11institutetext: Indian Institute of Technology (IIT), Varanasi (BHU)
22institutetext: National Institute of Technology (NIT), Tiruchirappalli,
India.

# Brain Image Segmentation of 2D Images using U-Net

Angad R.S. Bajwa 1122

###### Abstract

Brain tumour segmentation is an essential task in medical image processing.
Early diagnosis of brain tumours plays a crucial role in improving treatment
possibilities and increases the survival rate of the patients. Manual
segmentation of the brain tumours for cancer diagnosis, from large number of
MRI images, is both a difficult and time-consuming task. There is a need for
automatic brain tumour image segmentation. The purpose of this project is to
provide an automatic brain tumour segmentation method of MRI images to help
locate the tumour accurately and quickly.

###### Keywords:

image segmentation brain tumor classification u-net deep machine learning

##  1 Introduction

Tumors may start in the brain, or cancer elsewhere in the body may spread to
the brain. Four standard MRI modalities used in brain image study are
T1-weighted MRI (T1), T2-weighted MRI (T2), T1-weighted MRI with gadolinium
contrast enhancement (T1-Gd), and Fluid Attenuated Inversion Recovery
(FLAIR).Generally, T1 images are used for distinguishing healthy tissues, and
in this project, we use T1 brain images for our study.

Often automatic brain tumor segmentation methods use hand-crafted features
such as edges, corners, histogram of gradient, local binary pattern. These
features are extracted and then given to the classifier. The training
procedure of the classifier is not affected by the nature of those features.
Semantic segmentation is commonly used in medical imaging to identify the
precise location and shape of structures in the body and is essential to the
proper assessment of medical disorders and their treatment. Recently, deep
convolutional neural networks (CNNs) have led to drastic improvements in
numerous computer vision tasks.[1]

In this project, we try to predict the brain tumor location using one such CNN
architecture- U Net. Open source code for the project is available (link in
Acknowledgement section). We study three types of U-Net architectures. One is
the simple U-Net, the second is a U-Net with additional skip connections, and
lastly, an M- Net, a new unique approach to semantic segmentation. All
previous works on this dataset are for the classification of tumor types—none
of the previous works performed on this dataset are intended for segmentation
purposes.

Figure 1: Four different MRI modalities showing a high grade glioma, each
enhancing different subregions of the tumor. From left; T1, T1-Gd, T2, and
FLAIR [2]

##  2 Dataset and Evaluation Metrics

This brain tumor T1-Weighted MRI image-dataset consists of 3064 slices. There
are 1047 coronal images. Coronal images are those which are captured from the
back of the head. Axial images, those taken from above the skull, are 990 in
number. The dataset also contains 1027 sagittal images that are captured from
the side of the skull. This dataset has a label for each image, identifying
the type of the tumor. These 3064 images belong to 233 patients. The size of
each image is 512*512 pixels. The dataset includes three types of tumors- 708
Meningiomas, 1426 Gliomas, and 930 Pituitary tumors, which are publicly
available:  here

We evaluate the segmentation results using the Dice coefficient. The Dice
coefficient, also called the overlap index, is a metric used for validation in
medical image segmentation. The pair-wise overlap of the segmentations is
calculated by:

$\centering Dicecoefficient=\frac{2\cdot TP}{2\cdot TP+FP+FN}\@add@centering$
(1)

where TP is true positive results or correctly segmented tumour pixels, FP is
false positive, and FN is the false negative results of the segmentation.
False positive results are obtained when a pixel which is non-tumorous is
classified as tumorous. Also, FN refer to the number of pixels that are
tumorous and are falsely labelled as non-tumorous.[7]

$\centering Diceloss=1.0-Dicecoefficient\@add@centering$ (2)

$\centering
BCEloss=\frac{-1}{outputsize}\sum\limits_{i=1}^{outputsize}{y_{i}log(y^{*}_{i})+(1-y_{i})log(1-y^{*}_{i})}\@add@centering$
(3)

Where y is the target value (in our case 1 or 0, tumor or non-tumor pixel) and
y* is the predicted value for that pixel. Output size is the image size in our
512*512. We use a loss function which is a sum of both the binary cross
entropy and dice loss.

Figure 2: Brain MRI slices captured from different directions [6]

##  3 Deep Architectures

Figure 3: U-Net architecture [3]

####  3.0.1 U-Net:

U-net architecture was introduced in the paper” U-Net: Convolutional Networks
for Bio-medical Image Segmentation” in March 2015. It was developed by Olaf
Ronneberger for Bio-Medical Image Segmentation.[3] The U-net architecture
contains two paths. The first path is the contraction path (encoder part)
which is used to capture the context of the image. The encoder is a stack of
convolutional and max pooling layers. The second path is an expanding path
(decoder part) to enable precise localization using transposed convolutions.
It is an end to end fully convolutional network which only contains
convolutional layers and no Dense layers because of which such architecture
cannot accept images of any size. Two consecutive convolutions layers are
applied before max-pooling. This is done as when pooling is applied, lot of
information is lost as the dimensions of the data are halved. So, convolution
layers are stacked before each pooling, so it can build up better
representations of the data without quickly losing all the spatial
information.[4]

####  3.0.2 U Net with skip(both short and long) connections:

To the U-Net architecture I have added skip connections between the first
convolution layer and the max pooling layer on each level. The main purpose of
this was to increase accuracy and uniform distribution of parameters in the
layers.

####  3.0.3 M-Net:

M-net mainly has four pathways of 2D filters: two main encoding and decoding
paths and two side paths, which give our architecture deep supervision. Each
pathway has four steps. In the encoding path, each step has a cascade of 2D
convolution filters of size 3x3 and maxpooling by 2x2, reducing the input size
by half and allowing the network to learn contextual information. In the
cascade of convolution filters, skip connection is introduced to enable the
network to learn better features. The decoding layer is identical to encoding
layers with one exception: maxpooling is replaced by an upsampling layer to
double the input size and recover an output image of the original size.
Similarly, skip connections are also implemented between corresponding
encoding and decoding layers to ensure that the network has sufficient
information to derive fine grain labeling of an image without post-processing.
The left leg operates on 256*256 with four maxpooling layers of size 2x2, and
the outputs are given as input to the corresponding encoding layers. The right
leg upsamples the output of each of the decoding layers to the original size
of 256*256. Finally, the decoding layer and the right leg output is processed
by a 1x1 convolution layer with L channels, where L is the number of
structures of interest, including background. The advantage of the M-net is
that barring one 3D convolution filter, all other filters are 2D filters which
allows end-to-end training of the network with considerably low memory
requirement ( 5GB).[5]

Figure 4: M-Net architecture [5] Table 1: Performance measure of models | Model
---
Epochs | Loss | Dice Score
U-Net | 50 | 0.0577 | 67.28
U-Net with additional skip connection | 50 | 0.0537 | 61.02
M-Net | 50 | 0.0765 | 64.55

###  3.1 Training

We split the data into training, validation and test sets in the ratio of
80:10:10. Thereby having 2451 training images, 307 validation images and 306
test images. We implement our method using Keras with a Tensorflow backend
using ADAM optimizer (learning rate=0.0001). The model is trained for 50
epochs.

Due to limited RAM and GPU we split the training into 2 compartments. The
first with 30 epochs, saving the model and then training further for the next
20 epochs.Once the model is trained, we evaluate it on the test set.

##  4 Results:

We use Google Colab to implement the model. Colab uses Intel(R) Xeon(R) CPU @
2.20GHz, 12 GB of RAM and NVIDIA Tesla K80 TPU.

On comparing the three models, we find that the simple U- Net performs the
best with a dice score of 67.2.

Figure 5: U-Net: prediction vs ground truth for unseen data

##  5 Conclusion

We experimented with three different U-Net architectures and tried two loss
functions to derive the best results. The simple U- Net performs the best on
the given dataset. We also introduced skip connections between convolution
filters and deep supervision functionality in our network, which allows it to
learn better features. We have come up with a memory-efficient way of
detecting tumors that can reduce both the time and difficulty faced in
clinics. Early discovery can prevent patients from entering severe stages of
the tumor.

Figure 6: U-Net + additional skip connection: prediction vs ground truth for
unseen data Figure 7: M-Net: prediction vs round truth for unseen data

##  6 Acknowledgement

This work was done under the supervision of Professor Rajeev Srivastava during
my research internship with his group at Indian Institute of Technology
(Varanasi) in 2020. I am grateful for the training and mentoring I received
from Prof Srivastava during the project. Open source code for the project is
available at https://github.com/angadbajwa23/Segmentation-of-2D-Brain-MR-
Images-using-Deep-Neural-Architectures.

