_{0\sigma}^{2}+4r^{2}\absolutevalue{t_{L}}^{2}}.$
(17)

Hence within the band we do not get any divergences. The real part of the
roots, Equation 16, indicates resonances of increased amplitude in the
spectrum.

The results are summarized in Figure 2 which shows the spectral function,
Equation 12, as a function of the energy and the coupling, $t_{01}$. Note that
in this representation the spectral function does not depend on the lattice
structure of the layers, as the horizontal axis refers to
$\omega-\varepsilon_{\parallel}$.

Figure 2: Dependence of the spectral function of the central region,
$A_{00\sigma}(\omega,\varepsilon_{\parallel})$, on $t_{01}$ for a selection of
$\epsilon_{0}$. The parameters are $\epsilon_{L\sigma}=0$, $t_{L}=0.25$, and
$U_{0}=0$. For a non-interacting system, the spectral function is a function
of $\omega-\varepsilon_{\parallel}$ only:
$A_{00\sigma}(\omega,\varepsilon_{\parallel})=A_{00\sigma}(\omega-\varepsilon_{\parallel})$.
The white line indicates the position of the delta peak outside the band,
$\absolutevalue{\omega-\varepsilon_{\parallel}}>2t_{L}$, while the red-dotted
line is determined by the real-part of $\xi_{\sigma}$ within the band
$\absolutevalue{\omega-\varepsilon_{\parallel}}<2t_{L}$,
$\omega-\varepsilon_{\parallel}=\real\xi_{\sigma}$, resulting in an enhanced
spectral intensity; cf. Equation 16.

The representative feature of the spectral function is a continuum band in the
energy range of $[-2t_{L},+2t_{L}]=[-0.5,+0.5]$, which corresponds to the band
of the leads. The left graph in the figure ($\epsilon_{0}=0$) includes the
homogeneous “bulk” case, namely the setup in which $t_{01}=t_{L}$ ($=0.25$),
i.e., all hopping parameters as well as the on-site energies in every layer
are the same, $\epsilon_{0}=\epsilon_{L}$ ($=0$). The homogeneous case,
$\epsilon_{0}=0$, $t_{01}=0.25$, is the point where the white lines touch the
continuum.

In addition to the continuum, the spectral function displays a set of up to
two high-intensity lines. The existence of these states entails an enhancement
of the spectral function, and corresponds to bound and resonance states
generated by the coupling of the semi-infinite leads and the central region.
We note that a similar distinction between bound and resonance states can be
made within the single impurity Anderson model [32, 33]. The analysis of bound
versus resonance states can also be based on the assessment of the poles of
the spectral function, Equation 12.

For larger values of the coupling,
$\absolutevalue{t_{01}}^{2}>\absolutevalue{t_{L}}^{2}/2-\epsilon_{0}^{2}/8$,
bound states are located outside the continuum (white line). Since the
transmission is determined by the overlap of the spectral functions of the
leads with that of the central region, the bound states outside the continuum
do not contribute to transmission. With decreasing $t_{01}$ values, the high-
intensity states approach the continuum, and depending on the on-site energy
$\epsilon_{0}$ they may enter the continuum region. This leads to an
enhancement of the transmission as is apparent in Figure 3, where we show the
transmission for a given on-site energy of $\epsilon_{0}=0.25$ in the central
square-lattice layer. Up to $t_{01}\approx 0.1$, the maximum of the
transmission is given by the position of the resonance, which is given by the
black line with red dots which is
$T_{\sigma}(\omega=\real\xi_{\sigma}(t_{01}),t_{01})$ as a function of
$t_{01}$.

Figure 3: Transmission for various values of $t_{01}$ for $\epsilon_{0}=0.25$,
corresponding to the central plot of Figure 2. The black line with red dots is
the transmission at the position of the resonance (red dotted line in Figure
2). For small coupling, up to $t_{01}\approx 0.1$, the maximum of transmission
corresponds to the position of the resonance.

