# Chemical Rules for Stacked Kagome and Honeycomb Topological Semimetals

Liqin Zhou  These authors contribute equally to this work. Beijing National
Laboratory for Condensed Matter Physics and Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  Fazhi Yang  These authors
contribute equally to this work. Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Shuai Zhang  Institute of Theoretical Physics, Chinese Academy
of Sciences, Beijing 100190, China  Tiantian Zhang<EMAIL_ADDRESS>Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190,
China

###### Abstract

We study the chemical rules for predicting and understanding topological
states in stacked kagome and honeycomb lattices in both analytical and
numerical ways. Starting with a minimal five-band tight-binding model, we sort
out all the topological states into five groups, which are determined by the
interlayer and intralayer hopping parameters. Combined with the model, we
design an algorithm to obtain a series of experimentally synthesized
topological semimetals with kagome and honeycomb layers, i.e., IAMX family (IA
= Alkali metal element, M = Rare earth metal element, X = Carbon group
element), in the inorganic crystal structure database. A follow-up high-
throughput calculation shows that IAMX family materials are all nodal-line
semimetals and they will be Weyl semimetals after taking spin-orbit coupling
into consideration. To have further insights into the topology of the IAMX
family, a detailed chemical rule analysis is carried out on the high-
throughput calculations, including the lattice constants of the structure,
intralayer and interlayer couplings, bond strengths, electronegativity, and so
on, which are consistent with our tight-binding model. Our study provides a
way to discover and modulate topological properties in stacked kagome and
honeycomb crystals and offers candidates for studying topology-related
properties like topological superconductors and axion insulators.

##  I Introduction

The simplest model with topological phases arises from the honeycomb lattice,
with the most famous material being graphene, in which the band structure
features a Dirac cone at K and a van Hove singularity (vHs) at M Castro Neto
_et al._ (2009); Geim and Novoselov (2007); Polini  _et al._ (2013);
Mecklenburg and Regan (2011); Zhang  _et al._ (2011); Sheikhi  _et al._
(2021); Weng  _et al._ (2015a), as shown in Figs. 1 (a-b). Recently, the
kagome lattice, a two-dimensional (2D) grid structure composed of corner-
sharing triangles, has garnered significant attention in condensed matter
physics due to its intriguing and diverse array of exotic phenomena Syôzi,
Itiro (1951); Jovanovic and Schoop (2022); Yin  _et al._ (2022); Neupert  _et
al._ (2022); Ptok  _et al._ (2023, 2021), including charge density waves Wang
_et al._ (2013); Yu and Li (2012); Tan  _et al._ (2021); Kiesel  _et al._
(2013), topological states Guo and Franz (2009); Liu  _et al._ (2019); Morali
_et al._ (2019); Yin  _et al._ (2018, 2019); Ye  _et al._ (2018), and quantum
spin liquids Yan  _et al._ (2011); Kelly  _et al._ (2016); Hermele  _et al._
(2008); Peng  _et al._ (2021). The kagome lattice shares similar properties in
the band structure with the honeycomb lattice, including the Dirac cone and
the vHs known as saddle point. Nonetheless, it also manifests distinct
characteristics of flat band Kiesel and Thomale (2012), as depicted in Figs. 1
(c-d). Recently discovered kagome superconductors, such as CsV3Sb5, have also
sparked curiosity regarding the potential of unconventional superconductivity
and intertwined phase in kagome lattice Ortiz  _et al._ (2020); Jiang  _et
al._ (2021); Li  _et al._ (2021); Chen  _et al._ (2021); Zhao  _et al._
(2021); Hu  _et al._ (2022); Li  _et al._ (2022); Mielke III  _et al._ (2022);
Zheng  _et al._ (2022); Zhong  _et al._ (2023).

Figure 1: Schematic diagrams of three lattices and their band structures.
(a-b) are for the honeycomb lattice, (c-d) are for the distorted kagome
lattice, and (e-f) are for the system with stacked kagome and honeycomb
lattice.

In the past, topological materials with kagome lattice and honeycomb lattice
were studied separately, including nodal-line semimetals and Weyl semimetals
Ozawa and Nomura (2019); Nie  _et al._ (2017); Zhang  _et al._ (2017, 2022a);
Lu  _et al._ (2017), among which studies on kagome lattices were mainly
focused on magnetic materials, such as magnetic Weyl semimetals Co3Sn2S2 Wang
_et al._ (2018); Liu  _et al._ (2019); Morali  _et al._ (2019) and Fe3Sn2 Ye
_et al._ (2019); Yao  _et al._ (2018); Biswas  _et al._ (2020). Thus, the
interplay between the kagome lattice and honeycomb lattice is still in the
early stage and needs to be explored on both the analytical model level and
materials level.