## References

  * [1] Arnab Kumar Mondal, Jose Dolz and Christian Desrosiers,” Few-shot 3D Multi-modal Medical Image Segmentation using Generative Adversarial Learning”
  * [2] Ali Isin, Cem Direkoglu , Melike Sah ,”Review of MRI-based brain tumor image segmentation using deep learning methods”
  * [3] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation,2015.
  * [4] Malav Bateriwala and Pierrick Bourgeat,” Enforcing temporal consistency in Deep Learning segmentation of brain MR images”
  * [5] Raghav Mehta and Jayanthi Sivaswamy, “M-NET: A CONVOLUTIONAL NEURAL NETWORK FOR DEEP BRAIN STRUCTURE SEGMENTATION”
  * [6] Zahra Sobhaninia, Safiyeh Rezaei, Alireza Noroozi, Mehdi Ahmadi, Hamidreza Zarrabi, Nader Karimi, Ali Emami,Shadrokh Samavi , “Brain Tumor Segmentation Using Deep Learning by Type Specific Sorting of Images”
  * [7] Adrian V. Dalca, Evan Yu, Polina Golland, Bruce Fischl,Mert R. Sabuncu, and Juan Eugenio Iglesias,” Unsupervised Deep Learning for Bayesian Brain MRI Segmentation”
  * [8] Jan Funke , Fabian Tschopp , William Grisaitis , Arlo Sheridan ,Chandan Singh, Stephan Saalfeld , and Srinivas C. Turaga, “Large Scale Image Segmentation with Structured Loss Based Deep Learning for Connectome Reconstruction”
  * [9] https://github.com/zhixuhao/unet
  * [10] https://github.com/Jack-Etheredge/Brain-Tumor-Segmentation-3D-UNet-CNN
  * [11] https://github.com/Issam28/Brain-tumor-segmentation
  * [12] https://github.com/IntelAI/unet
  * [13] https://github.com/polo8214/Brain-tumor-segmentation-using-deep-learning


# Effect of Dispersion and Different Carrier Transitions on Absorption
Characteristics of GaAs

1st Subarna Alam  Department of EEE
Bangladesh University of Engineering and Technology
Dhaka, Bangladesh
<EMAIL_ADDRESS>2nd Mohammad Jahangir Alam  Department of EEE
Bangladesh University of Engineering and Technology
Dhaka, Bangladesh
<EMAIL_ADDRESS>
###### Abstract

One of the basic optoelectronic characteristics is the absorption of any
optoelectronic device or material. We present the characteristic of pure GaAs
and the deviation of ideal spectra. Due to crystal defects, vacancies,
impurities, etc., energy states exist in the semiconductor other than the
allowed bands (conduction and valence bands). In this research, the effect of
transitions between these states on absorption spectra and how the absorption
coefficients change with the density and location of those states are analyzed
numerically. Among the cases of 0.07 $eV$, 0.12 $eV$, and 0.17 $eV$ with the
density of 2%, 5%, 10%, 20%, and 30%, the minimum deviation occurs for the
states located 0.07 $eV$ below the conduction band and when the energy states
have the minimum density of 2% and the deviation increases for the cases of
0.12 $eV$ and 0.17 $eV$ with higher densities.

###### Keywords:

Optoelectronic characteristics, Optoelectronic device, absorption coefficient,
Energy states

##  I Introduction

Over the past decades, researchers have been focusing more on studying
compound semiconductors, especially GaAs and other III-V materials, as the
demand is increasing rapidly [1]. GaAs provides many superior properties to
those of Si. Because of being direct band gap material, GaAs can absorb and
emit light more efficiently. Moreover, this material is less susceptible to
radiation damage because of its wider bandgap [2]. Many annealing techniques
have been applied to minimize the damage caused by electron and proton
radiation [3]. Another of its many uses is in producing laser diodes [4].

(a)

(b)

Figure 1: Band diagram and absorption in (a) pure direct band gap material,
(b) direct band gap material with energy states exist at $E_{i}$

In any ideal semiconductor, there is no energy state in the forbidden gap.
Fig. 1a illustrates the energy band diagram and absorption for a pure direct
band gap material. But in practical semiconductors, there are impurities,
defects, vacancies, etc. As a result, there exist forbidden energy states.
Fig. 1b shows the same diagram if energy states exist at level $E_{i}$. In
this case, transitions between $E_{v}$ and $E_{i}$ will also exist. Any ideal
semiconductor characteristic gets deviated because of those transitions.

In this paper, the absorption spectra of GaAs and different factors that alter
the ideal spectra have been studied. The absorption coefficients for the case
of Fig. 1a and Fig. 1b have been evaluated.

##  II Methods and Experimental Analysis

There are upward (absorption) and downward (emission) transmissions between
energy bands in optical devices or semiconductors. Absorption allows electrons
to transit from a lower energy state to a higher state. In direct band-to-band
absorption, the electrons will transit from the valence band to the conduction
band. Direct band-to-band absorption in intrinsic GaAs is related by Equation
(1) [5].

$\alpha\propto\dfrac{\sqrt{hf-E_{g}}}{nhf}$ (1)

Where $\alpha$ is the absorption coefficient of GaAs, $f$ is the incident
frequency, $E_{g}$ is the band gap of GaAs which is 1.42$eV$ in this analysis,
and $n$ is the refractive index. The refractive index is also an energy-
dependent parameter [6]. The absorption spectra are analyzed again,
considering this dispersion. Finally, the effects of forbidden states as
illustrated in Fig. 1b on absorption are analyzed as a function of forbidden
state density at different levels.

Figure 2: Absorption coefficient vs wavelength considering no dispersion or
keeping refractive index constant

(a)

(b)

Figure 3: (a) Wavelength dependency of Refractive index of GaAs [6, 7] and (b)
Absorption co-efficient vs wavelength considering dispersion (variable
refractive index)

##  III Result and Discussion

Fig. 2 shows the absorption spectrum of intrinsic GaAs with constant
refractive index, i.e., no dispersion. The refractive index is a nonlinear
function of wavelength as per Fig. 3a, and so is the absorption coefficient.
Fig. 3b shows the absorption characteristics of pure GaAs where the refractive
index also changes with the wavelength. No energy below 1.42 $eV$ (bandgap of
GaAs) gets absorbed in pure GaAs. So, no coefficient is achieved beyond 872.45
nm.

(a)

(b)

(c)

Figure 4: Absorption coefficients vs wavelength considering the presence of
energy states (a) 0.07$eV$, (b) 0.12$eV$, and (c) 0.17$eV$, below the
conduction band where ‘sd’ represents the density of those states

Fig. 4 shows the characteristics when energy states exist at different levels
other than the conduction band and the valence band. Curves get deviated in
the presence of those states. Lower energies than 1.42$eV$ are also absorbed
in the presence of those states. Fig. 4a indicates that there is absorption up
to the wavelength of 925 nm. The same wavelengths for the cases of Fig. 4b and
Fig. 4c are 956 nm and 993 nm, respectively. The following two important
observations can be stated from Fig. 4:

  * •

The wider the gap between the conduction band and impurity energy states, the
more the absorption characteristics will get deviated from the ideal ones.

  * •

The higher the density of those energy states, the more the deviation will
occur.

##  IV Conclusion

In this study, absorption characteristics and the various factors that affect
the characteristics have been analyzed. In device processing, 100% of purity
cannot be achieved. So, the ideal characteristics cannot be achieved either.
But the characteristic of available practical materials or devices can be
compared with that of pure materials or devices. Absorption spectra studied
above can be an indicator to measure the energy level of impurities,
vacancies, and defects and the density of those states.

## Acknowledgment

The authors would like to thank Dr. Muhammad Anisuzzaman Talukder, Professor,
Department of EEE, BUET, for his instructions in the optoelectronics lab.

## References

  * [1] C. Zeng, D. Fu, Y. Jin, and Y. Han, “Recent progress in iii–v photodetectors grown on silicon,” in _Photonics_ , vol. 10, no. 5. MDPI, 2023, p. 573.
  * [2] B. Anspaugh and R. Downing, “Radiation effects in silicon and gallium arsenide solar cells using isotropic and normally incident radiation,” Tech. Rep., 1984\.
  * [3] R. Y. Loo, G. S. Kamath, and S. S. Li, “Radiation damage and annealing in gaas solar cells,” _IEEE Transactions on Electron Devices_ , vol. 37, no. 2, pp. 485–497, 1990.
  * [4] R. Hall, G. Fenner, and J. Kingsley, “Tj soltys i ro carlson, coherent light emission from gaas junctions,” _Physical Review Letters_ , vol. 9, p. 366, 1962.
  * [5] P. Bhattacharya, _Semiconductor optoelectronic devices_. Prentice-Hall, Inc., 1997.
  * [6] D. Marple, “Refractive index of gaas,” _Journal of Applied Physics_ , vol. 35, no. 4, pp. 1241–1242, 1964.
  * [7] A. D. Rakić and M. L. Majewski, “Modeling the optical dielectric function of gaas and alas: Extension of adachi’s model,” _Journal of applied physics_ , vol. 80, no. 10, pp. 5909–5914, 1996.


S

<EMAIL_ADDRESS>leave of absence from undergraduate study
at the University of Chicago, Chicago, IL, U.S.A Poudyal McChesney Zhang Qiao
Welp<EMAIL_ADDRESS>
[a]X-Ray Science Division, Advanced Photon Source, Argonne National Laboratory
Lemont, IL, U.S.A [b]Materials Science Division, Advanced Photon Source,
Argonne National Laboratory, Lemont, IL, U.S.A

# A general method for multiresolutional analysis of mesoscale features in
dark-field x-ray microscopy images

I  J  Z  Z  U

###### Abstract

Dark-field x-ray microscopy utilizes Bragg diffraction to collect full-field
x-ray images of ’mesoscale’ structure of ordered materials. Information
regarding the structural heterogeneities and their physical implications is
gleaned through the quantitative analyses of these images. Namely, one must be
able to extract diffraction features that arise from lattice modulations or
inhomogeneities, quantify said features, and identify and track patterns in
the relevant quantitative properties in subsequent images. Due to the
necessity to track features with a wide array of shapes and length scales
while maintaining spatial resolution, wavelet transforms was chosen as a
potent signal analysis tool. In addition to addressing multiple length scales,
this method can be used in conjunction with other signal processing methods
such as image binarization for increased functionality. In this article, we
demonstrate three effective use of wavelet analyses pertraining to DFXM. We
show how to extract and track smooth linear features—which are diffraction
manifestations of twin boundaries—as the sample orientation changes as it is
rotated about momentum transfer. Secondly, we show that even the simplest
wavelet transform, the Haar transform, can be used to capture the primary
features in DFXM images, over a range of length scales in different regions of
interest within a single image enabling localized reconstruction. As a final
application, we extend these techniques to determine when a DFXM image is in
focus

Dark-field x-ray microscopy, which utilizes Bragg diffraction to collect full-
field x-ray images on a crystalline sample, is a powerful tool being developed
for determining the mesoscale structure of crystals. Information regarding the
lattice structure and its physical implications is gleaned through the
quantitative analyses of these images. Namely, one must be able to extract
diffraction features that arise through contrast variations due to lattice
heterogeneities, quantify said features, and track and identify patterns in
the relevant quantitative properties in subsequent images. Due to the
necessity to track features with a wide array of shapes and length scales
while maintaining spatial resolution, we have employed wavelet transforms as a
potent signal analysis tool. In addition to addressing multiple length scales,
this method can be used in conjunction with other signal processing methods
such as image binarization for increased functionality.

