# A Standardized Pipeline for Colon Nuclei Identification and Counting
Challenge

Jijun Cheng†, Xipeng Pan†, Feihu Hou, Bingchao Zhao, Jiatai Lin, Zhenbing Liu,
Zaiyi Liu, Chu Han Chu Han, Xipeng Pan, Jiatai Lin, Bingchao Zhao and Zaiyi
Liu are with the Department of Radiology, Guangdong Provincial People’s
Hospital, Guangdong Academy of Medical Sciences, Guangzhou, Guangdong, 510080,
China.Jijun Cheng, Xipeng Pan, Feihu Hou and Zhenbing Liu are with the School
of Computer Science and Information Security, Guilin University of Electronic
Technology, Guilin, Guangxi, China.† Equal contribution.Corresponding author:
Han Chu (email<EMAIL_ADDRESS>
###### Abstract

Nuclear segmentation and classification is an essential step for computational
pathology. TIA lab from Warwick University organized a nuclear segmentation
and classification challenge (CoNIC) [1] for H&E stained histopathology images
in colorectal cancer with two highly correlated tasks, nuclei segmentation and
classification task and cellular composition task. There are a few obstacles
we have to address in this challenge, 1) limited training samples, 2) color
variation, 3) imbalanced annotations, 4) similar morphological appearance
among classes. To deal with these challenges, we proposed a standardized
pipeline for nuclear segmentation and classification by integrating several
pluggable components. First, we built a GAN-based model to automatically
generate pseudo images for data augmentation. Then we trained a self-
supervised stain normalization model to solve the color variation problem.
Next we constructed a baseline model HoVer-Net with cost-sensitive loss to
encourage the model pay more attention on the minority classes. According to
the results of the leaderboard, our proposed pipeline achieves 0.40665 mPQ+
(Rank 49th) and 0.62199 r2 (Rank 10th) in the preliminary test phase.

###### Index Terms:

Nuclear segmentation and classification, Computational pathology, Imbalanced
annotations, Data augmentation, Stain normalization

##  I Introduction

Automated nuclei segmentation and classification in whole slide images can
help objective assessment of tumor microenvironment [2]. Many previous works
have been proposed for this task in different technical perspectives, multi-
task learning [3, 4], weakly-supervised learning [5] and etc [6]. Recently,
TIA lab from Warwick University organized a nuclei segmentation and
classification challenge (CoNIC) [1] on Grand Challenge Platform for
colorectal cancer. This challenge is based on the recently proposed Lizard
dataset [7], which is the largest nuclei segmentation and classification
dataset for colorectal cancer with 495,179 nuclear instances with
classification labels. Different from the other existing datasets like Pannuke
[8], Lizard only focuses on colorectal cancer and it further divides
inflammatory nuclei into several sub-categories, lymphocyte, plasma,
neutrophil or eosinophil. There are two tasks in this challenge, the
segmentation and classification task as well as the cellular composition task.

Figure 1: Data distribution of Lizard dataset. Figure 2: Overview of our
proposed standardized pipeline for nuclei segmentation and classification.

Fig. 1 shows the data distribution and some examples of Lizard dataset. We
face some inherent difficulties. 1) The classification labels are extremely
imbalanced. This challenge comprised of 244,563 epithelials, 101,413
lymphocytes, 28,466 plasmas, 4,824 neutrophils and 3,604 eosinophils and
112,309 connectives. The number of epithelials is almost 70 times larger than
the number of eosinophils. Lack of training samples make neural network models
hard to train a well feature representation for the minority classes. 2) Since
the images are from multiple datasets in multiple centers, there exist some
color variation because of different staining protocols or scanners. 3) Even
it is the largest dataset so far for only one specific cancer, we can say the
samples are still not sufficient for training a robust model because of the
heterogeneous of malignant tumor. 4) The morphological appearances among
different types are similar, such as lymphocytes and plasma cells, tumor
epithelial nuclei and normal epithelial nuclei. Such difficulties not only
exist in CoNIC challenge, they are also the most common challenges in the
nuclei segmentation and classification task for all the cancers. How to solve
them might be the key to build a stable and general model.

Figure 3: Generated pseudo images and pseudo labels.

In this paper, we achieve nuclei segmentation and classification by a
systematic design other than developing a single model. With the help of our
previous works in digital pathology, we proposed a systematic and standardized
pipeline for nuclei segmentation and classification by integrating several
independent components, including data augmentation, stain normalization, CNN
backbone and loss functions. First, we applied a GAN-based model [9] to
generate paired pseudo masks and images to extend the training set. Next, we
introduced a self-supervised stain normalization model [10] to make the color
style consistent. Finally, we applied a strong baseline model HoVer-Net [4] as
the backbone with an additional cost-sensitive loss [11] to tackle the class
imbalanced problem. Our standardized pipeline is pluggable and all the
components can be replaced by more superior ones. On the leaderboard in the
preliminary test phase, we can see all the components have proved its
effectiveness according to the quantitative results. Finally, our proposed
standardized pipeline achieves 0.40665 mPQ+ (Rank 49th) and 0.62199 r2 (Rank
10th) in the preliminary test phase.