In this work, we first construct a minimal five-band tight-binding (TB) model
to study the topology of stacked honeycomb and distorted kagome lattice. Then,
based on the results of the TB model, we sweep through the inorganic crystal
structure database (ICSD) Hellenbrandt (2004) and discover 298 experimentally
synthesized new ideal topological materials, i.e., IAMX family with the
stacked distorted kagome and honeycomb lattice (IA = Alkali metal element, M =
Rare earth metal element, X = Carbon group element) Zhang  _et al._ (2019);
Song  _et al._ (2018a, b); Vergniory  _et al._ (2019); Po  _et al._ (2017).
The crystal structure of IAMX family is characterized by the hexagonal lattice
with Fe2P-type, with the space group of $P$-62$m$ (No. 189) Czybulka  _et al._
(1979, 1990), and the interlayer interactions of the IAMX family are chemical
bonds rather than the van der Waals forces comparing to the previous studied
kagome materials. We perform a high-throughput calculation for IAMX compounds
and discover four distinct topological node-ring configurations in the absence
of spin-orbital coupling (SOC), which is closely associated with different
chemical conditions, including the lattice constants of the structure,
interlayer and intralayer interactions, bond strengths, electronegativity, and
so on. Our numerical results match our TB model very well. Further calculation
shows that LiNdGe is not a superconductor with $T_{c}<10^{-4}$ K under the BCS
scenario, which implies IAMX family materials may be candidates for
unconventional (topological) superconductors when the superconductivity is
achieved by experiments. Our theory and findings offer a method for the stable
realization of topological states, particularly Weyl semimetals, in the
stacked kagome and honeycomb lattices. Combining the topological states with
the charge instability arising from vHs in kagome material Wang  _et al._
(2013); Yu and Li (2012); Tan  _et al._ (2021); Kiesel  _et al._ (2013), we
provide an effective pathway for achieving charge density wave (CDW) in the
Weyl semimetal state Wang and Zhang (2013); Roy and Sau (2015); Sehayek  _et
al._ (2020); Gooth  _et al._ (2019); Shi  _et al._ (2021). Our findings
present an efficient approach for exploring the axion insulator state in
kagome materials.

Figure 2: The flowchart for screening compounds with kagome lattice, honeycomb
lattice, and their stacking lattice. First, the lattice should be a hexagonal
one. Then, the crystal is classified based on their Wyckoff positions. If the
Wyckoff position has 2? (? stands for different letters to distinguish
different Wyckoff positions with the same site symmetry), the compound has an
isolated honeycomb lattice; If the Wyckoff position has 3?, the compound has
an isolated kagome lattice; If the Wyckoff positions have both 2? and 3?, the
compound has the stacked structure of kagome and honeycomb lattice. We obtain
298 IAMX compounds with stacked lattices and ideal topological properties,
which will be further proceeded with high-throughput calculations based on
DFT.

##  II method

The minimal five-band TB model for stacked kagome and honeycomb lattice
without SOC is shown in Eq. (1), with $p_{z}$ orbital at Wyckoff position 1a,
forming a honeycomb lattice, and $d_{z^{2}}$ orbital at Wyckoff position 3f,
forming distorted kagome lattice. The model belongs to space group $P$-62$m$
(No.189) and with the next-next-neighbor hopping, as shown in Fig. 3 (a) Zhang
_et al._ (2022b). Fig. 3 (c-h) are the node-ring configurations obtained in
the tight-binding model without SOC with different parameters, as marked by
the blue lines. After considering SOC, these nodal rings will evolve into Weyl
points marked by red dots.

The material screening process follows the flowchart shown in Fig. 2. We first
pick out all compounds with hexagonal lattices, then diagnose whether the
compounds are stacked kagome and honeycomb lattices by the atomic Wyckoff
positions. If atoms are located at Wyckoff positions of 3? or 2?, there will
be a kagome and honeycomb sublattice. Here, 3? means there will be 3 symmetry-
related atoms in the unit cell, and ? represents different letters for
different Wyckoff positions with the same site symmetry. In this work, we
discover 400 IAMX family candidates for DFT calculation, of which 298 are
ideal topological materials for further study.