##  1 Introduction

Mesoscale defects [crabtree2012quanta] in a lattice structure greatly affect
the physical properties of the associated crystal, such as its tensile
strength, electromagnetic behavior, and emergent properties in neuromorphic
and quantum materials [simons2018long]. Therefore, it is imperative to
identify and understand how defects transform when a material undergoes
various physical effects such as thermal cycling and applied fields.

In this work, we are concerned with the images gathered using dark-field x-ray
microscopy (DFXM), which utilizes Bragg diffraction to identify strain,
misorientation, and other local heterogeneities [simons2015dark,
poulsen2018reciprocal, yildirim2020probing, bucsek2019sub,
dresselhaus2021situ]. By analyzing the contrast variations in the collected
image that arise from these disparities, local variations in the lattice
configuration can be discovered.

There are several possible approaches for acquiring information about
crystalline structures from dark-field x-ray images, all of which require
quantitative extraction, classification, and tracking of features within a
collection of two-dimensional (2D) diffraction images [brennan2022ana]. The
two main approaches are to collect projection images from a single perspective
as a function of time [poudyal2022pump] or external perturbations
[dresselhaus2021situ], or to collect a series of projections [ludwig2001three]
from different perspectives and reconstruct a three-dimensional (3D) image
[flannery1987three].

A collection of diffraction images representing stand-alone two-dimensional
projections of the crystal as it is subject to external stimuli—such as a
change in temperature, application of strain [dupraz20173d], or being subject
to an electromagnetic field [karpov2017three]—is alone a powerful method for
characterizing a material. As the defects in the lattices transformed, it
would be necessary to quantitatively analyze these features to understand the
physical implications of their motion and compare the results to the
theoretical expectation. If necessary, one could reconstruct this collection
of projections into a full three-dimensional model of the defect network in
the sample, from which two-dimensional “slices” of interest can be isolated to
study separately. Observing these slices again requires an algorithm to
quantify the structure and motion of important features (e.g. dislocations,
magnetic or ferroelectric domain walls, charge density wave phases).

In any of the general strategies discussed above, one invariably requires the
quantitative extraction and tracking of features with elevated intensities in
diffraction images. In this paper, we develop a methodology of performing said
analysis. We first codify the required breadth and depth of the methodology
through observation of data and domain-science knowledge. We then explore
potential signal analysis tools that would enable the methodology to complete
the analyses most accurately and efficiently. We ultimately develop these
techniques into a full algorithm, and we apply this algorithm to a set of DFXM
images, in order to analyze performance and determine the next steps to be
taken.

##  2 Experimental Technique

In order to convert the collected DFXM image analysis data into meaningful
scientific information, it is necessary to apply the algorithms with an
understanding of the crystallographic techniques used (e.g. an understanding
of how particular crystal defects would manifest in diffraction images). The
details of DFXM—the motivating experimental technique of this article—and the
images we are concerned with analyzing are discussed in this section.

###  2.1 Dark-field X-ray Microscopy

Figure 1: A generic schematic of DFXM setup. An x-ray beam of a fixed
wavelength illuminates a crystalline sample. A diffracted beam then passes
through an objective lens to form a magnified image on a detector.

DFXM is a full-field imaging technique which creates real-space images on
mesoscale structures in a sample of ordered material [simons2015dark,
poulsen2018reciprocal]. Observing contrast variations caused by local
disparities in the lattice structure enables the capturing of various physical
properties (such as phase, orientation, and strain) of crystal deformities
that are deep below the sample surface which can be directly probed by hard
x-rays. Detectors are typically placed several meters away (in our case, 2.5
m) from the sample, with objective-lens focal length long enough to provide
sufficient working distance for the use of large sample environments in situ.

The inclusion of an x-ray objective lens in the diffracted beam, as shown in
Figure 1, is an essential feature of DFXM. By adjusting the focal length of
the lens, the user can zoom in and out to investigate a wide range of length
scales, thus capturing a broad array of mesoscale features while maintaining a
precise spatial as well as angular resolution. The objective lens also acts as
a filter for stray diffracted beams, thereby isolating elements of interest
within the sample.[yildirim2020probing]

###  2.2 Experimental Details

The DFXM images in Figures 2A and 2B were collected on 33-ID-D at the Advanced
Photon Source using 10 keV x-rays with a bandwidth of 1.3 eV. In this case a
KB mirror pair was used as the condenser lens and an x-ray zone plate as the
objective lens. An Andor Neo sCMOS camera with a pixel size of 6.5 $\mu$m
$\times$ 6.5 $\mu$m was used as the detector in combination with a thin-film
scintillator and a 20x optical lens, yielding a total magnification of 400x.
The images were captured with a viewing angle of $80^{\circ}$ about the
horizontal axis from surface normal, yielding an effective pixel size of 16 nm
$\times$ 33 nm.

The DFXM image in Figure 2C was collected on 6-ID-C at the Advanced Photon
Source using 13 and 20 keV x-rays, repectively. The microscope utilized either
a polymeric (20 keV) or Be (13 keV) refractive lens as a condenser, and a
polymeric (20 keV) refractive lens as an objective, without any harmonic
rejection mirrors. An Andor 5.5MP Zyla sCMOS camera was used with a
scintillator followed by a 5X optical objective lens, yielding a total
magnification of $\sim$80X [qiao2020large].

###  2.3 DFXM Images

Figure 2: A collection of various DFXM images from single-crystal samples.
Images A and B contain a perspective distortion caused by a viewing angle of
$80^{\circ}$ about the horizontal axis from surface normal.
A: DFXM image of twin boundaries in yttrium barium copper oxide (YBCO)
crystal.
B: DFXM image of twin boundaries in YBCO after an $84^{\circ}$ rotation of the
sample relative to Figure A.
C: DFXM image of single crystal strontium titanate (STO).

The features captured in relevant DFXM images take on a diverse array of
shapes, sizes, and regularity—contrast variations range from linear features
with an underlying smooth variation (Figures 2A and 2B), to point-like details
[gonzalez2020methods], to features with nondescript shapes [bucsek2019sub], to
a combination of the three at varying length scales (Figure 2C), all of which
represent different configurations of irregularities and their orientations.
Thus, in order to function properly for the multitude of possible defects and
their network , the algorithm used to analyze DFXM images must be versatile
with respect to the sizes, shapes, locations, and frequency of features.
Furthermore, to identify the boundaries of both arbitrary gross features and
the details contained within them, the signal analysis tool used must be able
to capture smooth and rapid variations of up to three orders of magnitude
apart, encompassing the mesoscale. For example, as depicted in Figure 3, the
tool must capture both the smooth variations along the twin boundaries and the
rapid oscillations between them in order to separate features of interest from
the overall profile of the illuminating X-rays. It also must be capable of
extracting features and oscillations of varying shapes, and must be localized
in space so that the position of said features can be resolved. These
requirements are satisfied by wavelets and the wavelet transform
[walker2008primer].

Figure 3: Near-orthogonal line profiles of Figure 2B. The line profile between
boundaries (red) captures more rapid oscillations, whereas the changes
depicted by the line profile along an image column (green) shows smoother
variations. This demonstrates the need for a versatile signal analysis tool,
one which can extract large- and small-periodicity features.

##  3 Image Processing

While the wavelet transform will be the primary technique featured in this
paper, other tools to be used in conjunction with it are also considered,
specifically image binarization. This section discusses the nature and
application of said tools. As the primary signal analysis technique used in
this project, the applications of the wavelet transform were explored first.

###  3.1 Discrete Wavelet Transform

The wavelet transform is a method of expressing a signal in a basis of
“wavelet” functions, which are localized in space and periodicity. The
wavelets can be scaled and translated to provide multi-resolutional analysis
[heil1989continuous]. In contrast, Fourier Analysis would give only
periodicity without space localization. While this may be sufficient for
images that contain single features, it would not be ideal when desiring to
extract features of different length scales in different regions of an image.
Furthermore, the sinusoidal restriction of Fourier analysis leaves less
versatility when compared to the myriad of possible wavelets available with
the wavelet transform, which complicates analyses for features than may be
particularly well-extracted by a matching wavelet.

Applying a wavelet transform to an image yields two sub-images—one is an
approximation of the original image, identifying smooth variations, while the
other is a detail image, which identifies rapid oscillations or abrupt changes
in features. The wavelet transform can be repeated on the approximation signal
to yield yet another set of sub-images. In developing this methodology, we
sought to first develop an intuition with the simplest images and wavelets and
utilize it to optimize the algorithm, before moving on to more complex data
[shensa1992discrete]. Accordingly, we first analyzed Figure 2A with the Haar
wavelet.

####  3.1.1 The Haar Transform

Having only two values, the Haar wavelet is generally thought to be the
simplest wavelet. Furthermore, due to its shape (essentially a single period
of a square wave), the Haar wavelet excels at detecting and storing sharp
edges within features. Therefore, the horizontal boundaries with clear
distinctions present in Figure 2A would hypothetically be aptly captured by
the Haar transform.

The Haar wavelet is one of few wavelets with simple and intuitive associated
transforms. Adjacent data in a signal are connected in a line. The midpoint of
the line is stored in the approximation signal, and the slope of the line
(scaled by a factor of $-\frac{1}{2}$) is stored in the detail signal. Both
subsignals are then scaled by an additional factor of $\sqrt{2}$, to preserve
the total energy of the signals. Further details on the Haar transform are
discussed in walker2008primer.

Figure 4: Left: A graph of the Haar wavelet in the space domain. Right: An
example of a Haar transform on an inverted parabola, with approximation and
detail signals.

