# Topological Quadrupole Phase in Two-dimensional C2N

Z. H. Li  Hunan Key Laboratory for Micro-Nano Energy Materials and Devices,
School of Physics and Optoelectronics, Xiangtan University, Hunan 411105,
People’s Republic of China  P. Zhou<EMAIL_ADDRESS>Key Laboratory of
Low-dimensional Materials and Application Technology, School of Material
Sciences and Engineering, Xiangtan University, Xiangtan 411105, People’s
Republic of China  Q. H. Yan  Hunan Provincial Key laboratory of Thin Film
Materials and Devices, School of Material Sciences and Engineering, Xiangtan
University, Xiangtan 411105, People’s Republic of China  X. Y. Peng
<EMAIL_ADDRESS>Hunan Key Laboratory for Micro-Nano Energy Materials
and Devices, School of Physics and Optoelectronics, Xiangtan University, Hunan
411105, People’s Republic of China  Z. S. Ma  Hunan Provincial Key laboratory
of Thin Film Materials and Devices, School of Material Sciences and
Engineering, Xiangtan University, Xiangtan 411105, People’s Republic of China
L. Z. Sun<EMAIL_ADDRESS>Hunan Provincial Key laboratory of Thin Film
Materials and Devices, School of Material Sciences and Engineering, Xiangtan
University, Xiangtan 411105, People’s Republic of China

###### Abstract

Quadrupole phase, as a novel high-order topological phase, exhibits nontrivial
gapless states at the boundaries whose dimension is lower than bulk by two.
However, this phase has not been observed experimentally in two-dimensional
(2D) materials up to now. In this work, using first-principles calculations
and tight-binding (TB) model, we propose that the experimentally synthesized
C2N is a 2D quadrupole topological insulator with one-dimensional gapped edge
states and zero-dimensional gapless corner states. C2N is found to have a
large bulk gap of 2.45 eV and an edge gap of 0.32 eV, making it an excellent
candidate to evidently present the nontrivial corner states in experiments.
The robustness of the corner states against the edge disorders has been
explicitly identified. Moreover, another three C2N-like materials are also
found to host the nontrivial quadrupole phase including an experimentally
synthesized material aza-fused microporous polymers (CMP). The four 2D
quadrupole topological phases proposed in our present work provide excellent
candidates for studying the novel high-order topological properties in future
experiments.

## INTRODUCTION

In recent years, topological insulators have attracted enormous attentions due
to their intrinsic topological nature and potential applications in many
fascinating areas, such as low-power electronic devices1, 2, 3, 4 and quantum
computing5, 6. A conventional $d$-dimensional topological insulator supports
gapless states on its ($d$-1)-dimensional boundaries, and the states are
insensitive to the local perturbations that preserve the nontrivial topology
of the bulk7, 8, 9. Recently, the notion of second-order topological
insulators (SOTIs) has been proposed to extend the bulk-boundary
correspondence, in which the gapless states appear in the corners of a 2D
system or the hinges of a three-dimensional system10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22. From a physical point of view, the low-dimensional
topological boundary states are determined by electric dipole or quadrupole
moments10, 11. So far, the topological corner states were mainly observed in
artificial 2D systems23, 24, 25, 26, 27, 28, such as square, Kagome, or
hexagonal lattices, in which the nontrivial phases are mainly derived from the
difference between intercell and intracell hopping parameters. However, the
hopping in electronic materials are generally more complicated. Therefore, we
need to go beyond the simplified model to find more potential SOTIs. Moreover,
although some experimentally synthesized materials, such as monolayer
graphdiyne29, 30, $\gamma$-graphyne31, large-angle twisted bilayer graphene32,
and honeycomb antimony33 have been predicted as SOTIs in their low-energy
gaps. Interestingly, recent research predicts twisted bilayer graphene and
twisted bilayer boron nitride34 as SOTIs in high-energy gaps. Unfortunately,
none of them has been experimentally verified yet. The main obstacle for the
confirmation of SOTI in a realistic system is that its bulk and edge states
gaps should be large enough so that the corner states can be isolated from the
bulk and edge states and not be destroyed by external perturbations.
Therefore, finding a SOTI fulfilling above requirements is an urgent task in
the area.
Recently, a 2D monolayer material with evenly distributed holes and nitrogen
atoms, named C2N-h2D (briefed as C2N hereafter), has been synthesized via a
bottom-up wet-chemical reaction35. Because of its tremendous application
prospects in electronics36, 37, 38, 39, 40, gas separation41, 42, 43, and
catalysis44, 45, 46, 47, 48, 49, 50, C2N has drawn much attention from both
theoretical and experimental researchers. It is generally believed that
monolayer C2N is just an ordinary semiconductor with an experimental band gap
of 1.9 eV. In this work, using density functional theory (DFT) and TB model,
we theoretically predict that the C2N monolayer actually is a realistic
example of 2D SOTI with non-zero bulk quadrupole moment, gapped edge states,
and topological corner states. The most desirable aspects of C2N are its large
bulk and edge state gaps, which will facilitate the detection of the corner
states in future experiments. Furthermore, we demonstrate that the topological
corner states are robust against disorder and C6 symmetry breaking. Similar
nontrivial two-order topological phase has also been found in other three
C2N-like 2D monolayers and one of them was synthesized experimentally.
Considering two of them have been synthesized in the experiments, C2N and its
analogues are excellent platforms for studying the excellent properties of
SOTIs.