TABLE I: Quantitative Results - Preliminary Test Phase TASK 1 - Preliminary
Test Phase (PQ)
---
Model | Data | Stain | Loss | mPQ+ | pla | neu | epi | lym | eos | con
HoVer-Net | - | - | - | 0.2874 | 0.2703 | 0.0321 | 0.4298 | 0.4195 | 0.2789 | 0.2938
- | $\checkmark$ | - | 0.3889 | 0.4353 | 0.1855 | 0.5234 | 0.4115 | 0.3794 | 0.3985
$\checkmark$ | $\checkmark$ | $\checkmark$ | 0.4067 | 0.4403 | 0.2005 | 0.5368 | 0.4394 | 0.4301 | 0.3928
TASK 2 - Preliminary Test Phase (r2)
Model | Data | Stain | Loss | overall | pla | neu | epi | lym | eos | con
HoVer-Net | - | - | - | 0.0756 | 0.4742 | 0.2843 | 0.5531 | -1.8573 | 0.6670 | 0.3346
- | $\checkmark$ | - | 0.4369 | 0.7570 | 0.5104 | 0.6862 | -0.4704 | 0.7554 | 0.3831
- | $\checkmark$ | $\checkmark$ | 0.4514 | 0.7485 | 0.5048 | 0.6912 | -0.3870 | 0.7939 | 0.3570
$\checkmark$ | $\checkmark$ | - | 0.6220 | 0.8022 | 0.8265 | 0.7858 | 0.3562 | 0.7191 | 0.2420

##  II Methodology

Fig. 2 demonstrates the standardized pipeline for nuclei segmentation and
classification. It includes four components, GAN-based data augmentation,
self-supervised stain normalization, CNN backbone and loss functions.

###  II-A GAN-based Data Augmentation

Besides the most common data augmentation ways like rotation, flipping and
etc, we introduced our previously proposed GAN-based model for data
augmentation [9]. This model used the masks from Lizard dataset as the input
image and generated pseudo histopathology images. With a well trained
generator, we randomly construct new pseudo labels and generated pseudo
pathological images.

In order to avoid inputting exactly the same masks to the generator while
keeping the real distribution of the nuclei, we introduced some slight
vibrations for each real mask. First of all, we extract every nuclear instance
in the semantic segmentation masks in the Lizard dataset. For each nuclear
instance, we introduced random $x$ and $y$ axis translations within the range
of (-3,3) pixels and a random rotation within the angle of $-3^{\circ}$ and
$+3^{\circ}$ while keeping the class and the shape unchanged. Such slight
changes aim to conforms to the real distribution of cells in the real
histopathological images. We generated 1000 pseudo images and masks pairs for
data augmentation. Fig. 3 shows the generated pseudo pathological images using
pseudo labels.

###  II-B Self-supervised Stain normalization

Color variation problem may greatly hurt the model generalization ability.
Since this challenge does not allow to use other public pathology datasets. We
introduced our previously proposed self-supervised stain normalization
approach RestainNet [10]. We used the CoNSeP dataset, which is a part of
Lizard dataset, to train our stain normalization model. Since RestainNet is a
self-supervised model, paired training images are not necessary, which shows
great flexibility in preparing training data. By applying RestainNet, all the
images were mapped into the same staining style with CoNSeP dataset, as
demonstrated in Fig. 2 (b).

###  II-C CNN Backbone

To the best of our knowledge, HoVer-Net [4] is the strongest baseline model
for nuclear segmentation and classification. Thanks to the horizontal and
vertical distance maps design, HoVer-Net is able to separate the overlapping
nuclei. So in this challenge, HoVer-Net was employed as our baseline method,
which was proposed to segment and classify nuclei in histology images
simultaneously. The network consisted of three branches: Nuclear Pixel (NP),
HoVer, Nuclear Classification (NC). NP was designed for nuclear semantic
segmentation, and the HoVer learned the horizontal and vertical map of the
nuclei. Combined with the results of NP, the markers were available for
watershed algorithm that was used for post-processing. NC predicted the
category of nuclear pixels, and merged the segmentation results of the above
two branches to obtain the final results.