####  3.1.2 Selective Reconstruction

In the process of reconstructing an image using the inverse wavelet transform
on sub-images, undesired detail signals can be omitted by being set to zero.
This is beneficial because it allows the isolation or omission of features of
particular lengths or periodicities. For example, in Figure 2A, there is a
near Gaussian envelope over the image signal, causing central features to be
brighter than those at the edges, which is likely the results of the
illuminating X-ray profile. Furthermore, the camera is subject to small-
periodicity noise across the entire image (caused by experimental
imperfections in background light omission) which complicates the quantitative
extraction of gross features and their properties. Utilizing selective
reconstruction allows the removal of the smooth near Gaussian profile (i.e.
normalizing out the illumination function) and the rapid noise oscillations,
preserving the mid-periodicity features of importance, as shown in Figure 5.
Upon examining the line profiles of the Haar approximations and details at
different levels—with detail level 1 representing changes on the smallest
length scale and detail level 11 representing the largest—levels 7-11 were
shown to encapsulate the details of the broad near Gaussian envelope, while
levels 1-2 contained high-frequency noise that did not significantly change
the contour of the image. Thus, levels 3-6 were determined to be the ideal
detail levels for capturing the mid-frequency features of interest. More
details regarding the specific mathematics of selective reconstruction, as
well as the analysis of appropriate detail levels, are discussed in Appendix
B.1.

Figure 5: Left: YBCO image twin boundaries image (Figure 2A) after selective
reconstruction and renormalization is applied, with a column of interest
highlighted. Right: Plot of original signal, selective reconstruction, and
normalized selective reconstruction of the column. Selective reconstruction
removes the near Gaussian profile, resulting in the features of interest
oscillating about the same mean value.

The wavelet transform excels at bringing the diffraction features of interest
into focus. However, to see how this way of encapsulating features is
beneficial, we must determine our quantitative extraction method. This brings
us to image binarization.

###  3.2 Image Binarization

While the wavelet transform is powerfully suited to the requirements of
diffraction image analysis, it is not the only tool available. One of the most
common techniques utilized for extracting quantitative properties of features
with elevated intensity is image binarization, which can be used in
conjunction with the wavelet transform to yield more useful results.

Binarizing an image is the act of converting each pixel into a boolean
variable, based on whether the value of the pixel satisfies a given condition.
Typically, the condition used to determine binarization is whether a pixel is
above or below a certain threshold. Pixels above the threshold are set to
“on”, while pixels below are set to “off”. Thus, binarization rigorously
defines previously ambiguous shapes and boundaries, thus enabling them to be
quantitatively analyzed. Properties of regions of interest that can be
measured include area, centroid, diameter, and many other geometric
quantities. The algorithm for binarization was adopted from tamraz.

As indicated in Figure 6, when attempting to binarize the original image,
distinctions between high-intensity features are lost if low-intensity
features are captured, and vice versa. However, upon selectively
reconstructing the image, the features of interest are made to oscillate about
a common mean, enabling all of them to be captured without losing
distinctions. Thus, applying selective reconstruction before binarization
improves the accuracy of the feature extraction. Ultimately, while
binarization is used to quantify the features of interest, the chosen wavelet
transform is what truly captures them and brings them into focus.

Figure 6: Left: Profile of column of interest in original signal with
attempted thresholds. Right: Profile of column of interest in selectively
reconstructed signal with potential threshold. Applying selective
reconstruction before binarizing allows low-intensity features to be captured
without sacrificing distinctions between high-energy features.

##  4 Discussion

The image analysis algorithm can be separated into two broad
sections—extracting features within an image, and tracking how features change
between contiguous images. In this section, results for feature extraction are
discussed in-depth, while an outline of a potential feature tracking algorithm
is given.

###  4.1 Feature Extraction

Selective reconstruction with the Haar transform was used to denoise the image
and separate the low-frequency near Gaussian envelope. The feature clustering
algorithm was then used to remove the exterior background of the image. The
image was then slightly refined using a Gaussian filter to smooth out the
peaks and was finally extracted using binarization. Before feature
quantification, a final refinement was implemented by removing low-area
features. The results are shown in Figure 7. The method successfully outlines
gross features within the image and is able to extract important quantitative
properties, as shown in Table 1.

The majority of notable linear features within the image were successfully and
accurately captured by the algorithm. The most significant error is in feature
7, which absorbed the tail of feature 6 and was connected on the right side of
the image with a narrow bridge to what is seemingly a different feature but
was not extracted separately. These errors may be mitigated or resolved with
the use of a more intricate wavelet.

Figure 7: Results of the binarization using selective reconstruction, with the
binary region outlines overlaid on the original image. Most linear features of
note are successfully encapsulated by the outlines in a way that stays true to
their orientation, shape and size.

Feature Number |  Area ($\mu$m2) |  Centroid ($\mu$m) |  Diameter ($\mu$m) |  Eccentricity |  Orientation (degrees from $x$-axis) |  Average Intensity (0-1) |  Integrated Intensity (pixels)
---|---|---|---|---|---|---|---
1 |  70.3 |  (22.8, 9.44) |  15.8 |  0.996 |  -25.2 |  0.0307 |  1420
2 |  49.5 |  (23.4, 23.6) |  16.4 |  0.999 |  -27.8 |  0.0397 |  1290
3 |  91.0 |  (20.3, 34.0) |  25.4 |  0.998 |  -33.5 |  0.0804 |  4820
4 |  137 |  (14.9, 40.5) |  35.7 |  0.999 |  -33.1 |  0.104 |  9340
5 |  166 |  (14.2, 55.0) |  30.0 |  0.998 |  0.348 |  0.191 |  20900
6 |  42.2 |  (13.5, 64.5) |  12.4 |  0.998 |  2.96 |  0.265 |  7350
7 |  249 |  (16.8, 76.0) |  30.4 |  0.995 |  8.18 |  0.431 |  70500
8 |  96.7 |  (16.9, 90.4) |  27.7 |  1.00 |  1.13 |  0.126 |  8000

Table 1: Quantitative properties of extracted features, corrected for the
perspective distortion caused by the viewing angle of $80^{\circ}$ from
surface normal. Each of these properties can be utilized to determine the
size, shape, location, orientation, and brightness of the feature. Tracking
features across frames will allow the investigation of how these features
change over time.

Upon developing the methodology with the simple and intuitive Haar wavelet,
other wavelet transforms were tested to compare performance. Figure 8 shows
the extraction results in a region of interest of a randomly-selected image of
the YBCO sample undergoing rotation for nine different wavelets: Daubechies 1
through 3, coiflet 1 through 3, and symlet 1 through 3. The Daubechies 1
wavelet is equivalent to the Haar wavelet.

Figure 8: Extraction results of randomly-selected YBCO image after selective
reconstruction with Daubechies wavelets 1-3, coiflets 1-3, and symlets 1-3.

The central feature in the images undergoes significant variance in the
extraction, with certain wavelets mistakenly extracting it as two separate
features. Ultimately, Daubechies 2 was thought to provide sufficient
extraction potency without significant computational demand—all subsequent
analyses are done with the Daubechies 2 wavelet transform.

###  4.2 Localization of Reconstruction

The spatial resolution of the wavelet transform allows distinct regions within
images to be selectively reconstructed using different detail levels. This
provides the ability to analyze different length scales in separate parts of
the image, to accord with the sizes of features of interest in different
regions. To demonstrate this capability, Figures 2A and 2B were concatenated
into a single image. This concatenation was then locally reconstructed with
location-dependent detail levels, and the results are shown in Figure 9. The
central concatenated image in the figure shows a small-length-scale
reconstruction of Figure 2A and a larger-length-scale reconstruction of 2B,
while the bottom image shows the inverse.

Figure 9: Top: Concatenation of Figure 2A and 2B. Middle: Selective
reconstruction of concatenated image, with rapid-variation reconstruction on
the left half and smooth-variation reconstruction on the right. Bottom:
Selective reconstruction of concatenated image, with smooth-variation
reconstruction on the left half and rapid-variation reconstruction on the
right.

###  4.3 Feature Tracking

Upon subjecting a sample of ordered material to physical perturbations, images
may be taken in quick succession to construct a DFXM video showing how the
Bragg signature of features changes over time. In this case, features only
undergo minor changes from image-to-image, and thus an extracted feature may
be tracked throughout the process to observe how its physical properties
change over the course of their “lifetime” of visibility. Thus, an algorithm
to track features was developed. Upon extracting features in adjacent images,
the algorithm computes their “feature distance”, a measure of dissimilarity of
their quantitative properties. Each feature is then paired with the feature
most similar to it in the opposite image, provided that their feature distance
falls below a maximum threshold. Unpaired features are deemed to have newly
disappeared or appeared. Diffraction features between images that the
algorithm claims to be linked are outlined in the same color. The results of
this tracking algorithm implemented through a 45∘ rotation of the YBCO sample
are shown in Figure 10.

Figure 10: DFXM images on YBCO twins through a $45^{\circ}$ sweep ($9^{\circ}$
increments) with multicolored outlines to show tracking algorithm results. The
red ovals encapsulate a contrast feature that the tracking algorithm correctly
identifies to be produced by the same real-space mesoscale heterogeneity in
each image, as indicated by the consistent blue highlight around the feature.

The algorithm successfully tracked a feature throughout the sweep, and several
others were successfully tracked throughout broad increments. The algorithm
misidentifies some features as new despite their clear relation to previously
extracted features, which is likely due to an erroneously low feature distance
cut-off between paired features.

###  4.4 Finding Image Focus

While feature extraction in DFXM images motivated this analysis technique, it
does not mark the full extent of its capabilities. Wavelet analysis can be
used to track the sharpness of an image, thereby determining when a DFXM image
is in focus. The technique utilizes the fact that sharper images have more
energy in detail signals of smaller length scales compared to blurred images.
A comparison of detail level energies in a set of artificially blurred DFXM
images on strontium titanate (STO) is shown in Figure 11. The center column of
the figure demonstrates a clear peak in low-level detail energy and a
corresponding valley in high-level detail energy for the focused image.
Furthermore, the left column demonstrates that the proportional change in
energy between the focused and blurred images is most pronounced in the first
detail level. Thus, it was concluded that finding the image which contained
the maximum energy in its first detail level would offer the most robust
distinction between images in and out of focus. Visual depictions of detail
levels for simulated images are shown in Appendix B.3. The Haar transform was
found to provide sufficient analysis for this algorithm.