High-throughput DFT calculations for the IAMX family are performed by Vienna
ab initio simulation package (VASP) Kresse and Hafner (1993). The exchange-
correlation functional is described by the generalized gradient approximation
(GGA) with the Perdew-Burke-Ernzerhof (PBE) functional type Blöchl (1994);
Perdew  _et al._ (1996). The cutoff energy for plane-wave basis is set to 520
eV and the BZ is sampled by 7$\times$7$\times$11 $\Gamma$-centered _k_ mesh.
The structural optimization is performed until forces on each atom are less
than 0.001 eV/$\AA$. To further obtain the topological information, we
constructed the tight-binding model with _d_ orbitals of M atoms and _p_
orbitals of X atoms based on the maximally-localized Wannier functions (MLWF)
method Mostofi  _et al._ (2014). The nodal-ring configurations are obtained by
WannierTools package Wu  _et al._ (2018).

Figure 3: (a) Lattice model, (b) band structure, and (c-h) topological
configuration for the stacked lattice tight-binding model. The blue lines
represent the nodal ring in the non-SOC case, while the red dots represent the
Weyl points in the SOC case. The shape of the nodal ring changes with various
$t_{2}$, with $\Delta e_{0}=e_{1}-e_{2},\Delta t_{z}=t_{z1}-t_{z2}$ as
depicted in the figure. The results of tight-binding model with parameters
$e_{1}=1,e_{2}=-1,t_{1}=-0.2,t_{z}=0.2,t_{z1}=-0.5,t_{z2}=0.05$, and $t_{2}=$
0.7 (d), 0.3 (b)-(c), -0.3 (e), -0.7 (f), -0.9 (g), -4 (h).

##  III Results and discussions

###  III.1 TB models

A minimal five-band TB model is constructed for the stacked honeycomb-
distorted kagome lattice, with $p_{z}$ orbital at Wyckoff position 1a (atomic
coordinate of $(1/3,2/3,1/2)$, forming the honeycomb lattice), $d_{z}^{2}$
orbital at Wyckoff position 3f (atomic coordinate of $(x,0,0)$, forming the
distorted kagome lattice), in the non-SOC case:

$\displaystyle H_{nSOC}$
$\displaystyle=\sum_{i\in\alpha,\beta}\epsilon_{i}c_{i}^{{\dagger}}c_{i}+\sum_{<i,j>\in\alpha}t_{1}c_{i}^{{\dagger}}c_{j}+\sum_{<i,j>\in\beta}t_{2}c_{i}^{{\dagger}}c_{j}$
(1)
$\displaystyle+\sum_{i\in\alpha}t_{z1}c_{i}^{{\dagger}}c_{i}+\sum_{i\in\beta}t_{z2}c_{i}^{{\dagger}}c_{i}+\sum_{i\in\alpha,j\in\beta}t_{z}c_{i}^{{\dagger}}c_{j}+h.c.$

where $\alpha$ and $\beta$ are the labels for kagome lattice and honeycomb
lattice. $t_{1}$ and $t_{2}$ are the hopping between nearest sites in two
sublattices, i.e., the intralayer hoppings in each sublattice, as illustrated
in Fig. 3 (a). $t_{z}$ is the interlayer hopping between two sublattices.
$t_{z1}$ and $t_{z2}$ are the hopping for the same sublattices in different
unit cells along the stacking direction, respectively. The Hamiltonian can be
expressed in the matrix form below:

$H_{nSOC}=\begin{pmatrix}H_{h}&H_{hk}\\\ H_{hk}^{{\dagger}}&H_{k}\\\
\end{pmatrix}$ (2)

$\centering H_{h}=\begin{pmatrix}e_{2}+2t_{z2}\text{cos
}k_{z}&t_{2}e^{-1/3i(2k_{x}+k_{y})}(1+e^{ik_{x}}+e^{i(k_{x}+k_{y})})\\\
t_{2}e^{-1/3i(k_{x}+2k_{y})}(1+e^{ik_{y}}+e^{i(k_{x}+k_{y})})&e_{2}+2t_{z2}\text{cos
}k_{z}\end{pmatrix}\@add@centering$ (3)

$H_{hk}=\begin{pmatrix}t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{1}}\cdot\mathbf{k})}&t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{2}}\cdot\mathbf{k})}&t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{3}}\cdot\mathbf{k})}\\\
t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{4}}\cdot\mathbf{k})}&t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{5}}\cdot\mathbf{k})}&t_{z}(-1+e^{ik_{z}})e^{i(\mathbf{d_{6}}\cdot\mathbf{k})}\end{pmatrix}$
(4)