Figure 4: Qualitative results of different sub-datasets. TABLE II:
Quantitative Results - Development Set Experiments on Different Configurations
---
Model | Data | Stain | Loss | mPQ+ | pla | neu | epi | lym | eos | con
HoVer-Net | - | - | - | 0.4735 | 0.4960 | 0.2121 | 0.5936 | 0.6704 | 0.2928 | 0.5760
$\checkmark$ | - | - | 0.4808 | 0.4951 | 0.2458 | 0.5921 | 0.6664 | 0.3160 | 0.5694
- | $\checkmark$ | - | 0.4658 | 0.4934 | 0.1997 | 0.5914 | 0.6656 | 0.2854 | 0.5597
- | - | $\checkmark$ | 0.4762 | 0.4869 | 0.2154 | 0.5946 | 0.6728 | 0.3136 | 0.5741
$\checkmark$ | $\checkmark$ | - | 0.4664 | 0.4831 | 0.2048 | 0.5914 | 0.6574 | 0.3044 | 0.5571
- | $\checkmark$ | $\checkmark$ | 0.4635 | 0.4877 | 0.1472 | 0.5919 | 0.6667 | 0.3271 | 0.5605
$\checkmark$ | - | $\checkmark$ | 0.4792 | 0.4969 | 0.2318 | 0.5946 | 0.6694 | 0.3112 | 0.5716
$\checkmark$ | $\checkmark$ | $\checkmark$ | 0.4642 | 0.4794 | 0.2024 | 0.5904 | 0.6591 | 0.2997 | 0.5543
Cross-validation on Stain Normalization
Model | Data | Stain | Loss | mPQ+ | pla | neu | epi | lym | eos | con
HoVer-Net | - | - | - |  | 0.4042
---
$\pm$ 0.0506
| 0.4001
---
$\pm$ 0.0554
| 0.1308
---
$\pm$ 0.1204
| 0.5368
---
$\pm$ 0.0371
| 0.5995
---
$\pm$ 0.0158
| 0.2676
---
$\pm$ 0.1204
| 0.4906
---
$\pm$ 0.0640
- | $\checkmark$ | - |  | 0.4121
---
$\pm$ 0.0526
| 0.4132
---
$\pm$ 0.0740
| 0.1762
---
$\pm$ 0.1245
| 0.5293
---
$\pm$ 0.0536
| 0.6006
---
$\pm$ 0.0306
| 0.2609
---
$\pm$ 0.1117
| 0.4923
---
$\pm$ 0.0222
$\checkmark$ | $\checkmark$ | $\checkmark$ |  | 0.4140
---
$\pm$ 0.0424
| 0.4181
---
$\pm$ 0.0759
| 0.1688
---
$\pm$ 0.0943
| 0.5340
---
$\pm$ 0.0520
| 0.5987
---
$\pm$ 0.0323
| 0.2619
---
$\pm$ 0.0647
| 0.5024
---
$\pm$ 0.0222

###  II-D Loss Functions

Besides the original losses introduced in HoVer-Net, we also introduced a
cost-sensitive loss in the NC branch. Imbalanced data is a regular occurrence
in target identification and classification tasks. The imbalance of different
cell types is particularly pronounced in the CoNIC challenge. For example,
epithelials, lymphocytes and connective cells contribute for a substantial
portion of the ratio, whereas eosinophils and neutrophils account for a
negligible portion. The imbalance of cell types will result in a decrease in
segmentation and classification performance. We proposed a cost-sensitive loss
function for multi-class nuclear segmentation and classification to address
this problem. By incorporating the cost-sensitive matrix into the loss
function, the impact of sample imbalance could be alleviated, and the
classification accuracy of small categories can be improved. For multi-class
tasks, it is assumed that the number of categories is $\boldsymbol{N}$. The
cost-sensitive matrix $\boldsymbol{M}$ with the size of
$\boldsymbol{N}\times\boldsymbol{N}$ can be defined as follows:

$\boldsymbol{M}=\begin{bmatrix}m_{(0,0)}&m_{(0,1)}&\ldots&m_{(0,N-1)}\\\
m_{(1,0)}&m_{(1,1)}&\ldots&m_{(1,N-1)}\\\ \vdots&\vdots&\ddots&\vdots\\\
m_{(N-1,0)}&m_{(N-1,1)}&\ldots&m_{(N-1,N-1)}\\\ \end{bmatrix}$

where $m_{(j,k)}$ represents the cost when class $k$ is incorrectly classified
as class $j$. The diagonal element of matrix $\boldsymbol{M}$ is 0. The values
of non-diagonal elements on the cost-sensitive matrix are set according to the
ratio of the number of samples.

##  III Experimental Results

We first demonstrate the results of both tasks on the leaderboard. Then we
conduct a local development set to further evaluate the importance of each
component in the proposed pipeline. Finally we demonstrate some qualitative
results in Fig. 4.

In the preliminary test phase, the quantitative results of both tasks are
shown in Table I. Note that, we do not specifically design a regression model
for the task 2, the results of the task 2 are directly generated from the
results of the task 1. According to the results of the leaderboard, the mPQ+
and r2 of our proposed pipeline achieves 0.40665 (Rank 49th) and 0.62199 (Rank
10th), respectively. We can observe that both stain normalization and data
augmentation can greatly benefit the nuclear segmentation and classification
task for all the classes. Since each team can only submit the algorithm for
once, we do not have chances to test all the configurations of different
components. Therefore, we conduct a local development set of further
evaluation.

Since task 2 is highly related to task 1. In this experiment, we only evaluate
the proposed pipeline in nuclei segmentation and classification task. We split
the Lizard dataset into a training set and a test set. All the models were
trained for 100 epoches with no validation set. As demonstrated in the upper
part of Table II, HoVer-Net alone achieves moderate performance. Both Data
augmentation and cost-sensitive loss improve the segmentation and
classification performance. Specifically, the result of eosinophils, the one
with the fewest labels, increases from 0.2928 to 0.3136 by cost-sensitive
loss. Combining data augmentation and cost-sensitive loss can also get an
obvious improvement for the minority classes, neutrophils and eosinophils.
However, when equipped with stain normalization, the quantitative performance
decreases in all the three models with it, which in contradiction with the
leaderboard result. The reason is that when we construct the local development
set, we mix up the data from all five datasets. So the stain style of the test
set if not blinded for the CNN model. In the meanwhile, GAN-based stain
normalization model may cause information loss. So stain normalization in this
experiment does not bring positive effect which is not as expected.