Figure 11: The left column shows contrast-normalized depictions of the most
“out-of-focus” (artificially blurred) renditions of Figure 2C, with heavy
(top), medium (middle), and light (bottom) blur. The middle column shows plots
of the energy of corresponding Haar decomposition detail levels as a function
of detector to sample distance, and the right column shows the ratio of
minimum to maximum energies of the STO images for a given detail level. This
essentially indicates the capacity for a given detail level to distinguish
between the blurred and focused images. At all three levels of blur, detail
level one has an energy ratio that is significantly higher than all other
detail levels.

##  5 Outlook

The choice of a particular wavelet in this paper was based off of qualitative
and subjective measures—a quantitative method of discerning the most efficient
wavelet for performing a given analysis of DFXM images of partricular system
needs to be further developed. We note that this work utilized wavelets solely
on single images as two-dimensional signals. Further applications and a more
robust analysis may be performed if images were analyzed in sequence, forming
a three-dimensional signal with two spatial dimensions and one temporal. This
may allow the isolation of temporal variations to observe real-time phase
transformation under dynamic physical perturbations in a spatially resolved
way. One might use wavelet analyses to extract how specific domains shrink or
grow under these effects, or how their orientation changes. Finally, one may
utilize this algorithm to determine when a DFXM microscope is in focus, or
whether a rotation axis for a sample is oriented and its location. However,
the algorithm to determine image focus herein made use of data that was
idealistic in nature—the algorithm should be further tested and developed on
real, experimental images collected at various distances with respect to
expecgted image-plane location. At next generation near diffraction-limited
x-ray sources DFXM is poised to become a potent imaging modality for materials
science, applied materials, and condensed matter physics research. Further
exploitation of wavelet based analyses then are likely to enhance and benefit
these research fields.

##  Appendix A Optical Image of YBCO Sample

Figure 12: Left: Image of YBCO sample on Al. This sample was used to collect
the data in Figure 2A. The red square indicates the region of interest, with
the crosshairs intersecting at the rotational axis. Right: Polarized
microscopic imaging of region of interest.

##  Appendix B More on Techniques and Methods

The following section contains more in-depth information about the methods
utilized in this report.

###  B.1 Selective Reconstruction

With successive wavelet transforms, one can select which detail signals to
utilize in the reconstruction, thus selectively omitting undesired features of
particular frequencies in the newly reconstructed image. For example, take the
signal

$S=(1,3,2,4),$

which, under a Haar transform, has first-level approximation and detail
signals

$A_{1}=\sqrt{2}(2,3)\text{ and }D_{1}=\sqrt{2}(-1,-1).$

Applying a Haar transform to $A_{1}$, we have

$A_{2}=2(2.5)\text{ and }D_{2}=2(-0.5).$

Upon resizing $A_{2}$ and scaling by a factor of $\frac{1}{\sqrt{2}}$ to
become

$A_{2}^{\prime}=\sqrt{2}(2.5,2.5),$

reconstructing $S$ using only $A_{2}^{\prime}$ and $D_{1}$ yields

$S_{rec}=(1.5,3.5,1.5,3.5),$

thereby removing the smooth growth of the values of $S$.
The decision of which detail levels to utilize in the selective reconstruction
of Figure 2A came from inspecting Figure 13 and preserving the levels in which
desirable features were emergent. By observing the scale on which desirable
changes these occur, we find that they align with the expectations of how
detail levels and the scale of changes are related, namely that a preserved
detail level $\ell$ captures change on the order of $2^{\ell-1}$ pixels.
Figure 14 shows that the features in Figure 2A have a width of approximately
64 pixels, meaning that the relevant changes between the edges and center are
roughly 32 pixels in length. These are the largest desired changes, and are
captured by detail level 6 as expected. The smallest desired changes are the
fine details that comprise the characteristic shape of a feature, without
being artifacts of noise. Figure 14 indicates that changes that supersede the
erratic jumps present within features are roughly 4 pixels in length, and are
thus captured by detail level 3.

Figure 13: Plots of central column approximation and detail signals for levels
0-11 of a Haar decomposition. In the early stages of decomposition, the small-
scale features disappear first, and the amplitude of the detail signal is
small compared to the approximation signal. As the decomposition level grows,
the larger-scale features begin to vanish, and the detail signal begins to
encapsulate smoother oscillations within the signal. Figure 14: Line profile
of Figure 2A, with highlighted relevant changes and their corresponding length
scales.

###  B.2 Feature Tracking Algorithm

Given two features $F_{1}$ and $F_{2}$ with $n$ properties, indicated by
$F_{1}(1)\dots F_{1}(n)$ and $F_{2}(1)\dots F_{2}(n)$, the feature distance
$\Delta F_{12}$ between them is given by

$\Delta
F_{12}=\sqrt{\sum_{i=1}^{n}\left(\frac{F_{1}(i)-F_{2}(i)}{M_{i}}\right)^{2}},$
(1)

where $M_{i}$ is a normalization parameter. For bounded properties (e.g.
orientation, centroid), $M_{i}$ is given by the maximum possible difference in
the $i$th property; for unbounded properties (e.g. area, diameter),
$M_{i}=\max(|F_{1}(i)|,|F_{2}(i)|)$.

###  B.3 Detail Images on STO

Figure 15: Grid of details of simulated STO images. The row outlined in yellow
indicates the detail signals of the focused image. The images were generated
with the stationary wavelet transform which differs slightly from the discrete
wavelet transform in that it preserves image size by altering the downsampling
and upsampling of the discrete wavelet transform. More details on the
stationary transform can be found in nason1995stationary. The images shown are
the averages of the absolute values of the horizontal, vertical, and diagonal
details of a given image. The periodicity of high-detail-level images is an
artifact of how Matlab addresses the boundary-value problem.

Acknowledgements

We extend our gratitude to Doga Gursoy for the insights provided in our
discussion on the wavelet transform and other signal analysis tools. This work
was supported in part by the U.S. Department of Energy, Office of Science,
Office of Workforce Development for Teachers and Scientists (WDTS) under the
Science Undergraduate Laboratory Internships (SULI) program. This research
used resources of the Advanced Photon Source, a U.S. Department of Energy
(DOE) Office of Science user facility at Argonne National Laboratory and is
based on research supported by the U.S. DOE Office of Science-Basic Energy
Sciences, under Contract No. DE-AC02-06CH11357.

[iucr]


# Moderate-temperature near-field thermophotovoltaic systems with thin-film
InSb cells

Rongqian Wang<EMAIL_ADDRESS>School of physical science and technology
& Collaborative Innovation Center of Suzhou Nano Science and Technology,
Soochow University, Suzhou 215006, China.  Jincheng Lu  School of physical
science and technology & Collaborative Innovation Center of Suzhou Nano
Science and Technology, Soochow University, Suzhou 215006, China.  Center for
Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics,
Shanghai Key Laboratory of Special Artificial Microstructure Materials and
Technology, School of Physics Science and Engineering, Tongji University,
Shanghai 200092 China  Jian-Hua Jiang<EMAIL_ADDRESS>School of
physical science and technology & Collaborative Innovation Center of Suzhou
Nano Science and Technology, Soochow University, Suzhou 215006, China.

###### Abstract

Near-field thermophotovoltaic systems functioning at 400$\sim$900 K based on
graphene-hexagonal-boron-nitride heterostructures and thin-film InSb $p$-$n$
junctions are investigated theoretically. The performances of two near-field
systems with different emitters are examined carefully. One near-field system
consists of a graphene-hexagonal-boron-nitride-graphene sandwich structure as
the emitter, while the other system has an emitter made of the double
graphene-hexagonal-boron-nitride heterostructure. It is shown that both
systems exhibit higher output power density and energy efficiency than the
near-field system based on mono graphene-hexagonal-boron-nitride
heterostructure. The optimal output power density of the former device can
reach to $1.3\times 10^{5}~{}\rm{W\cdot m^{-2}}$, while the optimal energy
efficiency can be as large as $42\%$ of the Carnot efficiency. We analyze the
underlying physical mechanisms that lead to the excellent performances of the
proposed near-field thermophotovoltaic systems. Our results are valuable
toward high-performance moderate temperature thermophotovoltaic systems as
appealing thermal-to-electric energy conversion (waste heat harvesting)
devices.