$H_{k}=\begin{pmatrix}e_{1}+2t_{z1}\text{cos
}k_{z}&t_{1}e^{-i\mathbf{A}}(1+e^{i(k_{x}+k_{y})})&t_{1}e^{-i\mathbf{B}}(1+e^{ik_{y}})\\\
*&e_{1}+2t_{z1}\text{cos }k_{z}&t_{1}e^{-i\mathbf{C}}(1+e^{ik_{x}})\\\
*&*&e_{1}+2t_{z1}\text{cos }k_{z}\end{pmatrix}$ (5)

where $\mathbf{d_{1}}$ \- $\mathbf{d_{6}}$ are the vectors connecting the
nearest neighbor kagome lattice and honeycomb lattice sites
($\mathbf{d_{1}}=(x-1/3,1/3,-1/2)$, $\mathbf{d_{2}}=(-1/3,x-2/3,-1/2)$,
$\mathbf{d_{3}}=(2/3-x,1/3-x,-1/2)$, $\mathbf{d_{4}}=(x-2/3,-1/3,-1/2)$,
$\mathbf{d_{5}}=(1/3,x-1/3,-1/2)$, $\mathbf{d_{6}}=(1/3-x,2/3-x,-1/2)$) and
$\mathbf{A}$ \- $\mathbf{C}$ are the vectors connecting the nearest kagome
lattice sites ($\mathbf{A}=(x,1-x,0)$, $\mathbf{B}=(2x-1,x,0)$,
$\mathbf{C}=(x,2x-1,0)$).

Figure 4: (a) Crystal structure of the IAMX family. The green, yellow, and
purple balls represent IA, M, and X atoms, respectively. (b) BZ for the IAMX
family, where the light green, light yellow, light blue, and dark gray planes
are the $M_{1-10}$, $M_{120}$ and $M_{210}$ and $M_{001}$ planes,
respectively. (c) The energy splitting in the bonding process for the orbitals
of Nd and Ge atoms in the crystal around the Fermi level. (d) The band
structure of LiNdGe in the non-SOC case, with the projections of Ge $p_{z}$
orbital and Nd $d_{z^{2}}$ orbital labeled in blue and red, respectively. (e)
Wavefunctions for the energy bands at $\Gamma$ around the Fermi level, showing
the compositions of Ge $p_{z}$ orbitals and Nd $d_{z^{2}}$ orbitals.

By fine-tuning the parameters of the TB model, nodal-ring semimetals with
different configurations can be obtained, as the five different configurations
shown in Figs. 3 (d-h). The node-ring topology is protected by mirror
symmetry, thus ensuring their stable presence on the mirror planes Bzdušek
_et al._ (2016); Yang  _et al._ (2018); Fang  _et al._ (2016); Song  _et al._
(2018a); Fang  _et al._ (2015). After considering SOC, these nodal-ring band
crossings will open energy gaps, giving rise to the emergence of Weyl points
at generic momenta, as shown by the red dots in Fig. 3. We note that we only
tune $t_{1}$ and $t_{2}$, i.e., the intralayer hopping strengths within two
sublattices for simplicity, and keep the others fixed. Fig. 3 (b) shows the
typical band structure without SOC depicted by solid blue lines and with SOC
depicted by dashed red lines. The nodal ring pinned on the $k_{z}=0$ plane
shown Fig. 3 (c) is the band crossing with a filling number of 2, which will
evolve into 6 pairs of Weyl points after SOC is taken into consideration.
Figs. 3 (c-h) show the evolution of nodal-ring configuration as the parameter
$t_{2}$ increases. The single nodal ring centered at $\Gamma$ will cross the
BZ boundary and become several nodal rings centered at K as $t_{2}$ increases,
and it gradually transforms into a nodal ring centered at the $\Gamma$ point
in Fig. 3 (c), further evolving into multiple nodal-ring patterns depicted in
Fig. 3 (e-h) as $t_{2}$ increases. Such an evolution process indicates the
intralayer hopping strength in the sublattice will play an important role in
the topological configuration.