Next, we introduce a ‘5-fold cross validation-like’ experiment to evaluate the
effectiveness of stain normalization. In this experiment, each sub-dataset is
regarded as one fold to ensure the testing fold is always blinded to the CNN
model. The lower part of Table II demonstrates the quantitative results of the
segmentation and classification task. When the color styles are blinded to the
model, stain normalization component generally help improve the results.

##  IV Conclusion

In this paper, we proposed a standardized nuclei segmentation and
classification pipeline for CoNIC challenge. The pipeline integrates several
pluggable components, including data augmentation, stain normalization, CNN
backbone and loss functions. Experiments on the preliminary test set and the
local development set demonstrate the effectiveness of this pipeline.

## Acknowledgment

This research was supported in part by National Science Fund for Distinguished
Young Scholars, China (Grant No. 81925023), National Natural Science
Foundation of China (Grant Nos. 62002082, 62102103),China Postdoctoral Science
Foundation (Grant No. 2021M690753), 2020 National Innovation and
Entrepreneurship Training Program for College Students (Grant No.
202010595023).

## References

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# Elastic properties and electronic structures of antiperovskite-type InNCo3
and InNNi3

Z. F. Hou  Department of Physics, Fudan University, Shanghai, 200433, P. R.
China

###### Abstract

We have performed the first-principles calculations to study the elasticity,
electronic structure, and magnetism of InNCo3 and InNNi3. The independent
elastic constants are derived from the second derivative of total energy as a
function of strain, and the elastic modulus are predicted according to the
Voigt-Reuss-Hill approximation. Our calculations show that the bulk modulus of
InNCo3 is slightly larger than that of InNNi3 due to a smaller lattice
constant for InNCo3. For InNCo3 the ferromagnetic state is energetically
preferable to the paramagnetic state, while the ground state of InNNi3 is a
stable paramagnetic (non-magnetic) state. This is due to the different
strength of 2p-3d hybridization for the N-Co and N-Ni atoms in InNCo3 and
InNNi3.

##  I Introduction

The ternary nitrides or carbides with the general formula AXM3 ( A: divalent
or trivalent element; X: carbon or nitrogen; and M: transition metal) are
already known for several decades Goodenough70 ; Chern92 ; Jager93 . These
compounds crystalize in a cubic anti-perovskite structure (A: cube-corner
position; X: body-center position; M: face-center position) and exhibit a wide
range of interesting physical properties Goodenough70 , such as giant magneto-
resistance Kim2001  and nearly zero temperature coefficient resistivity
Chi2001 . They have renewedly attracted considerable attention due to the
discovery of superconductivity at $\sim$8 K in intermetallic compound MgCNi3
He01 .

Considering the Ni-rich composition, it is expected that the ferromagnetism
could exist in MgCNi3. However, the absence of ferromagnetism was observed in
experiment He01  for MgCNi3. From the electronic structures obtained by the
first-principles calculations Singh01 ; Shim01 ; Rosner02 ; Wu2009251 , the
non-ferromagnetic ground state of MgCNi3 is ascribed to a reduced Stoner
factor that results from a strong hybridization between the Ni-3$d$ and C-2$p$
electrons. For other Ni-based ternary carbides ACNi3 (e.g., A = Al, Ca, In,
Zn, and Cd), the first-principles calculations Wu2009251 ; Okoye05 ;
Sieberer07 ; Zhong07 ; Wu20084232  also show that the ground states of these
compounds are non-magnetic and the C-Ni bonding exhibits nearly same
characteristics as the one in MgCNi3. Therefore, this indicates that the
change of composition A could not induce the ferromagnetism in ACNi3. On the
other side, it raises a question whether the change of composition X or M can
lead to the appearance of ferromagnetism in AXM3 or not.

Very recently, the antiperovskite-type compounds InNyCo3 and InNyNi3 ($y\sim$
1.0 and 0.8, respectively) have been synthesized by solid-gas reactions of
metal powders with NH3 and they have been reported to have spin-glass-like
properties based on the measurements of temperature dependence magnetization.
Cao093353  The recent first-principles calculations Sieberer07  showed that
the non-stoichiometry could affect the magnetic properties of ACNi3 (e.g.,
AlCNi3 and GaCNi3) and suggested that the tendencies toward magnetism found in
experiments Dong05 ; Tong06Al ; Tong06Ga ; Tong07  for these compounds should
be explained by the deviation of the Ni/C atomic ratio from the ideal
stoichiometry. To shed more light on the understanding on the magnetic
properties of InNCo3 and InNNi3 reported in experiment Cao093353 , it is of
great importance to theoretically study the electronic structures of these two
compounds as well as the nature of the N-Co and N-Ni bondings.