##  I Introduction

Thermophotovoltaic (TPV) systems are solid-state renewable energy resource
that are of immense potentials in a wide range of applications including solar
energy harvesting and waste heat recovery Shockley and Queisser (1961); Martín
and Algora (2004); Nagashima  _et al._ (2007); Fraas and Ferguson (2000);
Sulima and Bett (2001); Wu  _et al._ (2012); Chan  _et al._ (2013); Liao  _et
al._ (2016); Zhao  _et al._ (2017a); Tervo  _et al._ (2018). In the TPV
system, a photovoltaic (PV) cell is placed in the proximity of a thermal
emitter and converts the thermal radiation from the emitter into electricity
via infrared photoelectric conversion. However, the frequency mismatch between
a moderate-temperature thermal emitter and the PV cell leads to significantly
reduced performance of the TPV systems at moderate temperatures (i.e.,
400$\sim$900 K which is the majority spectrum of the industry waste heat). To
overcome this obstacle, materials which support surface polaritons have been
used to introduce a resonant near-field energy exchange between the emitter
and the absorber Wu  _et al._ (2012); Svetovoy  _et al._ (2012); Ilic  _et
al._ (2012); Svetovoy and Palasantzas (2014); Basu  _et al._ (2015); Zhao  _et
al._ (2017a). As a consequence, near-field TPV (NTPV) systems have been
proposed to achieve appealing energy efficiency and output power Narayanaswamy
and Chen (2003); Laroche  _et al._ (2006); Park  _et al._ (2008); Ilic  _et
al._ (2012); Bright  _et al._ (2014); Molesky and Jacob (2015); St-Gelais  _et
al._ (2017); Jiang and Imry (2018); Papadakis  _et al._ (2020). Near-field
systems based on graphene, hexagonal-boron-nitride (h-BN) and their
heterostructures have been shown to demonstrate excellent near-field couplings
due to surface plasmon polaritons (SPPs), surface phonon polaritons (SPhPs)
and their hybridizations [i.e., surface plasmon-phonon polaritons (SPPPs)]
Svetovoy  _et al._ (2012); Messina and Ben-Abdallah (2013); Svetovoy and
Palasantzas (2014); Woessner  _et al._ (2015); Zhao and Zhang (2015); Zhao
_et al._ (2017b); Shi  _et al._ (2017). It was demonstrated that a
heterostructure consisting of graphene-h-BN multilayers performs better than
the monocell structure and the heat flux is found to be three times larger
than that of the monocell structure Zhao and Zhang (2015); Zhao  _et al._
(2017b); Shi  _et al._ (2017). Here, we consider the graphene-h-BN-graphene
sandwich structure and graphene-h-BN-graphene-h-BN four-layer structure as the
near-field thermal emitters to provide the enhanced radiative heat transfer.

In order to convert the infrared radiation into electricity, the energy
bandgap ($E_{\rm gap}$) of the NTPV cell (i.e., the $p$-$n$ junction) must
match the radiative spectrum generated by the emitter. III-V group compound
semiconductors like Gallium Arsenide (GaAs), Gallium antimonide(GaSb), Indium
antimonide (InSb) and Indium Arsenide (InAs) have been used due to the small
bandgap energy, high electron mobility and low electron effective mass Ilic
_et al._ (2012); Laroche  _et al._ (2006); Messina and Ben-Abdallah (2013);
Zhao  _et al._ (2017a). Recently, semiconductor thin-films have been explored
in NTPV systems. In Refs. Zhao  _et al._ (2017a) and Papadakis  _et al._
(2020), a NTPV system based on a InAs thin-film cell has exhibits appealing
performance operating at high temperatures. But the system suffers from low
energy efficiency (below $10\%$) when operating at moderate temperature due to
the parasitic heat transfer induced by the phonon-polaritons of InAs. Here, we
use InSb as the near-field absorber since the bandgap energy of InSb is lower
compared to InAs and its photon-phonon interaction is much weaker than InAs.
For the temperature of the InSb cell, $T_{\rm cell}=320~{}{\rm K}$, which has
been proved an optimal cell temperature in our previous work Wang  _et al._
(2019), the gap energy is $E_{\rm gap}=0.17$ eV and the corresponding angular
frequency is $\omega_{\rm gap}=2.5\times 10^{14}$ rad/s.

In this work, we examine the performances of two NTPV devices: the
graphene-$h$-BN-graphene-InSb cell (denoted as G-FBN-G-InSb cell, with the
graphene-$h$-BN-graphene sandwich structure being the emitter and the InSb
thin-film being the cell) and the graphene-$h$-BN-graphene-$h$-BN-InSb cell
(denoted as G-FBN-G-FBN-InSb cell, with the double graphene-$h$-BN
heterostructure being the emitter and the InSb thin-film being the cell). We
study and compare the performances of these two systems and reveal their
underlying physical mechanisms. We further optimize the performance of the
near-field TPV systems for various parameters.

In this work, we address two issues: First, we try to optimize the design of
the h-BN-graphene heterostructures as the emitter to improve the performance
of the NTPV system. Second, we try to discuss the effect of finite thickness
of the InSb cell on the performance of the NTPV system.

This work is structured as follows. In Sec. II, we describe our NTPV system
and clarify the radiative heat flux exchanged between the emitter and the
cell. In Sec. III, we study and compare the performances of the two NTPV
systems and analyze the spectral distributions of the photo-induced current
density and indicent heat flux. We also study the photon tunneling coefficient
to further elucidate the physical mechanisms. In Sec. IV, we examine the
performances of the two NTPV systems for various InSb thin-film thicknesses
and emitter temperatures to optimize the performances of the NTPV systems.
Finally, we summarize and conclude in Sec. V.

Figure 1: Schematic illustration of the NTPV systems. (a) A representative TPV
device. A thermal emitter of temperature $T_{\rm emit}$ is located at a
distance $d$ of a thermophotovoltaic cell of temperature $T_{\rm cell}$. Two
compositions of the thermal emitter with (b) graphene-h-BN-graphene structure
and (c) graphene-h-BN-graphene-$h$-BN structure. (d) The thermophotovoltaic
cell with a InSb thin-film of thickness $h_{\rm InSb}$ supported by a semi-
infinite substrate.

##  II System and Theory

In Fig. 1, we consider the graphene-h-BN-InSb NTPV systems. Fig. 1(a) is a
schematic presentation of a typical NTPV system, which consists of a thermal
emitter and a thermophotovoltaic cell. The emitter and the thermophotovoltaic
cell is separated by a vacuum gap with thickness $d$. The temperatures of the
emitter and cell are kept at $T_{\rm emit}$ and $T_{\rm cell}$, respectively.
The thermal radiation from the emitter is absorbed by the cell and then
converted into electricity via photoelectric conversion. Figs. 1(b) and 1(c)
present the two compositions of the emitter. Fig. 1(b) is a graphene-h-BN-
graphene sandwich structure and Fig. 1(c) is made of two graphene-h-BN
heterostructures. The thickness of h-BN thin film is $h_{\rm BN}$ and the
graphene monolayer is model as a layer of thickness $h_{\rm g}$. Fig. 1(d) is
a thin-film InSb $p$-$n$ junction supported by a substrate. The thickness of
the InSb thin-film is $h_{\rm InSb}$ and the substrate is set to be semi-
infinite intrinsic InSb.

When the InSb cell is located at a distance $d$ which is on the order of or
smaller than the thermal wavelength $\lambda_{\rm th}=2\pi\hbar c/k_{\rm
B}T_{\rm emit}$ from the emitter, the thermal radiation can be significantly
enhanced due to energy transfer via evanescent waves Ilic  _et al._ (2012).
The radiative heat exchange between the emitter and the cell is given by
Polder and Van H. (1971); Pendry (1999)

$\displaystyle Q_{\rm rad}=Q_{\omega<\omega_{\rm
gap}}+Q_{\omega\geq\omega_{\rm gap}}$ (1)

where $Q_{\omega<\omega_{\rm gap}}$ and $Q_{\omega\geq\omega_{\rm gap}}$ are
the heat exchanges below and above the band gap of the cell, respectively.

The below $Q_{\omega<\omega_{\rm gap}}$ and above-gap heat exchange
$Q_{\omega\geq\omega_{\rm gap}}$ are respectively given by

$\displaystyle Q_{\omega<\omega_{\rm gap}}=\int_{0}^{\omega_{\rm
gap}}\frac{d\omega}{4\pi^{2}}\left[\Theta_{1}\left(T_{\rm
emit},\omega\right)-\Theta_{2}\left(T_{\rm
cell},\omega\right)\right]\sum_{j}\int kdk\zeta_{j}(\omega,k),$ (2)

and

$\displaystyle Q_{\omega\geq\omega_{\rm gap}}=\int_{\omega_{\rm
gap}}^{\infty}\frac{d\omega}{4\pi^{2}}\left[\Theta_{1}\left(T_{\rm
emit},\omega\right)-\Theta_{2}\left(T_{\rm
cell},\omega,\Delta\mu\right)\right]\sum_{j}\int kdk\zeta_{j}(\omega,k),$ (3)

where $\Theta_{1}\left(T_{\rm
emit},\omega\right)={\hbar\omega}/[\exp{\left(\frac{\hbar\omega}{k_{\rm
B}T_{\rm emit}}\right)}-1]$ and $\Theta_{2}\left(T_{\rm
cell},\omega,\Delta\mu\right)={\hbar\omega}/{[\exp{\left(\frac{\hbar\omega-\Delta\mu}{k_{\rm
B}T_{\rm cell}}\right)}-1]}$ are the Planck mean oscillator energies of
blackbody at temperature $T_{\rm emit}$ and $T_{\rm cell}$, respectively.
$\Delta\mu$ is the electrochemical potential difference across the $p$-$n$
junction, which describes the effects of charge injection or depletion on the
carrier recombination processes and hence modify the number of photons through
the detailed balance. $k$ is the magnitude of the in-plane wavevector of
thermal radiation waves. $\zeta_{j}(\omega,k)$ is the photon transmission
coefficient for the $j$-th polarization $(j=s,p)$, which consists of the
contributions from both the propagating and the evanescent waves Mulet  _et
al._ (2002)

$\zeta_{j}(\omega,k)=\begin{cases}\frac{\left(1-\left|r_{\rm
emit}\right|^{2}\right)\left(1-\left|r_{\rm
cell}\right|^{2}\right)}{\left|1-r_{\rm emit}^{j}r_{\rm
cell}^{j}\exp(2ik_{z}^{0}d)\right|^{2}},&k<\frac{\omega}{c}\\\ \frac{4{\rm
Im}\left(r_{\rm emit}^{j}\right){\rm Im}\left(r_{\rm
cell}^{j}\right)\exp(2ik_{z}^{0}d)}{\left|1-r_{\rm emit}^{j}r_{\rm
cell}^{j}\exp(2ik_{z}^{0}d)\right|^{2}},&k\geq\frac{\omega}{c}\end{cases}$ (4)

where $k_{z}^{0}=\sqrt{\omega^{2}/c^{2}-k^{2}}$ is the perpendicular-to-plane
component of the wavevector in vacuum. $r_{\rm emit}^{j}$ ($r_{\rm cell}^{j}$)
with $j=s,p$ is the complex reflection coefficient at the interface between
the emitter (cell) and the air. Here, the reflection coefficients of the
emitter and cell are calculated by the scattering matrix approach Whittaker
and Culshaw (1999); Zhang (2007).