###  III.2 Scanning Materials with Stacked Kagome and Honeycomb Lattice

By following the flowchart shown in Fig. 2 and the minimal five-band TB model,
we discover that the IAMX family (IA = Alkali metal element, M = Rare earth
metal element, X = Carbon group element) are ideal topological materials with
stacked kagome and honeycomb lattice by scanning the ICSD. They belong to the
space group $P$-62$m$ (No. 189), which has three equivalent mirror symmetries
_M_ 1-10, _M_ 120 and _M_ 210, an inequivalent mirror symmetry _M_ 001
perpendicular to the $z$-axis, a threefold rotation symmetry $C_{3z}$ along
$z$-axis and a twofold rotation symmetry $C_{2(110)}$. IA atoms occupy the
Wyckoff position 3g, and M atoms are located at position 3f, forming a
distorted kagome lattice, as depicted in Fig. 1 (c). X atoms occupy two
nonequivalent Wyckoff positions 1a and 2d, and the latter ones form the
honeycomb lattice, as shown in Fig. 1 (a). Thus, the crystal of the IAMX
family can be regarded as composed of distorted kagome lattice and honeycomb
lattice stacking along $z$ direction, as depicted in Fig. 4 (a). Fig. 4 (b) is
the first BZ of the IAMX family, with $M_{1-10}$, $M_{120}$ and $M_{210}$
represented by light green, light yellow, and light blue planes, respectively,
and the dark gray plane represents the non-equivalent (001) mirror plane. It
is worth highlighting that the interlayer interactions of the IAMX family are
chemical bonds rather than the van der Waals forces, which is distinct from
previously discovered two-dimensional or quasi-two-dimensional materials
featuring the kagome lattice. Thus, the interplay between the kagome lattice
and honeycomb lattice may bring new physical properties for their 3D nature.

###  III.3 Orbital Analysis for the IAMX family

Since all the IAMX family materials share similar elemental components, band
structures, and topology, we will take LiNdGe as an example for further
analysis. In LiNdGe, $p$ orbital from Ge forms a honeycomb lattice, and $d$
orbital from Nd forms a distorted kagome lattice, and they will split into
different bands under the crystal field, as shown in Fig. 4 (c). After the
orbital hybridization between $d_{z^{2}}$ and $p_{z}$, bonding and antibonding
bands can be obtained around the Fermi level, forming the band structure
depicted in Fig. 4 (d). Fig. 4 (e) shows the orbital charge density
distributions of different bands forming the kagome and honeycomb bands near
the Fermi level, both the band structure and the orbital components can be
captured by the aforementioned five-band TB model.

###  III.4 Topological Analysis for the IAMX family

To study the topological information of the IAMX family compounds, we
calculate band structures, surface states, nodal-ring configuration, etc.,
through a high-throughput calculation, with SOC taken into account or not,
respectively. 298 ideal topological materials in the IAMX family are proceeded
for further analysis. Within this dataset, we categorize the configurations of
nodal rings into four main groups, as shown in Fig. 5: (i) multiple nodal
rings centered at K point on $k_{z}$ = 0 plane; (ii) a closed nodal ring
located on the $k_{z}$ = 0 or $\pi$ plane centered at $\Gamma$ or A; (iii) a
nodal-chain composed by six nodal rings on the mirror planes and an additional
nodal ring on $k_{z}$ = 0 plane centered at $\Gamma$; (iv) three nested nodal
rings on the mirror planes, accompanied by a nodal line at the intersection of
the $\Gamma$-Z path. These four types of nodal-ring configuration highly
depend on the chemical rule of the IAMX family, which will be discussed in
detail below.

###  III.5 Chemical Rules for The Topology

####  III.5.1 Lattice Constant and Bonding strength

To study the influence of the overall bonding strength on the topological
configuration, we categorize the distributions of 298 IAMX family materials
based on their lattice constants $a$ and $c$ of the compounds in Fig. 5, with
type (i)-(iv) node-ring configurations fully separated in four regions,
denoted by blue, black, red, and brown dots, respectively. We notice that
different nodal-ring configurations can be separated by three linear dashed
lines, showing a strong correlation between the topological configuration and
the lattice constants.

Figure 5: The distribution of four nodal-ring configurations under different
lattice constants $a$ and $c$ for the IAMX family. Different nodal-ring
configurations are labeled in different colors, separated by linear slashes.

Lattice constant $a$ corresponds to the parameters $t_{1}$ and $t_{2}$ in our
TB model since it exerts influence on intralayer hopping in kagome or
honeycomb lattice, while lattice constant $c$ corresponds to the parameter
$t_{z}$ since it specifically impacts the interlayer coupling. If $c$ is fixed
and $a$ decreases in IAMX family materials, the increase of intralayer
coupling will lead to a change of the nodal-ring configuration from the right
to the left of the region in Fig. 5. As $a$ decreases, type-(i) configuration,
i.e., nodal rings centered at K points on the $k_{z}$ = 0 plane, shrink
gradually and transform into type-(ii) nodal ring, i.e., a closed nodal ring
centered at $\Gamma$. When the closed nodal ring on the $k_{z}$ = 0 plane
gradually shrinks and disappears, six nodal rings emerge on the vertical
mirror planes, forming type-(iii) node-ring configuration, progressively
approaching the $k_{z}$ axis, and eventually converging into type-(iv) node-
ring configuration, i.e., three nested nodal-rings. As discussed in the TB
model part, different node-line configurations can be obtained by tunning the
parameter $t_{1}$, which is consistent with the node-ring configuration
transition from (i) to (iv) by decreasing the lattice constant $a$, in which
process the bond strength in each sublattice will be enhanced.