Figure 1: (a) Crystal structure of C2N. The grey and blue spheres represent C
and N atoms, respectively. The orange and black dotted lines, respectively,
represent normal primitive and Wigner-Seitz unit cell of the monolayer; (b)
The 2D Brillouin zone of C2N; (c) The bulk electronic band structure of C2N
obtained by HSE06, blue and gray bands represent the subspaces originated from
pz and s+px+py orbitals, respectively; (d) The band structure of C2N obtained
from TB model.

## RESULTS

The 2D crystal structure of C2N monolayer is illustrated in 1 (a). It is a
typical 2D porous material and can be structurally obtained from a
$2\sqrt{3}\times 2\sqrt{3}$ graphene supercell by removing a carbon six ring
and then replacing the C atoms on the edge of the resulted holes with N atoms.
The original space group of C2N is P6/mmm (No. 191) and it can be generated by
the rotation operations C3z, C2z and mirror operations My, Mz. The lattice
constants relaxed by the first-principles calculations are a = b = 8.333 Å, in
good agreement with previous experiments35 and calculations37, 42, 43.
Considering the negligible spin-orbit coupling of C and N atoms, C2N can be
effectively treated as a spinless system. The bulk electronic band structures
of C2N obtained by HSE06 and PBE are presented in 1 (c) and Fig. S2 (b),
respectively. Since the different symmetry between pz and s plus px,y
orbitals, the bands derived from them form two independent subspaces, as shown
in 1 (c). Moreover, a global band gap of 2.45 eV is formed between the valence
band maximum (VBM) and conduction band minimum (CBM) at $\Gamma$ point. The
VBM and CBM mainly originate from the pz orbitals of both C and N atoms.
To study the topological properties of C2N, a TB model was adopted. In space
group P6/mmm, the inversion operation _I_ can be considered as the product of
C2z and Mz. Because the eigenvalues of Mz for the pz states are odd, the
eigenvalues of C2z and _I_ differ by a sign. Therefore, the eigenvalues of C2z
can also be used to calculate the quadrupole moment. Under the premise of
ignoring spin, there are 39 valence electrons in C2N, including 9 pz orbitals.
By analyzing the C2z eigenvalues of the occupied Bloch states at $\Gamma$ and
M points, we discover that the numbers of the states with eigenvalues of -1
are 18 and 20, respectively. Furthermore, as listed in Tab. S2, this
difference is entirely contributed by the pz orbital, indicating that the pz
states in C2N may form a nontrivial topological phase. To simplify the
analysis about the topological properties, we build a TB model with the basis
of pz orbitals as follows:

$\hat{H}=\sum_{i}\epsilon_{i}+\sum_{ij}\left(t_{ij}\hat{c_{i}}^{{\dagger}}\hat{c_{j}}+h.c.\right)$
(1)

In the model, only on-site energies ($\epsilon$C and $\epsilon$N) and nearest-
neighbor hopping between the pz orbitals ($\emph{t}_{C-C}$ and
$\emph{t}_{C-N}$) are considered. By fitting the TB bands to the occupied DFT
bands, we obtain the TB parameters $\epsilon$C = 0.78 eV, $\epsilon$N = -1.60
eV, $\emph{t}_{C-C}$ = -3.20 eV and $\emph{t}_{C-N}$ = -3.21 eV. As shown in 1
(d), the occupied TB bands are very close to the DFT bands derived from pz
orbitals. Furthermore, the irreducible representations (irreps) of the Bloch
states at high symmetry points obtained from first-principles method and TB
model match with each other well, as listed in Tab. S1. Namely, the TB model
can be effectively used to investigate the electronic topological properties
of C2N. The irreps of the Bloch states at the high symmetry points of
$\Gamma$, K, and M are closely related to the nontrivial second-order
topological phase of C2N, the details will be discussed below.
Firstly, we examine the topological properties of bulk C2N. Because of the
coexistence of the time-reversal and spatial inversion symmetries, the Chern
number of the 2D system must be zero. Moreover, the Z2 is always trivial for
spinless systems. Except the two topological indexes, the bulk quadrupole,
which is dependent on the dipole moments, can also be taken as a topological
invariant to characterize a SOTI. For 2D hexagonal materials, the bulk dipole
moment p can be defined as21, 51

$p_{1,2}=\frac{1}{2}\left(\sum_{n}q^{n}\text{ modulo
}2\right)e,\quad(-1)^{q^{n}}=\frac{\eta^{n}\left(M\right)}{\eta^{n}(\Gamma)}$
(2)

where _p_ 1 and _p_ 2 are the bulk dipole moments along the two reciprocal
lattice directions and the term of $\eta^{n}$ denotes the C2 eigenvalues of
the $n$th occupied band at $\Gamma$ or M. As for pz states, the occupied
number $n$ is 9. The numbers of the negative C2 eigenvalues for $\Gamma$ and M
points are 6 and 4 (as listed in Tab. S2), respectively, resulting in
vanishing bulk dipole moments11. The bulk quadrupole moment of C2N can be
further calculated from the bulk dipole moments by11

$q_{12}=\sum_{n}p_{1}^{n}p_{2}^{n}/e\text{ modulo }e.\\\ $ (3)

where the terms of $p_{1,2}^{n}=\frac{1}{2}\left(q^{n}\text{ modulo
}2\right)e$. The bulk quadrupole moment of C2N from the TB model is $q_{12}$ =
1/2 e, which is matched with the first-principles calculation with the basis
set of all _s_ and _p_ orbitals. There are two origins for the appearance of
SOTIs: nonzero polarization or quadrupole moment. In some lattices, the
Wannier centers of the occupied states are mainly used to analyze the
polarization of the systems. For example, in the breathing kagome lattice, the
mismatch between the Wannier centers and the lattice site is used to explain
the nontrivial corner states22. In these systems, the polarization is nonzero
and their quadrupole polarization is not defined. The corner states in these
systems are characterized by nonzero polarization. However, for a system with
C6 rotation, the polarization must be zero52, and the bulk quadrupole can also
be taken as a topological invariant to characterize a SOTI. Hence, C2N is a 2D
SOTI with a nontrivial bulk quadrupole moment.

Figure 2: (a) The sketch of Wyckoff positions of 1a (green dot) and 3c (red
pentagram); (b)The DFT band structure of pz orbital (blue solid line) and TB
band structure (red dashed line) from Wyckoff position 3c ($\epsilon$=-6.39
eV, tppπ=-0.27 eV).