Via the infrared photoelectric conversion, the above-gap radiative heat
exchange is then converted into electricity. Based on the detailed balance
analysis, the electric current density of a NTPV cell is given by Shockley and
Queisser (1961); Ashcroft and Mermin (1976)

$\displaystyle I=I_{\rm ph}-I_{0}[\exp(V/V_{\rm cell})-1],$ (5)

where $V=\Delta\mu/e$ is the voltage bias across the NTPV cell, $V_{\rm
cell}=k_{\rm B}T_{\rm cell}/e$ is a voltage which measures the temperature of
the cell Shockley and Queisser (1961). $I_{\rm ph}$ and $I_{0}$ are the photo-
generation current density and reverse saturation current density,
respectively. In Eq. (5), an ideal rectifier condition has been used to
simplify the nonradiative recombination Shockley and Queisser (1961). The
actual nonradiative mechanisms include Shockley-Read-Hall (RSH) and Auger
nonradiative processes. Here, for the sake of simplicity, we just follow the
Shockley-Queisser analysis Shockley and Queisser (1961).

The reverse saturation current density is determined by the diffusion of
minority carriers in the InSb $p$-$n$ junction, which is given by

$\displaystyle I_{0}=en_{\rm i}^{2}\left(\frac{1}{N_{\rm A}}\sqrt{\frac{D_{\rm
e}}{\tau_{\rm e}}}+\frac{1}{N_{\rm D}}\sqrt{\frac{D_{\rm h}}{\tau_{\rm
h}}}\right),$ (6)

where $n_{\rm i}$ is the intrinsic carrier concentration, $D_{\rm e}$ and
$D_{\rm h}$ are the diffusion coefficients of the electrons and holes,
respectively. $N_{\rm A}$ and $N_{\rm D}$ are the $p$-region and $n$-region
impurity concentrations, respectively Shur (1996). $\tau_{\rm e}$ and
$\tau_{h}$ are correspondingly the relaxation time of the electron-hole pairs
in the $n$-region and $p$-region. Numerical values of these parameters are
taken from Refs. Shur (1996) and Lim  _et al._ (2015).

The photo-generation current density is contributed from the above-gap thermal
heat exchange Laroche  _et al._ (2006); Messina and Ben-Abdallah (2013)

$\displaystyle I_{\rm ph}=\frac{e}{4\pi^{2}}\int_{\omega_{\rm
gap}}^{\infty}\frac{d\omega}{\hbar\omega}\left[\Theta_{1}\left(T_{\rm
emit},\omega\right)-\Theta_{2}\left(T_{\rm
cell},\omega,\Delta\mu\right)\right]$ (7) $\displaystyle\hskip
28.45274pt\times\sum_{j}\int kdk\zeta_{j}(\omega,k)\left(1-\exp{\left[-2\rm
Imag(\it k_{z}^{\rm InSb})\it h_{\rm InSb}\right]}\right),$

where $\it k_{z}^{\rm InSb}=\sqrt{\varepsilon_{\rm
InSb}\omega^{2}/c^{2}-k^{2}}$ is the perpendicular-to-plane component of the
wavevector in the InSb $p$-$n$ junction. $\varepsilon_{\rm InSb}$ is the
dielectric function of the InSb cell, which is given by $\varepsilon_{\rm
InSb}=\left(n+\frac{i\rm c\it\alpha(\omega)}{2\omega}\right)^{2}$. $n_{\rm
InSb}=4.12$ is the refractive index and $c$ is the speed of light in vacuum.
$\alpha(\omega)$ is a step-like function describing the photonic absorption,
which is given by Messina and Ben-Abdallah (2013) $\alpha(\omega)=0$ for
$\omega<\omega_{\rm gap}$ and
$\alpha(\omega)=\alpha_{0}\sqrt{\omega/\omega_{\rm gap}-1}$ for
$\omega>\omega_{\rm gap}$ with $\alpha_{0}$ = 0.7 $\mu\rm m^{-1}$ Messina and
Ben-Abdallah (2013). Since the NTPV cell is a thin film, the exponential decay
characteristic of the electromagnetic wave propagating in the InSb thin-film
must be considered. The term $\left(1-\exp{\left[-2\rm Imag(\it k_{z}^{\rm
InSb})\it h_{\rm InSb}\right]}\right)$ is the absorption probability of the
incident radiation in the InSb film of thickness $h_{\rm InSb}$, which
measures the actual availability of the above-gap photons in the photon-
carrier generation process.

The dielectric function of h-BN is described by a Drude-Lorentz model, which
is given by Kumar  _et al._ (2015)

$\varepsilon_{m}=\varepsilon_{\infty,m}\left(1+\frac{\omega^{2}_{\rm LO,\it
m}-\omega^{2}_{\rm TO,\it m}}{\omega^{2}_{\rm TO,\it
m}-i\gamma_{m}\omega-\omega^{2}}\right),$ (8)

where $m={\rm\parallel,\perp}$ denotes the out-of-plane and the in-plane
directions, respectively. $\varepsilon_{\infty,m}$ is the high-frequency
relative permittivity, $\omega_{\rm TO}$ and $\omega_{\rm LO}$ are the
transverse and longitudinal optical phonon frequencies, respectively.
$\gamma_{m}$ is the damping constant of the optical phonon modes. The values
of these parameters are chosen as those determined by experiments Geick  _et
al._ (1966); Caldwell  _et al._ (2014).

The effective dielectric function of the graphene monolayer is modeled as
Vakil and Engheta (2011)

$\varepsilon_{\rm g}=1+i\frac{\sigma_{\rm g}}{\varepsilon_{0}\omega h_{\rm
g}},$ (9)

where $\sigma_{\rm g}$ is the optical conductivity Falkovsky (2008).

The output electric power density $P_{\rm e}$ of the NTPV system is defined as
the product of the net electric current density and the voltage bias,

$\displaystyle P_{\rm e}=-I_{\rm e}V,$ (10)

and the energy efficiency $\eta$ is given by the ratio between the output
electric power density $P_{\rm e}$ and incident radiative heat flux $Q_{\rm
inc}$,

$\displaystyle\eta=\frac{P_{\rm e}}{Q_{\rm inc}},$ (11)

where the incident radiative heat flux is given by the radiative heat exchange
defined in Eq. (1). The second law of thermodynamics imposes an upper bound on
the energy efficiency, which is the Carnot efficiency,

$\displaystyle\eta_{c}=1-\frac{T_{\rm cell}}{T_{\rm emit}}.$ (12)

When studying the energy efficiency of various NTPV systems with different
temperatures, we use the Carnot efficiency to quantify and compare their
energy efficiencies.

##  III Performances of graphene-h-BN-InSb near-field systems

Figure 2: Performances of two NTPV cells. (a) Electrical power density and (b)
energy$-$conversion efficiency in units of Carnot efficiency ($\eta_{\rm c}$).
The temperatures of the emitter and the cell are set at $T_{\rm emit}=450$ K
and $T_{\rm cell}=320$ K, respectively. The vacuum gap distance is $d=20$ nm,
the h-BN film thickness is $h_{\rm BN}=20$ nm and the thickness of the InSb
thin film is $h_{\rm InSb}=1000$ nm. The chemical potential of graphene is
$\mu_{\rm g}=1.0$ eV. The Carnot efficiency is given by $\eta_{\rm
c}=1-{T_{\rm cell}}/{T_{\rm emit}}$.

We first examine the performances of the two types of NTPV cells. Fig. 2 shows
the output power density and energy efficiency as a function of voltage bias
for the two set-ups, respectively denoted as G-FBN-G-InSb cell (graphene-h-BN-
graphene heterostructure as the emitter and thin-film InSb $p$-$n$ junction as
the receiver) and G-FBN-G-FBN-InSb cell (graphene-h-BN-graphene-h-BN
heterostructure as the emitter and thin-film InSb $p$-$n$ junction as the
receiver). The output power density and energy efficiency are optimized for
various physical parameters, including the temperature of the cell, the
chemical potential of graphene and h-BN film thickness Wang  _et al._ (2019).
The analysis in Ref. [Wang  _et al._ , 2019] shows that setting $\mu_{\rm
g}=1.0$ eV, $h_{\rm BN}=20$ nm and $T_{\rm cell}=320$ K provides roughly
optimal performance, both in terms of power and efficiency, for the NTPV
systems considered in this work. Therefore, these parameters are kept as those
constants in the main text. The thickness of the InSb thin film is set as
$h_{\rm InSb}=1000$ nm, which is close to the thickness of experimental
samples Yang  _et al._ (2006).

As shown in Fig. 2, the maximum output power density of the G-FBN-G-InSb cell
is about $4.1\times 10^{2}~{}\rm{W\cdot m^{-2}}$, nearly 1.1 times of the the
maximum output power density of the G-FBN-G-FBN-InSb cell (about $3.9\times
10^{2}~{}\rm{W\cdot m^{-2}}$). For the energy-conversion efficiency, the
maximum values of the G-FBN-G-InSb cell is about $15\%\eta_{c}$, which
slightly higher than the maximum efficiency of the FBN-G-FBN-InSb cell (about
$14.9\%\eta_{c}$).