Figure 6: (a) The distribution of all IAMX family compounds with different
intralayer atomic distances in both the honeycomb lattice ($d_{h}$) and kagome
lattice ($d_{k}$). (b) The distribution of all IAMX family compounds with
different interlayer atomic distances between the honeycomb and kagome lattice
($d_{hk}$). Dots with different colors represent different node-ring
configurations, as shown in Fig. 5. All the data is distributed in a linear
way, showing a strong relation between the node-ring configuration and the
atomic distances in the IAMX family.

####  III.5.2 Intralayer and Interlayer bond strength

To explore the relationship between the topological configuration and the
intralayer and interlayer chemical bonds, we study the node-ring configuration
distribution of the IAMX family with different intralayer atomic distances in
the honeycomb lattice ($d_{h}$), kagome lattice ($d_{k}$) and the interlayer
lattice ($d_{hk}$), as shown in Fig. 6.

Figure 6 (a) illustrates the distribution relationship of all node-ring
configurations based on the distance between X atoms forming the honeycomb
lattice ($d_{h}$) and the distance between M atoms constituting the kagome
lattice ($d_{k}$). All the data points are predominantly aligned along a
straight line with the same gradient, which can be attributed to the linear
dependence of both $d_{h}$ and $d_{k}$ on the lattice constant $a$ for
different types of configurations. The intercepts of the lines, delineated by
points representing distinct node-ring configurations in the diagram, exhibit
correlation with the Wyckoff coordinates of the distorted kagome layer. Thus,
different types of configurations can be distinguished through different
magnitudes of intercepts, which highlights the significance of intralayer
hopping as a key contributing factor influencing the formation of nodal-ring
configurations.

Furthermore, we also analyze the distribution of topological configurations
based on the distance between the atoms comprising the kagome and honeycomb
lattice $d_{hk}$ and the distance between the atoms forming the kagome lattice
$d_{k}$, as depicted in Fig. 6 (b). The interlayer hopping leads to more
effective discrimination of various node-ring configurations, and
simultaneously, a continuous transition links the type-(i) and type-(ii) node-
ring configurations labeled by blue and black points, which is in alignment
with Fig. 5.

####  III.5.3 The Role of Chemical Bond Strength for Nodal-line Configuration

Figure 7:  The distribution of IAMX family materials with the type-(i)
configuration (blue points) and type-(ii) configuration (black points) with
respect to the lattice constant $a$ and nodal-ring radius, which are
distributed in the light blue and grey regions. These two kinds of
configurations are divided by a dashed line with a slope of
$\frac{1}{\sqrt{3}a}$, where $a$ is the lattice constant, showing that these
two configurations are similar in nature.  Figure 8: The distribution for the
radius of type-(iii) configuration materials (red points in Fig. 5) with
respect to lattice constant $a$.

Type-(i) configuration corresponds to a single closed nodal ring centered at
$\Gamma$. If the radius becomes larger, the nodal ring will cross the BZ
boundary and become six nodal rings centered at K, which will transform into
the type-(ii) configuration. In order to have a further study on the effect
role of chemical bond strength on node-ring configurations, we analyze the
relationship between the configurations of type-(i) and type-(ii) nodal rings,
i.e., the nodal-ring radius with respect to lattice constants $a$, as
illustrated in Fig. 7. These two configurations are well separated by a dashed
line with a slope of 1/($\sqrt{3}$a), and they show a continuity at the
boundary of these two regions. The increasing tendency for the radius of the
nodal ring as the lattice constant $a$ increases on a whole shows that there
is a strong correction between the topological configuration and the inplane
lattice constant. Since the chemical bond strength in the kagome and honeycomb
sublattices is also strongly corrected to the intralayer lattice constant, we
conclude that the intralayer chemical bond strength plays an important role in
modulating the topological configuration. However, we note that Fig. 7 shows a
downward trend in each local part, which will be further explored in Part.
III.5.4.