Based on these findings, we can discuss the model parameters for which the
investigated setup acts as an efficient spin filter. Clearly, choosing an on-
site energy $\epsilon_{0\sigma}$ within the continuum for one spin channel and
outside the continuum for the other, a high spin polarization of the
transmission can be achieved due to the resonant states only present in one
spin channel. As an example, we consider the parameters $\epsilon_{0}={0.5}$
and $h_{0}={0.5}$ ($\epsilon_{0\uparrow}=0.25,\epsilon_{0\downarrow}=0.75$).
In passing, we note that these are the same values as investigated recently
[30] for interacting bilayers. Hence, in spite of a finite spectral weight at
the Fermi level for the down-spin, for very small values of $t_{01}$ we obtain
a complete spin polarization in transmission, albeit with a small magnitude.
Thus, mediated by the resonant state an enhancement of the transmission spin
polarization is found. This finding is evident from Figure 4 which shows the
$k_{\parallel}$-resolved spectral function for the lead and the central region
at $\omega=0$ for a square lattice. The left (right) graph corresponds to
$t_{01}=0.05$ ($t_{01}=0.25$), while the upper (lower) part of each graph
shows the spectral function of the up-spins (down-spins), and the left (right)
part of each graph displays the lead (central region) spectral function,
respectively. The Brillouin zone (BZ) extends from $[-\pi,\pi]$; the plots are
taken in one of the four (identical) quadrants of the BZ.

The surface spectral function of the leads,
$A^{s}_{L}(\omega^{+}-\varepsilon_{\mathbf{k}_{\parallel}})$ (left half of
each graph), is identical for up- and down-spin since the leads are non-
magnetic. As the lead spectral function is calculated for the case where the
lead is decoupled from the central layer, it is independent of $t_{01}$. The
right half of each graph in Figure 4 shows the momentum resolved spectral
function in the central region, which is calculated in the presence of the
leads and thus changes with $t_{01}$.

For $t_{01}=0.05$, the maximum intensity of the spectral function is located
within the continuum for $\sigma=\;\uparrow$, respectively outside the
continuum for the $\sigma=\;\downarrow$ electrons. In spite of a significant
spectral weight in the local Green’s function, the transmission for the down-
spin almost vanishes, as the spectral weight comes from the bound states which
do not contribute to the current. Thus, we obtain a high polarization over a
large frequency range.

Increasing the coupling strength to $t_{01}=0.25$, we find that the spectrum
of the central region significantly changes. The high-intensity states of the
spin-up electrons are shifted out of the continuum, becoming sharp delta
peaks. Similarly, the sharp states (white line) in the down-spin channel are
repelled by the continuum and shift towards the edge of the BZ. Now for both
spin channels only the continuous spectrum contributes, which is of similar
magnitude for both spins.

This analysis shows that a change in the hopping amplitude between leads and
central region, $t_{01}$, significantly affects the central region spectral
function and consequently modifies the transmission qualitatively, beyond a
mere change of the prefactor in Equation 9. In real materials such a situation
is likely to happen as electronic states are significantly influenced by
structural reconstructions at the surfaces and the chemical bonding.

Figure 4: Spectral functions of the non-interacting setup at frequency
$\omega=0$. The left-hand-side, $k_{x}\in[-\pi,0)$, of each graph shows the
surface spectral function of the uncoupled lead,
$A_{L}^{s}(0-\varepsilon({k_{x},k_{y}}))$, and the right-hand-side,
$k_{x}\in(0,+\pi]$, shows the spectral function of the central region,
$A_{00\sigma}(0,\varepsilon({k_{x},k_{y}}))$, respectively. The top half,
$k_{y}\in(0,+\pi]$, and the bottom half, $k_{y}\in[-\pi,0)$, parts correspond
to the respective spin directions, as indicated. Left graph: weak coupling,
$t_{01}=0.05$; right graph: strong coupling, $t_{01}=0.25$.

###  3.2 Local electronic interaction in the central region

We model the local electronic interaction in the central layer employing
dynamical mean-field theory (DMFT) [18, 19, 20]. As was shown a while ago
[18], the Hubbard model simplifies considerably in the limit of infinite
spatial dimensions. However, DMFT provides a reliable (and non-trivial)
approximation also for two and three dimensions for a large range of model
parameters. Within this approach, the Hubbard model is self-consistently
mapped onto the single-impurity Anderson model (SIAM), thereby allowing the
use of various methods that are available for impurity problems [19, 34].