In order to analyze the physical mechanisms responsible for such performances
of these two setups, we show the spectral distributions of the photo-induced
current and incident radiative heat flux at the maximum electric power density
for the G-FBN-G-InSb cell and G-FBN-G-FBN-InSb cell in Fig. 3. The two shaded
areas in Fig. 3(a) and (b) are the two reststrahlen bands of h-BN Jacob
(2014). As exhibited in Fig. 3(a) that higher photo-induced current spectra of
the two systems appear at and near the reststrahlen band due to the
reststrahlen effect. The reststrahlen effect originates from the hyperbolic
optical properties of the h-BN film due to the photon—optical-phonon
interactions. In the reststrahlen band presented in Fig. 3(a), the in-plane
dielectric function of h-BN is negative, leading to strong reflection and
suppressed transmission of incident photons Jacob (2014). The near-field
radiation effects are essentially caused by the evanescent propagation of the
incident photons, which dramatically enhances the radiative heat transfer
between two closed spaced objects Wang  _et al._ (2019); Brar  _et al._
(2014); Kumar  _et al._ (2015); Woessner  _et al._ (2015); Zhao and Zhang
(2015); Zhao  _et al._ (2017b); Shi  _et al._ (2017). Such enhancement of the
radiative heat transfer eventually leads to the significant increase of the
input heat flux, output electric power and the energy efficiency Wang  _et
al._ (2019).

It is noticed that the overall photo-induced current spectrum of the G-FBN-G-
InSb cell is higher than the one of the G-FBN-G-FBN-InSb cell except in the
frequency range from $3.1\times 10^{14}~{}\rm{rad\cdot s^{-1}}$ to $4.1\times
10^{14}~{}\rm{rad\cdot s^{-1}}$. After integrating over $\omega$, this higher
photo-induced current spectrum gives rise to improved output power of the
G-FBN-G-InSb cell.

Figure 3: (a) Photo-induced current spectra $I_{\rm ph}(\omega)$ and (b)
incident radiative heat spectra $Q_{\rm inc}(\omega)$ at the maximum electric
power density for the two configurations. The temperatures of the emitter and
the cell are kept at $T_{\rm emit}=450$ K and $T_{\rm cell}=320$ K,
respectively. The vacuum gap is $d=20$ nm and the chemical potential of
graphene is set as $\mu_{\rm g}=1.0$ eV. The h-BN film thickness is $h_{\rm
film}=20$ nm. The chemical potential difference across the InSb $p$-$n$
junction $\Delta\mu$ is optimized independently for maximum output power for
each configuration.

Contrary to the photo-induced current spectra, the overall incident radiative
heat flux of the G-FBN-G-FBN-InSb cell is higher than the G-FBN-G-InSb cell.
Since the energy-conversion efficiency is defined as the ratio of the output
electric power density and the incident radiation heat flux, the higher output
power density and lower incident radiation heat flux give a higher energy
efficiency of the G-FBN-G-InSb cell.

To further elucidate the mechanism for the enhanced performance of the near-
field systems, we examine the photon tunneling coefficient
$\zeta_{j}(\omega,k)$ (given by Eq. 4). As shown in Fig. 4, the bright color
indicates a high transmission coefficient. The green dashed lines are the
light lines of vacuum and InSb, respectively. Note that the maximum
transmission coefficient is 2 due to the contribution of both $s$ and $p$
polarizations. In the above-gap range, both G-FBN-G-InSb cell and G-FBN-G-FBN-
InSb cell exhibit enhanced photon transmission, thanks to the SPPPs supported
by the graphene-h-BN heterostructures. Fig. 4 shows that the photon
transmission spectra of the two near-field NTPV systems do not differ much.
Therefore, the performances of the two NTPV systems are comparable.

Figure 4: Photon transmission coefficient $\zeta(\omega,k)$ for (a) G-FBN-G-
InSb cell and (b) G-FBN-G-FBN-InSb cell. The temperatures of the emitter and
the cell are kept at $T_{\rm emit}=450$ K and $T_{\rm cell}=320$ K,
respectively. The vacuum gap is $d=20$ nm and the chemical potential of
graphene is set as $\mu_{\rm g}=1.0$ eV. The h-BN film thickness is $h_{\rm
film}=20$ nm and the InSb thin-film thickness is $h_{\rm InSb}=1000$ nm.

Figure 5: Performances of two near-field TPV cells for various InSb thin-film
thickness. (a) Electrical power density and (b) energy$-$conversion efficiency
in units of Carnot efficiency ($\eta_{\rm c}$). The temperatures of the
emitter and the cell are set at $T_{\rm emit}=450$ K and $T_{\rm cell}=320$ K,
respectively. The vacuum gap distance is $d=20$ nm, the h-BN film thickness is
$h_{\rm BN}=20$ nm and the chemical potential of graphene is $\mu_{\rm g}=1.0$
eV.

Figure 6: Performances of two near-field TPV cells for various emitter
temperatures. (a) Electrical power density and (b) energy$-$conversion
efficiency in units of Carnot efficiency ($\eta_{\rm c}$). The temperature of
the cell is set at $T_{\rm cell}=320$ K. The vacuum gap distance is $d=20$ nm,
the h-BN film thickness is $h_{\rm BN}=20$ nm and the thickness of the InSb
thin film is chosen as an optimal value of $h_{\rm InSb}=10000$ nm. The
chemical potential of graphene is $\mu_{\rm g}=1.0$ eV.

##  IV Optimization of graphene-h-BN-InSb near-field systems

Figure 7: Optimal performances of two NTPV cells. (a) Maximum electrical power
density and (b) energy$-$conversion efficiency at maximum electric power in
units of Carnot efficiency ($\eta_{\rm c}$). The temperatures of the emitter
and the cell are set at $T_{\rm emit}=800$ K and $T_{\rm cell}=320$ K,
respectively. The h-BN film thickness is $h_{\rm BN}=20$ nm and the thickness
of the InSb thin film is $h_{\rm InSb}=10000$ nm. The chemical potential of
graphene is $\mu_{\rm g}=1.0$ eV. The Carnot efficiency is given by $\eta_{\rm
c}=1-{T_{\rm cell}}/{T_{\rm emit}}$.

We now study the performances of the two near-field systems for various InSb
thin-film thicknesses and emitter temperatures in Figs. 5 and 6. As presented
in Fig. 5 that both of the output power density and the energy-conversion
efficiency of the two setups are improved with increasing InSb thin-film
thickness. Especially for $h_{\rm InSb}=10000$ nm, the maximum power densities
of the G-FBN-G-InSb cell and G-FBN-G-FBN-InSb cell are about $4.5\times
10^{2}~{}\rm{W/m^{2}}$ and $4.0\times 10^{2}~{}\rm{W/m^{2}}$, respectively.
The maximum values of the corresponding efficiency are close to $16\%\eta_{\rm
C}$ and $15\%\eta_{\rm C}$, respectively. This enhancement can be attributed
to the increased absorption factor $\left(1-\exp{\left[-\rm Imag(\it
k_{z}^{\rm InSb})\it h_{\rm InSb}\right]}\right)$ because it is dependent on
$h_{\rm InSb}$. However, when the InSb thin-film thickness is increased from
4000 nm to 10000 nm, the increase of the output power density and the energy-
conversion efficiency for both systems are soon saturated. This is essentially
due to the near-field nature of the heat transfer: the photons transferred
from the emitter to the InSb cell is dominated by the evanescent
electromagnetic waves which are highly localized at the surface of the
emitter. Further increase of the thickness of the InSb cell does not change
the amount of photons received by the InSb cell.

Fig. 6 shows that by increasing the emitter temperature, both the performances
of the two setups can be improved by orders of magnitudes. Here, the InSb
thin-film thickness has been chosen as the optimal value with $h_{\rm
InSb}=10000$ nm. For the G-FBN-G-InSb cell, the maximum output power density
and energy efficiency at $T_{\rm emit}=800$ K are increased to $3.3\times
10^{4}~{}\rm{W/m^{2}}$ and $33\%\eta_{c}$, respectively. For the G-FBN-G-FBN-
InSb cell, the maximum output power density and energy efficiency at $T_{\rm
emit}=800$ K are also improved to $3.1\times 10^{4}~{}\rm{W/m^{2}}$ and
$32\%\eta_{c}$, respectively.

We now examine the maximum output electric power density and the efficiency at
maximum power. We consider specifically the situation with $h_{\rm
InSb}=10000$ nm and $T_{\rm emit}=800$ K. As shown in Fig. 7, both the maximum
electric power density and the efficiency at the maximum power of the two
near-field NTPV systems dramatically increase as the vacuum gap $d$ decreases.
The best performances are achieved when the vacuum gap is $d=10$ nm for both
systems. Specifically, the maximum electric power density reaches to
$1.3\times 10^{5}~{}\rm{W/m^{2}}$, while the efficiency at maximum power goes
up to $42\%$ of the Carnot efficiency. The two near-field NTPV systems perform
nearly equally well.

##  V Summary and conclusions

In conclusion, we investigated two NTPV devices, denoted as the G-FBN-G-InSb
cell and G-FBN-G-FBN-InSb cell, which have different emitters. The purpose of
the investigation is to find a high-performance emitter design with relatively
simple structure. Indeed, we find that both systems perform better than the
near-field NTPV system based on mono graphene-$h$-BN heterostructure, i.e.,
graphene-$h$-BN-InSb cell Wang  _et al._ (2019). The optimal output power
density of the G-FBN-G-InSb cell can reach to $1.3\times 10^{5}~{}\rm{W\cdot
m^{-2}}$, nearly twice of the optimal power density of the graphene-$h$-BN-
InSb cell. The optimal energy efficiency can be as large as $42\%$ of the
Carnot efficiency, which is $24\%$ higher than the optimal efficency of the
monocell-system. We analyze the underlying physical mechanisms that lead to
the excellent performances of the G-FBN-G-InSb cell. We further show that the
performance of the proposed NTPV system is affected negligibly by the finite
thickness of the InSb cell which is due to the near-field nature of the heat
transfer: the absorbed photons are highly localized at the surface of the InSb
cell. Our study shows that NTPV systems are promising for high-performance
moderate temperature thermal-to-electric energy conversion.

##  VI Acknowledgements

We acknowledge support from the National Natural Science Foundation of China
(NSFC Grant No. 11675116), the National Natural Science Foundation of China
(NSFC Grant No. 12074281), the Jiangsu distinguished professor funding and a
Project Funded by the Priority Academic Program Development of Jiangsu Higher
Education Institutions (PAPD). J.L thanks China’s Postdoctoral Science
Foundation funded project (No. 2020M681376).

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