Similarly, we also analyze the type-(iii) materials (red points in Fig. 5) on
the relationship between the node-ring radius and the lattice constant $a$, as
depicted in Fig. 8. Overall, their distribution exhibits an increasing
tendency as $a$ increases, showing a strong correction between the node-ring
radius and the intralayer bond strength, which matches the results shown in
Fig. 7.

Figure 9: Further analysis of type-(ii) nodal-ring configuration in Fig. 5,
which corresponds to a single closed nodal-ring centered at $\Gamma$. The
tendencies are different for the radius of the nodal ring and the lattice
constant $a$ for different IA elements.

####  III.5.4 The Influence of Electronegativity on Topology

As we discussed above, even within the same node-ring configuration, the
relationship between the node-ring radius and the lattice constant $a$ may
locally show an opposite tendency compared to the global tendency, as shown in
Fig. 7. To have a deep insight into such phenomena, we take materials in
type-(ii) configuration (black points) as an example, and further divide them
into six groups, as shown in Fig. 9, with X taking the carbon element and IA
taking Li, Na, K, and Rb, respectively. Overall, the radius of the nodal ring
and the lattice constant $a$ are linearly related, but with different slopes
in these six groups. The slope of the linear dispersion decreases and even
becomes zero when the IA element varies from Li to Rb. If we fix the IA
element to Na, then the slope of the distribution gradually becomes negative,
as shown in Fig. 9 (e-f).

To gain further insight, we randomly selected one material from each group for
further calculation. Our results reveal that each of the three elements in the
IAMX family shows different valence states in each group. When the valence
state of IV group element X is greater than -2, the slope of the distribution
curve is positive. By contrast, when the valence of the X atom is less than
-2, the slope is negative. While the valence state of X approaches -2, the
slope of the curve tends to be around zero. These results suggest that the
electronegativity or electron affinity of the X element significantly
influences the bonding strength, leading to variations in the node-ring radius
and the topological configurations.

###  III.6 Topological properties of LiNdGe with SOC

Since materials within the IAMX family share similar crystal structures, band
structures, and topological properties, we choose LiNdGe as an example to show
the electronic structures and topological properties after taking SOC into
consideration.

The band structure of LiNdGe without SOC is depicted in Fig. 4 (d), where a
nodal ring is located on the $k_{z}=0$ plane, protected by the $M_{001}$
symmetry. When SOC is taken into account, all band crossings on the nodal ring
are fully gapped, generating six pairs of Weyl points around the mirror plane,
making LiNdGe a Weyl semimetal with energy bands contributed from the
distorted kagome and honeycomb lattice. The distribution of these Weyl points
is depicted in Fig. 10 (a), where blue/red dots represent Weyl points with
negative/positive monopole charge, and they are constrained by the $M_{001}$
and $C_{3z}$ symmetries. Figure 10 (b) shows the band structure crossing two
time-reversal-symmetry-related Weyl points along $P-\Gamma-P^{\prime}$, and
the energy of Weyl points is just 1.4 meV above the Fermi level, which means
that IAMX family materials are ideal platforms to study physical properties
induced by the Weyl fermions.

To have a better understanding of the topological information, we perform the
mirror Chern numbers (MCN) calculations on the two non-equivalent $M_{001}$
and $M_{1-10}$ planes based on the Hamiltonian obtained from Wannier90. Fig.
10 (c-d) indicate the MCN of $M_{001}$ is 2, while that of $M_{1-10}$ is 1.
According to the previous studies on TaAs Weng  _et al._ (2015b); Lv  _et al._
(2015), if two non-parallel mirror planes have different MCNs, each of the
four regions that these two mirror planes divide will contain an odd number of
Weyl points. In this case, we find that there are three Weyl points in each of
the four regions divided by $M_{001}$ and $M_{1-10}$, and 12 Weyl points in
the whole BZ. Conversely, if the MCNs of the two non-parallel mirrors are the
same, there will be 0 or an even number of Weyl points. Thus, in the regions
divided by any two mirror planes among $M_{1-10}$, $M_{120}$, and $M_{210}$,
there are 2 or 4 Weyl points in each region, thus there will also be 12 Weyl
points in total.

Fermi arc is one of the distinctive hallmarks of Weyl semimetals, and it
connects the projected Weyl fermions with opposite chirality on the surface
Weng  _et al._ (2016); Armitage  _et al._ (2018). Thus, we perform a surface
states calculation based on semi-infinite Green’s function method for LiNbGe
along (001) direction to demonstrate the existence of the Weyl fermions. 12
Weyl points in LiNbGe project onto 6 momenta (marked by red dots) on the
surface BZ and each of the projected points is composed of two Weyl points
with opposite chirality. Figure 10 (e) shows that there are six Fermi arcs
connecting two Weyl points that are the second nearest neighbor to each other.