In combination with materials-specific input, we have previously applied the
DFT+DMFT [34] technique to heterostructures, more recently using supercells
[16, 17]. While these studies were based on a perturbative impurity solver, in
the present work we employ the recently developed FTPS solver [29], which is
non-perturbative and allows to accurately compute spectral functions [30]. We
emphasize that this solver works at zero temperature, and that there is no
need to perform an analytic continuation of the spectral function which is
otherwise a difficult technical issue [35, 36]. The hybridization function of
the SIAM is discretized using a large number of bath sites, in our case $249$
sites per spin. We calculate the ground state of the finite-size impurity
problem using the density matrix renormalization group (DMRG) [37, 38].
Subsequently, we perform the time evolution using the time-dependent
variational principle (TDVP) [39, 40, 41, 42] to obtain the local Green’s
function. For the time evolution, we choose time-steps $\Delta t=0.1$, and a
maximal time of $t=250$.

Presently we perform the scheme outlined above for the setup described in
Section 2, including the Hubbard term $U_{0}$ in the central region, see
Equation 3. Again, we consider the Hamiltonian with a square lattice of half-
bandwidth $D=4t_{\parallel}=1$ in-plane, non-magnetic half-filled leads,
$\epsilon_{L\sigma}=0$, with hopping $t_{L}=0.25$; in the central layer, we
assume $\epsilon_{0}=0.5$ and $h_{0}=0.5$.

Figure 5: Dependence of the spectral function of the central region,
$A_{00\sigma}(\omega,\varepsilon_{\parallel})$, on $t_{01}$. The parameters
are: $\epsilon_{L\sigma}=0$, $t_{L}=0.25$, $D=1$, $\epsilon_{0}=0.5$,
$h_{0}=0.5$, $U_{0}=2$. The red dotted line corresponds to the real-part of
equation Equation 16 within the band of the leads for the _non-interacting_
system, $U_{0}=0$.

The spectral function of the central region is given by the imaginary part of
Equation 10 on the real axis, $z=\omega^{+}$. This requires the knowledge of
the many-body self-energy for the various coupling strengths $t_{01}$ for a
fixed interaction strength, here $U_{0}=2$. We calculate the self-energy for
steps of $\Delta t_{01}=0.05$ and linearly interpolate in-between to obtain a
continuous function. Figure 5 shows the spectral function. We contrast this
with the non-interacting case by plotting the delta peak outside the band
(white line) and the resonance in the band (red dotted line) for $U_{0}=0$,
which are given by the real-part of Equation 16. We observe that the resonance
in the up-spin is hardly affected by the interaction. In the regime of small
$t_{01}$ the down-spin is almost depleted, consequently there are only small
interaction effects for the up-spin. The bound states, however, change in
slope compared to the non-interacting case. For the down-spin we observe a
shift of the bound-states to higher frequencies $\omega$, and a considerable
broadening.

Figure 6: Polarization of the transmission (top row) and local spectral
function (bottom row) for an interacting central layer with varying on-site
interaction $U_{0}$. Left column: small coupling, $t_{01}=0.05$; right column:
intermediate coupling, $t_{01}=0.25$. The spectral function for the down-spin
varies strongly with $U_{0}$ at the Fermi level $\omega=0$, the polarization
of transmission, on the other hand, varies strongly only for $t_{01}=0.25$.

Figure 6 shows the local spectral functions of the central region,
$A_{00\sigma}(\omega)$ (bottom), as well as the transmission polarization
(top) for weak, $t_{01}=0.05$ (left column), and strong, $t_{01}=0.25$ (right
column), coupling between leads and central region, for various interaction
strengths $U_{0}$. The local Coulomb interaction in the central region
increases the polarization of the spectral function, since the minority spin
states are shifted towards higher energies. For $U_{0}\geq 2$ the decoupled
central region $t_{01}=0$ is fully polarized, which means that it is a half-
metallic ferromagnet [30]. While for larger values of $t_{01}$ the leads
induce states for minority electrons in the central region, the spectral
functions remain strongly polarized. For both couplings, $t_{01}={0.05}$ and
$t_{01}=0.25$, the majority spectral function remains largely unchanged in the
presence of the interaction. This is a consequence of the almost depleted
spin-down channel. Away from the Fermi energy tails of the spectral functions
are formed, which are contributing to the high-energy satellite discussed
previously [30].