The nontrivial quadrupole moment of C2N can be understood from the symmetry
properties and Wannier centers of the occupied states. For a one-atom-thick 2D
system, the entire two-dimensional plane is mirror-invariant, which means that
all states in the plane can be classified into two groups: odd and even under
the mirror operation Mz. The pz states are odd, whereas the s, px and py
states are even. In other words, all pz states form an independent subspace
and the in-plane mirror symmetry Mz can be utilized to differentiate pz states
from s, px and py states. Therefore, it is sufficient to use the subgroup P6mm
(No. 183) to investigate the pz states, and this subgroup can be generated by
the operations C3z, C2z and My. There is a C6 rotation operation along z-axis.
All Wyckoff positions of the space group are listed in Tab. S3. The C and N
atoms of 2D C2N locate at the Wyckoff positions of 12f and 6d with the local
point group of E and Cs, respectively. The induced irreps from the $p_{z}$
orbitals at the two positions can be obtained with the theory of elementary
band representations (EBRs)53, 54, 55, and the results are listed in Tab. S4
and Tab. S5. However, by summing the number of the valence electrons of C and
N atoms, only half of the bands are occupied, which makes the centers of the
Wannier functions of the occupied bands may not coincide with the positions of
the ions. For the systems with C6 rotation symmetry, the Wyckoff positions of
1a, 2b, and 3c are the maximal, and we only consider the irreps of the wave
functions at the high symmetry points of $\Gamma$, K, and M induced from them.
The 1a and 3c Wyckoff positions are labeled with green dots and red pentagrams
in 2 (a), respectively. The EBRs induced by the Wannier functions on these
Wyckoff positions can be found in the Bilbao Crystallographic Server56, 57. As
demonstrated in the supplemental material’s formula (1), we establish a series
of linear equations using the EBRs at $\Gamma$, K, and M as the basis set and
all the irreps of C2N’s valence bands as the solutions. A set of coefficients
represents a collection of possible EBRs for the valence bands. Certainly, the
coefficients of these equations must be integers. After solving these
equations, we discover that the nine valence bands can only be induced by the
sum of the Wannier functions of A13c, A23c, and B13c. The mappings between
them are

$\displaystyle A1^{3\mathrm{c}}$
$\displaystyle\mapsto\Gamma_{1}+\Gamma_{5}+M_{1}+M_{3}+M_{4}+K_{1}+K_{3}$ (4)
$\displaystyle A2^{3\mathrm{c}}$
$\displaystyle\mapsto\Gamma_{2}+\Gamma_{5}+M_{2}+M_{3}+M_{4}+K_{2}+K_{3}$
$\displaystyle B1^{3\mathrm{c}}$
$\displaystyle\mapsto\Gamma_{3}+\Gamma_{6}+M_{1}+M_{2}+M_{3}+K_{1}+K_{3}$

where A1 induces the three bands with the lowest energies, whereas the other
six occupied bands with higher energies come from A2 and B1. The EBRs of the
occupied bands indicate that the centers of Wannier functions locate at the
Wyckoff positions of 3c that clearly deviate from the ion positions. According
to their eigenvalues of symmetry operation , we can obtain the invariant
indexes of the three groups of occupied states17. Each index forms a free
Abelian additive structure, the non-zero summation of the indexes can be taken
as non-trivial criterion of the system. According to the eigenvalues of the C2
at M and the C3 at K, the indexes are

$\chi^{(6)}(A1)=(-2,0);\ \chi^{(6)}(A2)=(-2,0);\ \chi^{(6)}(B1)=(2,0).$ (5)