Figure 10: (a) Nodal-ring configuration (green line, without SOC) and the
distribution of Weyl points (with SOC) in the BZ, where the blue and red dots
denote the Weyl points with opposite chirality. (b) The band structure with
SOC along the path $P-\Gamma-P^{\prime}$. (c-d) The average position of the
Wannier centers for occupied bands with mirror eigenvalue _+i_ for $M_{001}$
(c) and $M_{1-10}$ (d) planes, showing mirror Chern number of $-2$ and +1,
respectively. (e) The Fermi surface including Weyl Fermi arcs along (001)
direction for LiNdGe.

Here, we offer a perspective to understand the connection of Fermi arcs in
terms of MCN, since the number of Fermi arcs is constrained by the mirror
Chern numbers. The MCN of $M_{001}$ is 2, so there are two Fermi arcs cross
through half of $M_{001}$ on the projection surface that is perpendicular to
$M_{001}$. Likewise, the MCNs of $M_{1-10}$, $M_{120}$, and $M_{210}$ are all
equal to 1, so there is one Fermi arc crossing through the half of these three
mirror planes on the (001) surface, forming the Fermi arcs connection shown in
Fig. 10 (e).

##  IV Summary

We study the topological properties and their chemical rules for systems with
stacked kagome and honeycomb lattice, in both analytical ways based on the TB
model and numerical ways based on the DFT calculation. The minimal five-band
TB model shows that distinct node-ring semimetals and Weyl semimetals can be
obtained by tuning the parameters, in the non-SOC case and SOC case,
respectively. Combined with the TB model, we propose an algorithm for fast-
scanning the materials with stacked kagome and honeycomb lattices in the ICSD
and carry out a high-throughput calculation on their physical properties. 298
ideal topological semimetals of the IAMX family are proposed, and their
topological features are fully captured by the five-band TB model. Further
analyses reveal that different topological configurations in IAMX family
materials are determined by lattice constants, intralayer and interlayer bond
strength, electronegativity, and so on. Our comprehensive study on the stacked
kagome and honeycomb crystals paves the way for future studies on both
topology-related physical properties in such systems and their interplay with
CDW, superconductivity, axion insulators and other physical properties in
kagome and honeycomb lattices.

##  V Acknowledgement

We acknowledge the support from the National Natural Science Foundation of
China (Grant Nos. 12047503 and 12374165).

##  Appendix A The DFT calculation results of some red dots

In this section, we show the DFT results of two materials from red dots (type-
iii) as shown in Fig. 11.

Figure 11: (a, b) The band structures of LiLuSn (a) and LiTmPb (b) without
SOC. (c, d) The corresponding nodal-line configurations of LiLuSn (c) and
LiTmPb (d).

##  Appendix B The superconductivity of LiNdGe

We calculate the superconducting transition temperature ($\text{T}_{c}$) based
on the BCS theorem Bardeen  _et al._ (1957). We perform the phonon and the
electron-phonon coupling matrix using the Quantum Espresso package Giannozzi
_et al._ (2009, 2017, 2020). The PBE functional Perdew  _et al._ (1996) with
PAW method Blöchl (1994) is used with the $f$-shell electrons of Nd were
treated as the semi-core electrons. The cutoff energy for wavefunctions is set
as 60 Ry and a $7\times 7\times 11$ $k$-mesh is set to get converged total
energy and charge density. We calculate the dynamic matrix at a $2\times
2\times 4$ $q$-point mesh. We use the EPW Giustino  _et al._ (2007); Poncé
_et al._ (2016) and the Wannier90 package Mostofi  _et al._ (2014) to
interpolate the electron-phonon coupling matrix to the full Brillouin zone (a
$20\times 20\times 40$ fine mesh) and calculate the integrated electron-phonon
coupling strength $\lambda$, the Eliashberg spectrum function $\alpha^{2}F$,
as shown in Fig. 12, with phonon dispersion and DOS. According to the
McMillan-Allen-Dynes formula McMillan (1968); Allen and Dynes (1975), when we
assume the effective Coulomb repulsion $\mu*=0.16$, the Tc is less than
$10^{-4}$ K, which rule out the possibility of the BCS superconductor due to
the small electron-phonon coupling strength.

Figure 12: The phonon spectra, density of states, integrated electron-phonon
coupling strength $\lambda$, and the $\alpha^{2}F$ spectrum of LiNdGe.

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