In order to completely understand the electronic structures and magnetic
ground states of InNCo3 and InNNi3 with cubic anti-perovskite structure, we
carried out the first-principles calculations on these compounds using the
pseudopotentials method with plane-wave basis set within the local density
approximation and the generalized gradient approximation. Since the elastic
properties of a solid are highly associated with various fundamental solid-
state properties such as phonon spectra, specific heat, Debye temperature, and
so on, we have calculated the independent elastic constant and the elastic
moduli of InNCo3 and InNNi3.

##  II Computational details

All calculations on antiperovskite-type InNCo3 and InNNi3 were performed using
the Quantum ESPRESSO code Baroni , which is based on the density functional
theory (DFT) Kohn65 . The electronic exchange-correlation potential was
calculated within the local density approximation (LDA) Ceperley80 ; Perdew81
and the generalized gradient approximation using the scheme of Perdew-Burke-
Ernzerhof (PBE) Perdew96 . The spin polarization was also considered in the
calculation in order to assess the magnetic properties of these compounds.
Electron-ion interaction was represented by the norm-conserving optimized
Rappe90  designed nonlocal pseudopotentials. The 4 _d_ electrons are
explicitly included in the valence of In. The electronic wavefunctions were
expanded by the plane waves up to a kinetic energy cutoff of 55 Ry. The
k-point sampling in Brillouin zone (BZ) of simple cubic lattice was treated
with the Monkhorst-Pack scheme Monkhorst76  and a 20$\times$20$\times$20
k-point mesh (i.e., 286 irreducible points in the first BZ) was used. The
chosen plane-wave cutoff and number of k points were carefully checked to
ensure that the total energy was converged to be better than 1 mRy/cell. The
total energies are obtained as a function of volume and they are fitted with
the Birch-Murnaghan 3rd-order equation of states (EoS) Birch47  to give the
equilibrium lattice constant and other ground state properties. During the
calculation of density of states (DOS), a dense k-point mesh of
30$\times$30$\times$30 is used, the total DOS is computed by the tetrahedron
method Lehmann77 , and the atomic-projected DOS is calculated by the Löwdin
populations Portal95 .

For a cubic crystal, its independent elastic constants are $c_{11}$, $c_{12}$,
and $c_{44}$. To determine the elastic constants of InNCo3 and InNNi3 by means
of the curvature of the internal energy versus the strain curves Mehl93 ;
Wu07425 , three strain modes Hou081651  are adopted and their nonzero strains
are as follows: (1) $\epsilon_{11}=\epsilon_{22}=\delta$,
$\epsilon_{33}=(1+\delta)^{-2}-1$; (2)
$\epsilon_{11}=\epsilon_{22}=\epsilon_{33}=\delta$; and (3)
$\epsilon_{12}=\epsilon_{21}=\delta/2$,
$\epsilon_{33}=\delta^{2}/(4-\delta^{2})$. The deformation magnitudes $\delta$
from -0.012 to 0.012 in the step of 0.03 are applied in the first and second
strain modes, and $\delta$ from -0.04 to 0.04 in the step 0.01 are adopted in
the third strain mode. Once the independent elastic constants for single
crystal properties are obtained through the above procedure, the elastic
moduli (e.g., the shear modulus and the bulk modulus) of polycrystalline
aggregates can be estimated according to the Voigt-Reuss-Hill approximation
Voigt28 ; Reuss29 ; Hill52 . In the Voigt average Voigt28 , the shear modulus
and the bulk modulus of cubic lattice are given by

$G_{V}=\frac{1}{5}\left[(c_{11}-c_{12})+3c_{44}\right]$ (1)

and

$B_{V}=\frac{1}{3}(c_{11}+2c_{12}),$ (2)

while in the Reuss average Reuss29  they are given by

$G_{R}=\frac{5}{4(s_{11}-s_{12})+3s_{44}}$ (3)

and

$B_{R}=\frac{1}{3s_{11}+6s_{12}}$ (4)

with the relations $c_{44}=s_{44}^{-1}$, $c_{11}-c_{12}=(s_{11}-s_{12})^{-1}$,
and $c_{11}+2c_{12}=(s_{11}+2s_{12})^{-1}$ in the cubic lattice, where
$s_{ij}$ are the elastic compliance constants. Therefore, $G_{R}$ and $B_{R}$
in the cubic lattice can be rewritten as

$G_{R}=\left[\frac{4}{5}(c_{11}-c_{12})^{-1}+\frac{3}{5}c_{44}^{-1}\right]^{-1},$
(5)

and

$\displaystyle B_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left[(c_{11}+2c_{12})\right]$ (6) $\displaystyle=$
$\displaystyle B_{V}.$

In the Hill empirical average Hill52 , the shear modulus and the bulk modulus
are taken as $G=\frac{1}{2}(G_{V}+G_{R})$ and $B=\frac{1}{2}(B_{V}+B_{R})$,
respectively. Knowing $G$ and $B$, the Young’s modulus $E$ and Poisson’s ratio
$\nu$, which are frequently measured for polycrystalline materials when
investigating their hardness, can be calculated from the isotropic relations:

$E=\frac{9BG}{3B+G}$ (7)

and

$\nu=\frac{3B-2G}{2(3B+G)}.$ (8)