According to the theory of the index, the above three indexes actually
describe the same topological phase and they correspond to the same primitive
generator h${}_{3c}^{6}$17. Therefore, the pz occupied bands of C2N can be
seen as $h_{3c}^{6}\oplus h_{3c}^{6}\oplus h_{3c}^{6}$ . If the nine bands are
separated into the lowest three ones and the other six ones as shown in 1 (d)
, the nontrivial topological phase of C2N can be viewed as coming from the
lowest three bands because the index of other six bands is equal to (0, 0). If
we only consider the A1 orbitals at the 3c Wyckoff positions, the
configuration of the A1 orbitals is similar with a Kagome lattice, and the
three bands with the lowest energies for C2N are similar with the occupied
bands of the lattice22. The C2 eigenvalues at the $\Gamma$ and M for the
lowest three bands, as shown in 2 (b), indicate that the polarization of the
bands are $\frac{e}{2}$, 0 and $\frac{e}{2}$, giving rise to a zero total
polarization. However, the corresponding quadrupole moments are $\frac{e}{4}$,
0 and $\frac{e}{4}$, leading to a non-vanishing total quadrupole of
$\frac{e}{2}$. Moreover, the distribution of the C2 eigenvalues is the same as
the occupied bands of the Kekul$\acute{e}$-like hexagonal lattice, thus the
C2N and the Kekul$\acute{e}$-like hexagonal lattice21, 51, 58 share the same
second-order topological nontrivial properties.

Figure 3: Semi-infinite C2N structures of termination-a edge (a) and
termination-b edge (c). The projected edge density of states for termination-a
edge (b) and termination-b edge (d), respectively.

To study the edge states of C2N, we cut it into two semi-infinite structures
named as termination-a and -b, respectively, as shown in 3 (a) and (c). With
the help of the TB Hamiltonian, their edge density of states are calculated
and the results are presented in 3 (b) and (d). The results indicate that,
between the projected valence bands and conduction bands, there are two gapped
edge states for termination-a, whereas there is none for termination-b. The
termination dependent edge states can be explained by the filling anomaly17 in
C2N. As discussed above, the occupied states of C2N can only be induced from
the orbitals at the Wyckoff position 3c that clearly do not match with the
atomic positions of C2N. For the case of termination-a, the 3c positions just
locate on the boundary, which will produce a kind of charge imbalance
($\rho$). The value of the charge imbalance can be obtained based on
obstructed atomic limit approximation11 as
$\rho=\frac{\\#ions-\\#electrons}{2}|e|$, where #ions and #electrons are the
number of ions and electrons per unit cell, respectively. The value of the
$\rho$ is proportional to the number of the Wannier centers on one edge per
unit cell. If we assume the number of unit cell along the periodic direction
is Nx, for the termination-a edge, the #ions and #elec are 4Nx and 2Nx,
respectively and they are 2Nx and 2Nx for termination-b. After reducing the
number of primitive cells Nx, we can acquire $\rho=2|e|$ for the termination-a
and $\rho=0|e|$ for the termination-b. The value of $\rho/|e|$ is rightly
equal to the number of the edge states. Interestingly, the energy gap of the
two edge states, as shown in 3 (b), reaches up to 0.5 eV. Although the energy
gap of the two edge states will reduce to 0.32 eV when considering the Wannier
Hamiltonian of all orbitals, as shown in Fig. S1, the energy gap of the two
edge states will leave enough energy space to accommodate the corner states of
C2N that will be discussed below.

Figure 4: (a) Energy spectrum of the hexagonal-shaped C2N cluster with
different size. S is the number of holes in C2N cluster. (b) The total and (c)
the individual real-space wave function distributions of six in-gap
topological corner states, The radii of the spheres on each atom denotes the
absolute values of square modulus of wave functions.

Besides the gapped edge states, the existence of corner states is another
strong evidence for a SOTI. According to the theory of the filling anomaly,
the corner charge can be obtained from the symmetry properties of the occupied
bulk bands17:

$Q_{\text{corner
}}^{(6)}=\frac{e}{4}\left[M_{1}^{(2)}\right]+\frac{e}{6}\left[K_{1}^{(3)}\right]\bmod
e\\\ $ (6)

where $[M_{1}^{(2)}]$ ($[K_{1}^{(3)}]$) is the difference of the number of the
states whose C2 (C3) eigenvalue equal to 1 between $\Gamma$ and M (K) points.
We find that the terms $[K_{1}^{(3)}]$ and $[M_{1}^{(2)}]$ are equal to 0 and
2 for C2N, resulting in $Q_{\text{corner }}^{(6)}=1/2|e|$, in consistent with
the value of quadrupole moment q12 as discussed above. To reveal the
distributions of the charges, C2N is cut into a hexagonal cluster with
armchair edges, and its discrete energy levels are shown in 4 (a). Remarkably,
there are six in-gap states ($\psi_{0}$-$\psi_{5}$ as shown in the inset of 4
(a)) around the Fermi energy. The results also indicate that $\psi_{1}$
($\psi_{2}$) and $\psi_{5}$ ($\psi_{4}$) are degenerate. The real space
configuration of all the in-gap states can be visualized by their wave
function distributions as shown in 4 (b). The in-gap states are well localized
at the six corners. In 4 (c), we also show the real-space wave function
distributions of the six in-gap corner states separately. The distributions of
the corner states can be well explained by the symmetry of the cluster. The
point group of the hexagonal cluster is D6. Thus, we can use the irreps of D6
to analyze the characteristics of the eigenvalues and eigenvectors of the six
in-gap states. According to D6 point group, the in-gap states can be written
as the linear combination of the isolated pz states at the six corner as
follows:

$\displaystyle\psi_{m}$
$\displaystyle=a+\epsilon{{}_{m}}b+\epsilon{{}^{2}_{m}}c+\epsilon{{}^{3}_{m}}d+\epsilon{{}^{4}_{m}}e+\epsilon{{}^{5}_{m}}f$
(7) $\displaystyle\epsilon{{}_{m}}$
$\displaystyle=\mathrm{e}^{m\pi\mathrm{\emph{i}}/6}\qquad m\in{(0,1,2,3,4,5)}$

where _a_ -_f_ denote the isolated pz states at the six corners. According to
the theory of equivalence transformation59, the equivalence representation of
the pz states $\Gamma^{a.c.}$ can be written as the direct sum of
$\Gamma_{1}$$\oplus$$\Gamma_{3}$$\oplus$$\Gamma_{5}$$\oplus$$\Gamma_{6}$.
$\psi_{0}$ will remain unchanged under all the D6 symmetry operations, that
the irrep of $\psi_{0}$ is $\Gamma_{1}$. Analogous to the simple hypothetical
SH6 molecule59, $\psi_{0}$ state should be non-degenerate and its energy
should be the lowest one among the six in-gap states. The irrep of $\psi_{3}$
is $\Gamma_{3}$ with anti-symmetric characteristics, it should be non-
degenerate and its energy is the highest one among the six in-gap states.
$\psi_{1}$ and $\psi_{5}$ states conjugate each other and the irrep of the
combination of the two states is $\Gamma_{6}$, thus the two states should be
degenerate. Similarly, $\psi_{2}$ and $\psi_{4}$ are also degenerate because
they are complex conjugate each other, and the irrep of the two states is
$\Gamma_{5}$. It is worth noting that the splitting of the energy levels of
the in-gap states derives from the couplings between the corner states. The
splitting will reduce with the increase in the cluster size. Namely, the
corner states will become gapless in the limit of large size clusters, the
results can be found in 4 (a).

Figure 5: Crystal structures of C3N2 (a), C4N3 (b), and aza-CMP (c). The blue,
grey, and pink spheres represent N, C, and H atoms, respectively. (d)-(f) are
the corresponding energy spectra of the hexagonal-shaped clusters, and real-
space wave function distributions of six in-gap topological corner states, the
radii of spheres on each atom denote the absolute values of the square modulus
of wave functions.