The polarization of the transmission, however, shows a behavior different from
that of the local spectral function. For $t_{01}=0.05$, in spite of the
significant change of the minority spectral function at the Fermi energy
($\omega=0$), the polarization remains above $95\%$ for all values of $U_{0}$,
and changes only by a few percent. In this regime, the transmission of the
majority spin is dominated by the resonance, while the spectral weight in the
minority spin derives from the bound state which does not contribute to the
current. For $t_{01}=0.25$, on the other hand, the polarization of
transmission follows the interaction-induced change of the local spectral
function. It increases from $\approx 50\%$ at $U_{0}=0$ to $\approx 70\%$ at
$U_{0}=4$.

##  4 Conclusion

In the present work we study a one-band generic model to discuss the physics
of transmission through metallic heterostructures. This model consists of two
non-interacting leads sandwiching a central region that can be subject to
local Coulomb interactions. The spin-dependent transmission is computed within
the Meir-Wingreen formalism. The left and right leads in our model are assumed
to be identical, and the central region consists only of a single layer,
therefore the transmission decomposes into a product of the spectral function
of the central region and the surface spectral function of the uncoupled
leads. Independent of the presence of the local electronic interaction, small
variations of the on-site energies and the hopping amplitude between leads and
central region may strongly affect the shape of the spectral functions in the
central region, and hence the conductance.

We identify bound and resonant states that may appear in this model, depending
on the coupling strength between the leads and the central region. Generally
speaking, resonance phenomena reveal themselves in the electronic conduction
of mesoscopic condensed matter systems, however, a direct characterization of
the resonance energy and the line width remains non-trivial. Bound and
resonant states correspond to poles of the scattering matrix ($S$-matrix),
which relates the initial to the final state of a physical system undergoing a
scattering process [43]. For a bound state the binding energy is directly
given by a real eigenvalue, while the resonance energy and the line width are
obtained from the complex eigenvalues. By analyzing the pole structure of the
Green’s function corresponding to the embedded central region, we identify the
bound and the resonant states for the present setup. As a result, for a large
coupling between the leads and the central region bound states are formed
outside the continuum spectrum of the leads. These states do not contribute to
the transmission, thus we expect them to be localized. In contrast, for the
weak-coupling resonant state, the complex poles of the Green’s function entail
an enhancement of the spin polarization of the transmission.

Dynamical mean-field theory is used to solve the interacting problem in the
central region. The recently developed FTPS solver provides accurate results
for the spectral functions and for the position of the bound and resonant
states. For a certain set of parameters, electronic interactions lead to an
enhanced spin polarization of the spectral function, as the minority electrons
are shifted away from the Fermi level. As a consequence, a reduction of
electronic correlations for the majority spins is found. For the bound states
outside the continuum of the leads’ spectral function, electronic correlations
lead to a significant broadening.

Finally, we wish to emphasize that the results of the present study may
significantly advance computational endeavors directed at the transport
properties of (generally interacting) electronic systems coupled to leads. As
discussed above, for a central region sandwiched between non-interacting
leads, the effect of local electronic correlations on the resonant and bound
states in the central region can be studied by including the self-energy
produced by the real-space DMFT into the Meir-Wingreen approach. In addition,
the relevance of the coupling between the leads and the central region, and
the relation between bound and resonant states and the continuum beyond the
non-interacting picture, certainly requires further investigations, as well as
its dependence on model parameters. A particular interesting question concerns
the nature of the quasiparticle states in the central region. Furthermore,
departures from the Fermi liquid description could be relevant [30]. The
current approach also can be extended to multi-orbitals models, which are
predestined to generate further challenges at the numerical level, as well as
to nonlinear (e.g., finite voltage) transport properties [44, 45].

Acknowledgments

Financial support by the Deutsche Forschungsgemeinschaft (project number
107745057, TRR 80) is gratefully acknowledged.

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