##  III Results and Discussions

###  III.1 Structural properties

In experiment with the powder X-ray diffraction patterns, Cao et al Cao093353
have reported that InNyCo3 and InNyNi3 ($y\sim$ 1.0 and 0.8, respectively)
have the cubic anti-perovskite structure with the space group
221($Pm\bar{3}m$) and the corresponding lattice parameters were 3.854 Å and
3.844 Å, respectively. Starting from the experimental data, we have calculated
the total energies of unit cell at a series of volumes for each compound in
the paramagnetic (PM) and ferromagnetic (FM) states. The results are presented
in Fig. 1. It is found that the energy difference between the FM and PM states
is -0.0397 eV (-0.226 eV) in the LDA (GGA) calculations for InNCo3 and 0.0 eV
for InNNi3. The total magnetic moment of InNCo3 is about 2.14 $\mu_{B}$ (2.91
$\mu_{B}$) and the local magnetic moment of each In ion is about 0.69
$\mu_{B}$ (0.94 $\mu_{B}$) in LDA (GGA) calculations. The total magnetic
moment of InNNi3 and the local magnetic moment of each Ni atom are zero. These
indicate that the ferromagnetic state is energetically favorable to InNCo3 and
the ground state of InNNi3 is paramagnetic state (non-magnetic). The obtained
equilibrium lattice constant ($a_{0}$), bulk modulus ($B$), and first pressure
derivative of bulk modulus ($B^{\prime}$) of InNCo3 and InNNi3 are listed in
Table 1. In our calculations, the predicted lattice constant of InNCo3 is
slightly larger than that of InNNi3, which is oppsite to the trend reported in
experiment Cao093353 . This may be due to the deviation of the Ni/N atomic
ratio from the ideal stoichiometry in experiment for InNyNi3. In addition, it
can be seen that the deviations of the LDA (GGA) lattice constants of both the
InNCo3 and InNNi3 with respect to the experimental values are less than 2.6%
(1.0%). That is to say, the calculated equilibrium lattice constants of InNCo3
and InNNi3 are in excellent agreement with the experimental data Cao093353 .

###  III.2 Elastic properties

The calculated independent elastic constants for single crystal of InNCo3 and
InNNi3 are listed in Table 2. Based on the Voigt-Reuss-Hill approximation
Voigt28 ; Reuss29 ; Hill52 , the elastic moduli of InNCo3 and InNNi3 are
estimated and the results are listed in Table 2. For the bulk moduli of InNCo3
and InNNi3, the estimations based on the independent elastic constants agree
well those obtained by the fit of the Birch-Murnaghan 3rd-order EoS. To the
best of our knowledge, no experimental data or theoretical results for the
elasticity of InNCo3 and InNNi3 compounds have been reported up to now.
Considering that the elastic properties of ZnNNi3, InNSc3, and InCNi3 with
cubic anti-perovskite structure have been studied recently and the theoretical
results are available in literature Maurizio09 ; Wu20084232 ; lichong09 ;
Shein10 , it will be meaningful to compare them with those of InNCo3 and
InNNi3 compounds. The order of bulk moduli of these five compounds from low to
high is: B(InNSc3) $<$ B(InNNi3) $<$ B(InCNi3) $<$ B(InNCo3) $<$ B(ZnNNi3).
This could be understood from the trend in lattice constants ($a$) of these
compounds (i.e., $a$(InNSc3) $>$ a(InNNi3) $>$ a(InCNi3) $>$ a(InNCo3) $>$
a(ZnNNi3)) as well as the relationship between bulk modulus and equilibrium
volume (i.e., $B\sim V^{-1}$) Cohen85 . The GGA lattice constants of InNSc3,
ZnNNi3, and InCNi3 are 4.411 Å Maurizio09 , 3.77 Å lichong09 , and 3.880 Å
Wu20084232 , respectively. The order of shear modulus from low to high is:
G(ZnNNi3 and InNNi3) $<$ G(InCNi3) $<$ G(InNCo3). Pugh Pugh54  has proposed
that a high $B/G$ ratio may be associated with better ductility whereas a low
value would correspond to a more brittleness, and the critical value
separating ductile and brittle materials is around 1.75. From the results of
$B/G$ ratio for InNCo3, InNNi3, InNSc3, ZnNNi3, and InCNi3, it is found that
only InNSc3 can be classified as brittle materials and others may be ductile
materials. Furthermore, InNNi3 seems to be more ductile than InNCo3.

For a cubic crystal, its mechanical stability requires that its three
independent elastic constants should satisfy the following relations Wallace72
:

$(c_{11}-c_{12})>0,c_{11}>0,c_{44}>0,(c_{11}+2c_{12})>0.$ (9)

These conditions also lead to a restriction on the magnitude of $B$:

$c_{12}<B<c_{11}.$ (10)

The predicted $c_{ij}$ values (see Table 2) for InNCo3 and InNNi3 satisfy
these conditions, indicating that cubic antiperovskite-type compounds InNCo3
and InNNi3 are mechanically stable.