For a concrete C2N cluster, the influence of relaxation of the chemical bonds
for the boundary atoms and external atom absorptions on its electronic
properties should be considered. Therefore, the robustness of the corner
states against external perturbations is investigated by adding a random on-
site energy to the edge atoms in TB model, and the results can be found in
Fig. S3. Despite the intensity of the disorder (the difference between highest
and lowest values of random on-site energies) reaches up to 1.0 eV that is
much larger than the gap of topological edge states, the eigenvalues of the
corner states still remain in the gap of the edge states. However, the
degeneracy of the in-gap states are destroyed. Moreover, although the total
real-space wave function distributions of all the corner states is very
similar to the case without disorders, its distribution symmetry is broken, as
shown in Fig. S3 (c). It means that strict symmetries of the cluster are not
necessary for the appearance of topological corner states as long as the bulk
and edge band gaps are not closed. Namely, the corner states of C2N cluster is
robust against external influence, which is favorable for experimental
observation.
The Wigner-Seitz unit cell of C2N as shown in 1 (a) indicates that the C2N
monolayer can be obtained by applying C6 rotation and translational symmetries
to a three-atom ligand (blue shadow area in the figure). Moreover, the ligand
can be replace with other odd number C-N atom chains to produce new CN
monolayers, such as C3N2 and C4N3 as shown in 5 (a) and (b), respectively.
Furthermore, partial and total N atoms in the ligand can be replaced by the
C-H atomic group to obtain CNH or CH monolayers. A well-known one with this
replacement is the experimentally synthesized aza-fused microporous polymers
as shown in 5 (c). We take C3N2, C4N3, and C5N2H as examples to investigate
their electronic structures and topological properties. The band structures
from first-principles calculations are presented in Fig. S4. The results
reveal that these three materials are semiconductors with the band gaps of 2.0
eV, 1.49 eV, and 1.64 eV for C3N2, C4N3, C5N2H, respectively. The results of
bulk quadrupole moments of the occupied states for all the three 2D materials
indicate that they are two-order topological insulators with the quadrupole
moments of 1/2 e. With the help of TB models only considering the nearest-
neighbor hopping of pz orbitals (the detail TB model parameters can be found
in Tab. S6), we find that similar corner states appear in the three materials,
as shown in 5 (d)-(f).

## CONCLUSION

In summary, the nontrivial topology of C2N is revealed by the nonzero bulk
quadrupole moment, gapped edge states, and in-gap topological corner states.
Detailed symmetry analysis reveals that the nontrivial topological properties
come from the three bands with the lowest energies. Further calculations
verify that the corner states of C2N are robust against edge disorders.
Moreover, other three C2N-like materials were found to be similar SOTIs,
including the experimentally synthesized aza-CMP. Our results will promote the
experimental detection of the corner states and provide realistic candidates
for studying the 2D two-order topological insulators in experiments.

## METHODS

Our first-principles calculations were carried out in the framework of
generalized gradient approximation (GGA) for exchange-correlation potential in
the form of Perdew-Burke-Ernzerhof (PBE)60 implemented in Vienna ab initio
simulation package (VASP)61, 62. A vacuum space of 15 Å was used to avoid
couplings between two neighboring slabs. The first Brillouin zone was
represented by the Monkhorst-Pack special k-point scheme of
7$\times$7$\times$1 grid meshes. The energy cutoff, convergence criteria for
energy, and force were set to be 500 eV, 10-6 eV, and 0.01 eV/Å, respectively.
The bulk band structures were calculated using HSE0663. The edge states of the
semi-infinite structure were obtained with the tight-binding method and
maximally localized Wannier functions (MLWFs)64, 64 constructed with the
wannier90 code.
This work is supported by the National Natural Science Foundation of China
(Grant No. 11804287, 11874315, 11574260), Hunan Provincial Natural Science
Foundation of China (2019JJ50577) and the Scientific Research Fund of Hunan
Provincial Education Department (18A051).

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## AUTHOR CONTRIBUTIONS

Z.H.L. performed tight-binding calculations, Z.H.L. and Q.H.Y. performed
first-principles calculations, and P.Z., X.Y.P., and Z.S.M. did theoretical
analysis. L.Z.S. supervised the project. All authors analyzed the data and
wrote the paper.

## COMPETING INTERESTS

The authors declare no competing interests.

## DATA AVAILABILITY

The datasets generated during and/or analyzed during this study are available
from the corresponding author on reasonable request.

##  1 Table of Contents