###  III.3 Electronic structures

In order to understand the different magnetic ground states for InNCo3 and
InNNi3, we examined the electronic structures of these two compounds. For the
simplicity in discussion, only the GGA results are presented below. The
calculated electronic band structures along the high symmetry directions in
the Brillouin zone are shown in Fig. 2. As discussed above, for InNCo3 the
ferromagnetic state is energetically preferable to the paramagnetic state, and
hence it is clearly seen that the spin-splitting occurs in the bands around
the $E_{F}$. For InNCo3 the profile of majority spin bands looks roughly
similar to the one of minority spin bands, however only one band in the
majority spin bands crosses the Fermi level and the corresponding band with
same dispersion in the minority spin bands is unoccupied. In addition, two
bands in the minority spin bands across the Fermi level of InNCo3 (see Fig.
2(a)). For InNNi3 the ferromagnetic state is not energetically preferable to
the paramagnetic state, consequently, the majority and minority spin bands are
degenerated, that is to say, no spin-splitting occurs in the band structure
(see Fig. 2(b)). It is interesting to note that the whole feature of majority
spin bands of InNNi3 is very similar to the one of InNCo3. In order to reveal
the detailed character of band structure, the total density of states (DOS)
and the angular-momentum-projected DOS of each atom in InNCo3 and InNNi3 are
presented in Fig. 3. For both InNCo3 and InNNi3, the four bands from -9 eV to
-5 eV come mainly from the N 2 _p_ states and In 5 _s_ states, and the five
bands roughly from -5 eV to -2.5 eV have significant contribution from 3 _d_
-$t_{2g}$ states of transition metal atoms (Co/Ni) and the 5 _p_ state of In
atom. For the bands from -2.5 eV to 0 eV (i.e., the Fermi level), they are
dominated by the 3 _d_ states of transition metal atoms (Co/Ni) and have small
contribution from the 2 _p_ states of N atom. Because the number of 3 _d_
electrons in Co is one less than that of Ni, two minority spin bands composed
of the 3 _d_ -$t_{2g}$ states around the Fermi level are unoccupied in InNCo3,
while the counterpart in the InNNi3 are occupied. This also results in
different behavior of spin-splitting in InNCo3 and InNNi3. Due to the
significant spin-plitting around the Fermi level in InNCo3, the hybridization
between the Co 3d and N 2p states in InNCo3 are slightly weaker than the one
between Ni 3d and N 2p states in InNNi3(see Fig. 3).

The contributions of each kind of atoms to the DOS at the $E_{F}$ of InNCo3
and InNNi3 are listed in Table 3. The total DOS at the $E_{F}$ of InNCo3 is
about 2.761 states/eV per formula unit (f.u.) in the GGA calculations, which
is larger than that of InNNi3, and its main contribution comes from Co 3$d$
states which accounts for 87%. For InNNi3 in the GGA calculations, the
contribution of Ni 3d states to the the total DOS at the $E_{F}$ (i.e., 1.803
states/eV.f.u.) accounts for 72%. These indicate the 3d states of transition
metal atoms in InNCo3 and InNNi3 play dominant roles in the total dnesity of
states of these compounds.

In order to understand the bonding nature among the ions in InNCo3 and InNNi3,
we analyzed the charge density contours of InNCo3 and InNNi3 in the(110)
plane, as shown in Fig. 4. From Fig. 4, it is found that a certain amount of
charges are accumulated in the intermediate region between N and Co atoms in
InNCo3, and slightly more charges are accumulated intermediate region between
N and Ni atoms in InNNi3. This gives an evidence for the strong hybridization
between N and transition metal (Co/Ni) atoms, indicating that the N-Co and
N-Ni bondings exhibit strong covalent characteristics and the latter is
slightly stronger than the former. The similar bonding characteristics for
Ni-N atoms or Ni-C atoms were also found in other Ni-based ternary nitrides or
carbides AXNi3 Wu2009251 ; Wu20084232 ; lichong09 ; Shein10 . Therefore, our
results suggest that the magnetic properties of InNNi3 reported in experiment
Cao093353  are very likely due to the non-stoichiometry effect, which was also
found in the cases of AlCNi3 and GaCNi3 Dong05 ; Tong06Al ; Tong06Ga ; Tong07
; Sieberer07 .

##  IV Conclusions

In summary, we performed the first-principles calculations to study the
elastic and electronic properties of cubic antiperovskites InNCo3 and InNNi3.
Based on the Voigt, Reuss and Hill bounds, the shear, Young’s moduli and
Poisson’s ratio have also been estimated for the InNCo3 and InNNi3
polycrystals. The theoretically predicted equilibrium lattice parameters are
in good agreement with the available experimental data. Our calculations show
that the 3d states of transition metal atoms in InNCo3 and InNNi3 play
dominant roles near the Fermi levels. InNCo3 energetically prefers to the
ferromagnetic state. The magnetic ground state of InNNi3, which is same to
other Ni-based ternary nitrides or carbides with a cubic anti-perovskite
structure, is a stable paramagnetic (non-magnetic) state. This could be
understood from that the hybridization between Ni-3$d$ and N-2$p$ states in
InNNi3 is slightly stronger than the one between Co-3$d$ and N-2$p$ states in
InNCo3 because of the more 3d electrons in Ni.

## Acknowledgments

The author acknowledges support from National Natural Science Foundation of
China under Grant No. 10674028.

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Table 1:  Calculated lattice constants ($a$, in Å), bulk modulus ($B$, in GPa), and the first pressure derivative $B^{\prime}$ of bulk modulus for InNCo3 and InNNi3. $\Delta E_{\mathrm{tot}}^{\mathrm{FM-PM}}$ (in eV per formula unit)is the difference between the total energies of ferromagnetic (FM) and paramagnetic (PM) states for InNCo3 and InNNi3. The available experimental values are also listed.  | InNCo3 | InNNi3
---|---|---
| LDA | GGA | Expt. Cao093353  | LDA | GGA | Expt. Cao093353
| PM | FM | PM | FM |  | PM/FM | PM/FM |
$a$ | 3.744 | 3.753 | 3.835 | 3.855 | 3.8541 | 3.784 | 3.882 | 3.8445
$B$ | 255.78 | 243.06 | 211.29 | 194.17 |  | 226.91 | 179.93 |
$B^{\prime}$ | 4.497 | 4.483 | 5.570 | 5.568 |  | 4.761 | 4.281 |
$\Delta E_{\mathrm{tot}}^{\mathrm{FM-PM}}$ | - | -0.0397 | - | -0.226 |  | 0.0 | 0.0 |

Table 2: Calculated elastic constants ($c_{11}$, $c_{12}$, and $c_{44}$, in GPa), shear modulus ($G$, in GPa), Young’s modulus ($E$, in GPa), and Poisson’s ratio ($\nu$) of InNCo3 and InNNi3. The Voigt shear modulus ($G_{V}$, in GPa) and the Reuss shear modulus ($G_{R}$, in GPa) are also presented. For comparison, the elastic properties of InNSc3, ZnNNi3, and InCNi3 compounds with a cubic anti-perovskite structure are listed. Compound | Method | $c_{11}$ | $c_{12}$ | $c_{44}$ | $G_{V}$ | $G_{R}$ | $B$ | $G$ | $E$ | $\nu$ | $B/G$
---|---|---|---|---|---|---|---|---|---|---|---
InNCo3 | LDA | 389.11 | 171.12 | 102.55 | 105.13 | 105.03 | 243.78 | 105.08 | 275.63 | 0.311 | 2.320
| GGA | 317.54 | 126.76 | 94.98 | 95.14 | 95.14 | 190.35 | 95.14 | 224.67 | 0.286 | 2.001
InNNi3 | LDA | 356.77 | 164.23 | 69.06 | 80.35 | 78.12 | 228.41 | 79.24 | 212.94 | 0.344 | 2.895
| GGA | 274.08 | 131.20 | 60.01 | 64.58 | 64.11 | 178.83 | 64.35 | 172.37 | 0.339 | 2.779
InNSc3 | GGA Maurizio09  | 238.57 | 54.28 | 90.76 | 91.31 | 91.31 | 115.71 | 91.31 | 216.88 | 0.188 | 1.267
ZnNNi3 | GGA lichong09  | 354.28 | 134.01 | 48.06 | 72.89 | 62.05 | 207.43 | 67.47 | 182.61 | 0.353 | 3.074
| GGA Shein10  | 364.20 | 124.90 | 32.69 | 67.47 | 46.09 | 204.67 | 56.78 | 155.92 | 0.373 | 3.604
InCNi3 | LDA Wu20084232  | 414.67 | 135.72 | 68.02 | 96.60 | 85.56 | 228.71 | 91.08 | 241.22 | 0.324 | 2.511
| GGA Wu20084232  | 344.20 | 106.30 | 62.68 | 85.19 | 77.31 | 185.60 | 81.25 | 212.70 | 0.309 | 2.284

Table 3: Total and partial density of states at the Fermi level ($N$($E_{F}$), in states/eV.f.u.) for InNCo3 and InNNi3. |  | Total density | Partial density of states
---|---|---|---
|  | of states | In | N | Co3/Ni3
InNCo3 | LDA | 3.811 | 0.073 | 0.337 | 3.401
| GGA | 2.761 | 0.144 | 0.169 | 2.448
InNNi3 | LDA | 1.580 | 0.160 | 0.212 | 1.208
| GGA | 1.803 | 0.180 | 0.256 | 1.367

## Figure Captions

  * •

FIG. 1: Total energy (in eV per formula unit) versus the atomic volume (in Å3)
for InNCo3 (upper panel) InNNi3 (lower panel).

  * •

FIG. 2: (Color online) Electronic band structures obtained with GGA for the
majority (red solid line) and minority (blue dashed line) spins of (a) InNCo3
and (b) InNNi3.

  * •

FIG. 3: Total and partial density of states (DOS) of InNCo3 (left panel) and
InNNi3 (right panel) obtained with the spin-polarized GGA calculations.

  * •

FIG. 4: Charge density contours of the (110) plane for (a) InNCo3 and (b)
InNNi3 obtained with the spin-polarized GGA calculations.

Figure 1:

Figure 2:

|
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Figure 3:

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Figure 4: