# Deep learning approach to genome of two-dimensional materials with flat
electronic bands

A. Bhattacharya1,2<EMAIL_ADDRESS>I. Timokhin1 R.
Chatterjee3 Q. Yang1<EMAIL_ADDRESS>A. Mishchenko1
<EMAIL_ADDRESS>1Department of Physics and Astronomy, University of
Manchester, Manchester, UK  2Department of Mechanical Engineering, Indian
Institute of Technology Delhi, New Delhi, India  3Department of Physics,
Indian Institute of Technology Delhi, New Delhi, India

###### Abstract

Many-body electron-electron correlations play central role in condensed matter
physics; they govern a wide range of phenomena, stretching from
superconductivity to magnetism, and are behind numerous technological
applications. Two-dimensional (2D) materials with flat electronic bands
provide a natural playground to explore this rich interaction-driven physics,
thanks to their highly localized electrons. The search for novel 2D flat band
materials has since attracted intensive efforts, especially now with the
development of open science databases encompassing thousands of materials with
computed electronic bands. In this work, we automate the otherwise daunting
task of materials search and classification by combining supervised and
unsupervised machine learning algorithms. To this end, a convolutional neural
network was employed to identify 2D flat band materials, which were then
subjected to symmetry-based analysis using a bilayer unsupervised learning
algorithm. Such hybrid approach of exploring materials databases allowed us to
reveal completely new material classes outside the known flat band paradigms,
with high efficiency and accuracy. Our results present a genome of 2D
materials hosting flat electronic bands that can be further explored to enrich
the physics of electron-electron interactions.

Electrons in some materials have dispersionless spectrum, i.e. their energy
$E_{\mathbf{k}}$ is independent of momentum $\mathbf{k}$, resulting in the
formation of flat band, $E_{\mathbf{k}}\approx\mathrm{const}$. Vanishing group
velocity, $\nabla_{\mathbf{k}}E\approx 0$, suppresses kinetic energy, making
flat bands conducive to electron-electron interactions1. Flat bands originate
from spatial localization of wavefunctions in, for instance, $f$-orbitals of
lanthanides and actinides, lattice sites of atomic insulators, line- and
split-graphs of bipartite lattices, or twisted bilayer graphene
superlattices2, 3, where controlled engineering of the flat band has been
demonstrated. Flat bands often exhibit topological nontriviality, resulting in
a plethora of exotic physics like unconventional superconductivity in twisted
bilayer graphene 4, quantum anomalous Hall effect5, anomalous Landau levels6,
strongly correlated Chern insulators 7, Wigner crystallization 8, unusual
ferromagnetism 9, chiral plasmons10, or bulk photovoltaic effect11, and the
list goes on. Most of these were reported in two-dimensional (2D) materials
and have triggered intense search for new 2D materials hosting flat bands and
novel physics. Some have focused on theoretical 2D lattices hosting
dispersionless bands, like Lieb, kagome, or dice12. Meanwhile, progress in
open materials science platforms, such as Materials Project13 and Aflow 14,
provides thousands of possible exfoliable 2D candidates from their 3D
counterparts 15, 16, 17, forming large databases made solely of 2D materials,
like C2DB 18, 2D materials encyclopedia 19, and MC2D 16. 2D lattices hosting
flat bands, mostly theoretically, can then be identified for further
investigation. Alternatively, a symmetry-based approach resorting to known
flat dispersions in line- and split-graphs of bipartite lattices could also
help to predict new 2D lattices20, 1. However, with a limited number of known
2D flat band materials, current symmetry-based studies lack the statistical
significance to make accurate predictions.

In this work, we use a combination of these two approaches - first, identify
2D flat band materials in the vast database using the convolutional neural
network deep learning approach; the identified lattices are then classified
based on their structural fingerprints, employing symmetry-based clustering.
Our deep learning algorithm surveys the entire database with high throughput
and precision. The resulting flat band sublattices are presented as clustering
charts, serving as a roadmap for 2D flat band lattices in the future. Our
work, therefore, offers a comprehensive and efficient path towards 2D flat
band materials and, more importantly, the exciting physics they enable.

The search of flat dispersions in 2D materials has been attempted, where band
flatness was defined by an arbitrary fixed bandwidth 21, 22. However, the
identification of flat bands is not a straightforward task, even for simple
band structures. When an electronic band is parametrized, the band index is
determined by the order in which it appears in the energy scale. However,
crossings between bands can change the band index, which could lead to large
bandwidth even when a flat band exists. Hence, using parametrized bands and
predefined bandwidth to identify flat bands may largely underestimate their
number, therefore, risk overlooking potential flat band materials. To account
for this, we put forward an unusual yet more inclusive approach, utilizing the
band structure images from a database rather than the parametrized bands
themselves. Current databases still lack the complete description of
wavefunctions, which is a prerequisite for systematic classification of the
electronic bands using machine learning algorithm based on, for example,
topology, or other features23, 24, 25, 26. However, the easily accessible band
structure images offer a timely alternative to help identifying flat features
in the bands, allowing us to harvest these 2D databases already at such an
early stage while more development in data science to enrich their contents is
on the way.

Our deep-learning-assisted framework enables high-throughput identification of
flat bands from any database with images of band structures. Here we choose 2D
Materials Encyclopedia (2Dmatpedia), currently the largest open 2D materials
database, as the source of the band structure images. A convolutional neural
network (CNN) was trained and applied to identify a genome of flat band
materials across 2Dmatpedia. Afterwards, symmetry-based analysis using
unsupervised machine learning algorithm was employed to classify flat-band
materials into clusters, based on the structural fingerprints of identified
corresponding sublattices. This classification framework helps to retain
subtle connections among structural nuances, enabling the identification of
hierarchical classes within the flat band 2D materials genome. Perhaps more
importantly, it allows the prediction of entirely new flat band 2D materials
outside the known paradigm, quickly expanding our 2D flat band materials
toolbox. Our work, therefore, presents a new path towards the exploration of
the rich and exciting interdisciplinary field across data science and physics.

## Results

Our protocol for the hybrid approach (supervised deep learning for flat band
identification + unsupervised machine learning for symmetry-based clustering)
is outlined in Fig. 1. First, CNN is trained and used to identify flat bands
by examining all the band structure images across 2Dmatpedia, allowing us to
reveal materials with flat bands spanning the whole Brillouin zone (BZ).
Second, a crystal sublattice which has the maximum projected density of states
(DOS) near the flat band is assigned as the sublattice responsible for the
flat dispersion. Third, a structure descriptor algorithm
CrystalNNFingerprint27 is used to calculate structural similarity of the
sublattices with predefined local coordination templates and assign a
structure fingerprint. Finally, the vectorized fingerprints of the structures
were fed into a bilayer unsupervised clustering module, with density-based
hierarchical algorithm HDBSCAN28 as the inner layer and t-distributed
Stochastic Neighbourhood Embedding29 (t-SNE) as the output layer. This module
clustered the flat band material population into stratified isostructural
groups. All the developed code is available at https://github.com/Anupam-
Bh/ML_2D_flat_band.

Figure 1: Architecture of combined supervised and unsupervised machine
learning algorithms used in this work. CNN was trained to identify flat band
materials using segmented band structure images from the database, followed by
identifying sublattices that are responsible for flat dispersion using element
projected DOS. Then, density-based clustering combined with t-SNE was used to
classify the assigned structural fingerprints, and to identify classes of flat
band 2D materials.

### Identification of flat bands using CNN

For flat band recognition, we used CNN, which was trained on a subset of the
Materials Project database, see Methods (Convolutional neural network). Figure
2a highlights one of the essential steps, image segmentation, which provides
two main benefits. First, vertical segmentation allows identifying the
location of a flat band with respect to the Fermi energy. Secondly, horizontal
segmentation of the band structures at high-symmetry points enables us to find
whether the dispersion is plane-flat (flat in the whole plane) or line-flat
(flat only along one direction) across the BZ.

Figure 2: Identification of flat bands from band structure images using
supervised machine learning. (a) Segmentation of band structure images from
2Dmatpedia, and identification of flat band segments. Left panel: segmentation
of band structure image of SnCl4 [2dm-13] horizontally into four 0.5 eV energy
strips, and vertically along high symmetry points in k-space, as denoted in
green, blue, and yellow for $\Gamma\rightarrow\mathrm{X}$,
$\mathrm{X}\rightarrow\mathrm{M}$, and $\mathrm{M}\rightarrow\Gamma$ paths,
respectively. Right panel: segmented band structure in the [-1,1] eV range and
predicted outputs. Segments with flat bands are marked in red frames their
corresponding flatness score is shown on top. (c) Histogram of compound
flatness scores of all the materials in 2Dmatpedia.

In this study we focused on plane-flat 2D materials, as their electrons have
suppressed kinetic energy in wider momentum space, promoting electron-electron
interactions. Figure 3a shows an example of plane-flat band material. For
comparison, we show an example of line-flat band in Fig. 2a, where two flat
segments (outlined in red boxes) belong to different energy strips. After all
the materials with plane-flat bands have been identified (scored more than 0.5
along all the k-paths within an energy bandwidth), each material is assigned a
compound flatness score between 0 to 1, which can be used to estimate the
relative flatness of bands.

The number of materials as a function of their compound flatness score is
shown in Fig. 2b, with a major peak near flatness score of 1. We found 2127
plane-flat materials out of total 5270 materials in 2Dmatpedia.

Figure 3: Identification of flat band sublattices. (a) Band structure of an
identified flat band material example Pd3P2S8. (b) Element projected DOS from
(a) reveals the element responsible for flat band is Pd. (c) The crystal
structure of Pd3P2S8. (d) Kagome sublattice of Pd remains after stripping off
P and S elements from Pd3P2S8 structure.

### Identification of flat band sublattices

To build a connection between the identified flat band materials and their
structural features, we propose a conjecture based on spatial localization of
electrons forming a flat band. Let us assume a spatially localized region
outside which electronic density approaches zero, formed either by the
interference of wavefunctions or through other mechanisms. If this localized
state originates from orbitals from multiple types of atoms, to keep the
eigenvalue constant throughout the span of the localized state (plane-flat
materials), there has to exist an accidental degeneracy between the states of
those multiple types of atoms. Therefore, for most of the observed flat bands,
the constituent electrons originate from a single element in the compound,
assuming that the probability of accidental band degeneracy is low. Our
conjecture is supported by several studies showing that it is the elemental
sublattices which obey specific lattice and orbital symmetries that lead to
flat bands 30, 31, 32, 22. For example, in intermetallic CoSn, the flat band
is attributed to the kagome sublattice of the transition metal element 30, and
in HgF2, it is the diamond-octagon sublattice of mercury 21.

This conjecture allows us to use flat band element sublattice instead of full
crystal structure for further analysis, largely simplifying the process while
maintaining high accuracy. To this end, we first extract the element
sublattice with the highest orbital contribution to the flat band. Here,
element projected DOS (Fig. 3b) is used to identify the element with maximum
density contribution, ignoring the orbital mixing for classification purposes.
As shown in Fig. 3, using flat band material Pd3P2S8 as an example, after the
energy segment containing flat band is identified (green segment in Fig.
3a-b), the corresponding bandwidth in the element projected DOS is analyzed to
obtain the element which has maximum projected DOS, and its sublattice. In
this case, it is element Pd, and its sublattice, as shown in Fig. 3d. The
lattice structure of Pd3P2S8 (Fig. 3c) is stripped off of P and S atoms to
segregate the Kagome sublattice of Pd (Fig. 3d). As we conjecture, the
symmetry operations relevant to this elemental sublattice lead to the flat
band, thus only the sublattice with the chosen element for each compound is
kept for further analysis.

The extracted flat band sublattices are then subjected to a structure
descriptor, CrystalNNFingerprint 27 and represented as a 244-dimensional
vector to facilitate further classification using unsupervised machine
learning algorithms, see Methods (Sublattice extraction and vectorization).

### Unsupervised machine learning: bilayer clustering

The vectorized flat band sublattices were further classified based on their
structural fingerprints. We use a complementary bilayer classification
algorithm to achieve optimal clustering. Density-based algorithm HDBSCAN 28 is
used to obtain hierarchical information and to identify closely-packed
clusters, while t-SNE29 was compensating for the tendency of HDBSCAN to
overlook local neighborhood information of the clusters. Details are given in
the Clustering module section in Methods.

Two parameters of HDBSCAN, minimum cluster size (MS), and sample size for
density calculation (SS), were tuned to obtain optimal clustering solutions.
The MS parameter trims off clusters which are smaller than its value and marks
their members as unclassified, while the density around each point is
calculated as a reciprocal relation to the distance of the point from its SS-
th nearest neighbor. Both MS and SS were varied in a wide range, from 3 to 20,
to obtain optimal clustering solutions. Performance of the clustering
algorithm was quantified using two indices, density-based clustering
validation (DBCV) 33, and cluster validity index (S_Dbw) 34. DBCV calculates
the intra-cluster and inter-cluster densities to estimate affinity between
objects inside a cluster in comparison to connectivity between clusters. A
higher DBCV index indicates a better clustering solution. The S_Dbw, on the
other hand, is expressed as the sum of intra-cluster variance as a measure of
cluster compactness, and inter-cluster density as a measure of separation. A
smaller S_Dbw index marks a better clustering solution. Figure 4a-d shows the
DBCV and S_Dbw indices, the number of clusters, and the number of unclassified
materials for different values of MS and SS. We notice that smaller values of
MS and SS result in better DBCV and S_Dbw, as well as a higher number of
clusters. The number of unclassified materials in Fig. 4d, however,
demonstrates a non-monotonic behavior with MS and SS: it first increases,
caused by the existence of groups of fingerprints that are difficult to
classify; then drastically drops, implying that many such dispersed
fingerprints suddenly get included into clusters, resulting in bad
classification. Therefore, the choice of MS and SS parameters is expected to
be smaller than the values at which such sudden change in Fig. 4d occurs, in
order to obtain the optimal solution.

Figure 4: HDBSCAN algorithm optimization. Graphs showing (a) DBCV index, (b)
S_Dbw indices, (c) number of clusters and (d) number of unclassified materials
as functions of minimum cluster size (MS) and sample size for density
calculation (SS).

For HDBSCAN, we choose MS = 7 and SS = 6 to map the coordination patterns into
the structure fingerprint space, yielding a total of 51 clusters and 1448
unclassified materials. Our results are shown as a phylogenetic tree of
clusters in Fig. 5a, and as a 2D t-SNE representation in Fig. 5b. A relatively
small number of clusters facilitates the identification of major groups of
structure fingerprints, although it yields a higher number of unclassified
materials. We also tried to classify our results using a finer clustering
solution with MS = 4 and SS = 3. Reduced MS and SS values improve the
performance of the HDBSCAN and give a much higher number of clusters (131
clusters). Detailed classification of all the flat band sublattice structures
for MS = 4 and SS = 3 is provided in Supplementary Information Fig. S1 and
Tables LABEL:mono-LABEL:multilayer.

Identifying structural groups from the HDBSCAN clusters has been largely
possible because of the soft clustering feature of HDBSCAN. This density-based
algorithm attaches a probability (representing the chance of being a member of
a cluster; more about this membership probability in Methods) to each member
of a cluster. The members with perfect probabilities or ‘exemplars’ help the
direct identification of sublattice represented by a cluster. Thus, we find
exemplar sublattice structures from each cluster; however, often different
clusters which are in close proximity, represent almost identical sublattices.
To identify this local neighborhood, we use t-SNE. We combine the sublattice
structures from nearly identical cluster exemplars to obtain the total list of
lattice structures responsible for flat dispersion. For MS=7 and SS=6, we
identify 28 such sublattice structures which are given in the Supplementary
Note Table LABEL:tab_7_6.

For the optimized t-SNE representation, distance information of the 90 nearest
neighbors for each structure fingerprint is preserved using perplexity = 30.
The principal component analysis is used to calculate the projection plane to
retain maximal global neighborhood information. The learning rate is
automatically adjusted according to

$\mathrm{learning\ rate}=\frac{\mathrm{sample\ size}}{\mathrm{early\
exaggeration\ factor}}$ (1)

where the early exaggeration factor is 12 for the first 250 iterations35, 36.
The total number of iterations was kept at 10000 to ensure convergence of
t-SNE.

Figure 5: Clustering solution from HDBSCAN with MS = 7, SS = 6. (a)
Phylogenetic tree expression of the hierarchical relations among clusters
represented by colored circles. The size of each circle is proportional to the
corresponding cluster’s volume, while the cluster index is indicated by the
number nearby. (b) t-SNE 2D visualization of the structure fingerprint space,
different coordination patterns are color-coded; ip is in-plane and oop is
out-of-plane coordination. Insets are representative coordination templates,
where atoms in different atomic planes of the templates are colored
differently.

### Evolution of the identified coordination patterns and lattice structures

The phylogenetic tree in Fig. 5a shows the hierarchy among identified clusters
of structurally similar lattices, which maps into the t-SNE plot in Fig. 5b.

By analyzing the clusters, 12 coordination patterns are revealed, shown as
shaded regions in Fig. 5b. Fuzzy boundaries between different coordination
patterns develop, as the coordination in each sublattice evolves gradually
throughout the embedded space. Most well-defined isostructural clusters are
found near the edges in the t-SNE plot, where the structure fingerprints
provide a satisfactory description of underlying lattice geometries. On the
contrary, many dispersed unclassified fingerprints appear closer to the
central region in the t-SNE plot. This can be ascribed to lattices with large
unit cells, which can be described by multiple coordination patterns
(typically, more than 3–4) in their lattice-sites. The average of their
multiply-coordinated structure fingerprints results in lattices that struggle
to be properly included in any single coordination geometry. Examples of such
dispersed fingerprints are seen in LiSb(PO3)4 with 24 oxygen sites, or AlPI8
with 16 iodine sites. Overall, finding an adequate structure descriptor for
such large-unit-cell lattices is still an open problem37. Some unclassified
examples are also found lying near clusters despite sharing similar structures
to those within the well-defined clusters, because of density-based
segregation.

Let us highlight the main features of these plots. We first notice that
clusters #41-43, corresponding to honeycomb lattice materials with planar
3-fold coordination of atoms, clearly stand out at the end of the phylogenetic
tree, Fig. 5a, and are positioned at the bottom of the t-SNE plot, Fig. 5b.
Their separation from the rest of the charts marks the distinctness of their
structural fingerprints. Moving to the top of Fig. 5b, we see the structure
evolution from in-plane hexagonal sublattice (clusters #26, #30 and #31) to
out-of-plane hexagonal lattice (#0) on the left, further to the out-of-plane
6-fold coordination sublattices (#4, #5, #9 and #29). Similarly, the
transition from in-plane 4-fold coordination (e.g., #22, #27, and #28) to out-
of-plane 4-fold coordination (#11, #12, and #14) occurs in the same manner.

Our t-SNE chart also captures the gradual evolution between different
coordination patterns, for example, from hexagonal to rectangular sublattices.
Moving downwards from the in-plane hexagonal sublattice, we first encounter
elongated hexagons (#17), which can also be represented as planar centered
orthorhombic lattice. Further distortion of the central atom creates planar
monoclinic lattice (#13 and #20) and planar orthorhombic lattice (#22, #27,
and #28). From here, evolution into the square lattice (#24 and #25) is
straightforward. Even the subtle structural differences, such as differences
in stacking distance, within the same coordination are also well reflected in
our clustering plot. For example, stacked hexagonal bilayer in 9-fold
coordination (6-fold in-plane and 3-fold out-of-plane) with short stacking
distance (#45) is excluded from the branch for longer stacking distance
coordination (#48-50), as shown in the phylogenetic tree. Even among the
clusters #48-50, we could identify slight in-plane distortion in #48 leading
to coloring triangle lattices whereas #49 and #50 show no such distortion
leading to simple AB stacked hexagonal lattice. Contrary to the well-defined
isostructures, clusters that don’t show any regular or identifiable
coordination patterns (e.g., #10, #33, #38, and #46) lie around the dispersed
region in the center of t-SNE plot.

## Discussion

Using our hybrid machine learning approach, we identified many sublattice
structures responsible for hosting flat bands. Among these, some have already
been confirmed in the literature, endorsing the validity of our approach. For
example, kagome, breathing kagome, sawtooth, square-chain (Creutz) lattices
are known to host flat bands 30, 38, 39, 40. But many of the lattices
identified in this work are entirely not explored, and the origin of their
flat bands remains to be clarified. Some of such monolayer sublattices are
listed below, alongside example materials followed by their 2Dmatpedia id.
These include Lieb-square lattice (Fe4S5 [2dm-4626], Os4S5 [2dm-1815]),
centered orthorhombic lattice (Li4VF8 [2dm-4139], VZnF4 [2dm-4036]), diamond-
chain lattice (ReS2 [2dm-4006], ReSe2 [2dm-3917]), $(3^{2},4,3,4)$ Archimedean
lattice (TiP [2dm-2672]), vertex shared square chain lattice
(Tl2Te5[2dm-1601]), linear chain lattice (NbS3[2dm-1482]), isolated
quadrilaterals (MoSe2[2dm-2007]) and centered monoclinic lattice (Ta2I5
[2dm-5407]). More examples are given in Table LABEL:tab_7_6 and LABEL:mono in
the Supplementary Information.

Bilayer flat band sublattices, in many cases, consist of stacked monolayer
lattices with flat bands. For example, stacked kagome, stacked square, stacked
centered orthorhombic, AB and AA stacked hexagonal, and stacked orthorhombic
lattices belong to this category. Among bilayer sublattices, stacked centered-
orthorhombic-square chains (SmF3 [2dm-875]) and isolated tetrahedra lattice
(LiNiPO4 [2dm-5215], Si(HgO2)2 [2dm-4520]), are newly identified flat band
lattices.

Meanwhile, several few-layer sublattices made of stacked monolayer flat band
sublattices were also identified, including stacked hexagonal and
orthorhombic, kagome-hexagonal, and coloring-triangle-hexagonal lattices.
Apart from this, we observed many interesting new multilayer sublattices with
flat bands. For example, sublattices that consist of in-plane octahedras,
where different octahedra connections generate seven new sublattices. These
are vertex sharing octahedra chain (YFeF5 [2dm-3862]), edge sharing octahedra-
parallel chain (Sc5Cl8 [2dm-4448]), vertex sharing planar octahedra (ZrF4
[2dm-5172]), vertex-edge sharing planar octahedra (CaTlCl3 [2dm-4337]),
isolated twisted octahedra (ReSeCl [2dm-5461]), edge-sharing zigzag octahedra
chain (ZnMoO4[2dm-3090]) and edge sharing octahedra chain (Y2Cl3 [2dm-3786]).
Furthermore, the stacked $(3^{2},4,3,4)$ Archimedean sublattice (CoBr4
[2dm-1906], MnF4 [2dm-5281])with out-of-plane distortion, is also identified
to host flat bands. Other new multilayer sublattices, e.g. vertex sharing
triangular pyramid chain (V2CuO6 [2dm-4387]), triangular bipyramid lattice
(MoO3 [2dm-5495]), distorted Kagome lattice (MgCl2 [2dm-4672]), planar
hexahedra lattice (TaF5 [2dm-4011]), and stacked orthorhombic-square (HgI2
[2dm-3966]) are also found to show flat bands. Future research is expected to
reveal the origin and the rich physics behind the majority of the potential
flat band lattices listed here.

Our hybrid approach combining supervised and unsupervised machine learning has
successfully demonstrated its accuracy and efficiency in navigating through
enormous information space to hunt for new flat band materials. This approach
can be conveniently adapted in searching for materials with intended
properties, and to mitigate issues arising from the quality of high-throughput
density functional theory calculations in existing databases. An interesting
direction is to identify topologically nontrivial flat bands, which show
crossings with other dispersive bands at high symmetry points in the BZ 41.
Features like band crossings, which are both common and small, may require
developing automatic graphical pattern search tools42. Moreover, in addition
to using structural fingerprints, such as lattice coordination patterns as
described in the current work, to distinguish different flat band materials,
electronic fingerprints, e.g., the similarity of electronic states43, may also
be attempted in the future to offer new perspectives towards the discovery of
new materials.

## Methods

### Convolutional neural network

The CNN trained for the identification of flat bands consists of 6 layers (as
shown in Fig. 1) and 6.9 million learnable parameters. It is a regression
network which predicts the existence of a flat band in the image segment and
outputs as a single value – the flatness score. The resolution of each segment
is set to $96\times 96$ pixels in consideration of both image quality and
computing time. The first convolutional layer consists of 30 channels of
$10\times 10$ filters with [1,1] stride. The second convolutional layer has 12
channels of $3\times 3$ filters with [1,1] stride. Sigmoid activation was used
and found to give better performance than softmax and ReLU activation as the
output of the network lies within [0,1]. The first fully-connected layer has
80 output nodes, whereas the second has a single output. For training
purposes, Adam44 optimization algorithm was used with learning rate $5\times
10^{-5}$ and L2-regularization parameter 0.003.

To train the CNN, band structures of approximately 1900 materials from the
Materials Project database were used. There are several reasons for selecting
the Materials Project database for training. First, it offers more training
flexibility compared to the relatively small database of 2Dmatpedia ($\approx
5300$ materials), avoiding applying the trained algorithm back to the same
database. Second, training the algorithm on 3D compounds that have inherently
more complicated band structures could improve its accuracy when applied to
the 2Dmatpedia. Finally, training the model on a general 3D database allows
future application to many other available materials databases.

The developed CNN model was then tested on 7000 image segments from Materials
Project. The test set predictions match 92.1$\%$ with manual selection,
whereas false negative and false positive predictions were seen in 3.3$\%$ and
4.6$\%$ of all cases, respectively. This is quite high precision since even
manual selection of flat segments could not guarantee 100$\%$ consistency, as
partially flat segments can often lead to confused judgment from a human
selector.

Before applying the trained CNN to 2Dmatpedia, band structures from 2Dmatpedia
in the energy range [-1,1] eV were segmented at high symmetry points of
k-space. Each resulting segment has $96\times 96$ resolution. For example, the
band in Fig. 2b is divided horizontally into four 0.5 eV energy bandwidths,
and vertically along k-path into three segments (as denoted in green, blue and
yellow). A threshold flatness score predicted from the trained CNN is used to
determine whether a band-structure segment is flat (score $\geq 0.5$) or non-
flat (score $<0.5$). We find a significantly higher fraction of flat band
segments in 2Dmatpedia, approximately 25$\%$, than that in 3D materials from
Materials Project (approximately 13%). Dimensionality reduction and
accompanied restriction in degrees of freedom of electrons in 2D materials are
likely to be the reason. To identify plane-flat band 2D materials, which have
flat bands throughout high symmetry lines and are of particular interest, we
select the energy strip in which all the horizontal segments are recognized as
flat. In the rare cases where multiple energy strips contain all-flat
segments, priority is given to the energy bandwidth where electrons are more
easily accessible, i.e. closer to the Fermi level.

### Sublattice extraction and vectorization

We use the element projected DOS, as shown in Fig. 3a-b, to identify the
element corresponding to the flat band sublattice. Because flat electronic
bands are accompanied by corresponding peaks in DOS, an element with maximum
density contribution in the flat band is most likely to be responsible for the
flat dispersion, although orbital mixing among different species is also a
contributing factor. Here we conjecture that the symmetry operations
pertaining to this elemental sublattice lead to the flat band, thus only the
sublattice with the chosen element for each compound is kept for further
analysis. After the energy segment containing flat band is identified, the
corresponding bandwidth in the element projected DOS is analyzed to obtain the
element and, subsequently, its sublattice.

A structure descriptor was then applied to a selected sublattice to create a
vector representation for machine learning algorithms. Here we use a structure
descriptor CrystalNNFingerprint 27, implemented in the Matminer package 45.
Other structure descriptors, like Coulomb matrix, Ewald sum matrix, Smooth
Overlap of Atomic Positions (SOAP), Many-body Tensor Representation (MBTR),
usually yield a constant length vector representing the structure 37, 46, 47,
were found less suitable for our analysis. CrystalNNFingerprint retains more
information when the number of atomic sites is high, thus particularly
suitable for identifying local coordination patterns in crystals even in the
presence of small lattice distortions. For example, popular descriptors like
SOAP calculating spectral average of the local fingerprints, would fail to
retain local structural information. CrystalNNFingerprint first determines the
local environment for each atomic site using a neighbor-finding algorithm
based on Voronoi decomposition. The resulting coordination pattern is then
compared with Local Structure Order Parameters (LoStOPs) which are different
coordination templates, and a 61-dimensional fingerprint vector is assigned to
the atomic site. The global fingerprint for the whole structure is then
represented by arranging the mean, standard deviation, maximum and minimum of
each local (atomic site) CrystalNNFingerprint within a structure. The final
structure fingerprint lies in a 244-dimensional vector space.

### Clustering module

We then used unsupervised machine learning module consisting of density-based
clustering algorithm HDBSCAN 28 and t-SNE 29 to further cluster
244-dimensional structure fingerprints. HDBSCAN creates clusters with
fingerprints that are densely populated, acting as a strict identifier of
similarity in coordination patterns with many fingerprints designated as
outliers. Simultaneously, soft clustering feature of HDBSCAN yields a
probability for each material to be included in a cluster. Consequently,
members of clusters with very high inclusion probability (exemplars)
facilitate straightforward identification of corresponding sublattice
structures. The exemplars are the members of a cluster which persist in the
cluster for the largest range of density. Hence, these may be identified as
the central members even if the shape of the cluster is fairly complicated as
they do not depend on any distance metric. However, HDBSCAN tends to assign
fingerprints with even small discrepancies into different clusters, risking
overlooking local neighborhood information. As a complement, we use t-SNE to
visualize and identify similarities among HDBSCAN clusters, acting as a second
layer of unsupervised classification. The nonlinear dimensionality reduction
capability of t-SNE allows both local neighborhood information and global
distance relations to be preserved, in comparison to other algorithms like
Isomap 48, Locally Linear Embedding (LLE) 49, Hessian eigenmapping 50, etc. We
also tested UMAP 51, no significant improvement over t-SNE is observed
(results are discussed in the Supplementary Information section II).

The choice of HDBSCAN as the first layer of unsupervised clustering algorithm
is based on the fact that more commonly utilized clustering algorithms, like
K-means and hierarchical clustering, would yield non-optimum segregation.
K-means algorithm solely depends on the distance among inputs, generating good
results only for inputs that form ellipsoid clusters. At the same time,
hierarchical clustering is too sensitive to the presence of noise in the data,
making it unsuitable in this context since lattice distortion is an avoidable
source of noise. For our 244-dimensional data we used Manhattan distance
metric, $L_{1}$ norm, since Euclidean distance metric ($L_{2}$ norm) becomes
less effective in calculating distances among points in high-dimensional
spaces 52.

## Data Availability

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

## Code Availability

The codes generated during the current study are available in the Github
repository, https://github.com/Anupam-Bh/ML_2D_flat_band.

## Author contributions

A.B. carried out the calculations, implemented the algorithms in codes,
analyzed the data and drafted the manuscript. A.M. conceived the research
plan, analyzed the data, drafted the manuscript and supervised the research
work. R.C. co-supervised the research work. I.T. and Q.Y. analyzed the data
and contributed to writing the manuscript.

## Acknowledgements

This research was supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (Grant Agreement
No. 865590), and the Royal Society International Exchanges 2019 Cost Share
Program (IEC\R2\192001). A.B. acknowledges the Commonwealth Scholarship
Commission in the UK for financial assistance. Q.Y. acknowledges the funding
from Leverhulme Early Career Fellowship ECF-2019-612, Dame Kathleen
Ollerenshaw Fellowship from the University of Manchester, and Royal Society
University Research Fellowship URF\R1\221096.

## Competing Interests

The authors declare no Competing Financial or Non-Financial Interests.

## Supplementary Information

Table S1: Flatband sublattices obtained from Fig.5 Lattice type |  Cluster ID#[cluster size]:Example[sublattice] |  Remarks |  Structure
---|---|---|---
1.Monolayer hexagonal lattice |  0[10]: Rb2O [O], Ti2S [S] 26[12]: CrPb3 [Cr], Hg3AsS4Br [As] 30[13]: Co(ReO4)2 [Co], BaCr4O8 [Ba] 31[21]: CdClO [O], MnCl2 [Mn] |  |
2.Sawtooth-chain lattice |  32[7]: Ta2TiZn2O8 [Ti], Ta2TeO8[Te] 3[8]:V2ZnS5[V], V2ZnO5[V] 23[9]:CuS4BrN4 [Cu] 44[10]: BaI2 [I] 45[33]: PdI2 [Pd], NiI2 [Ni] |  |
3.Planar Centered monoclinic lattice |  15[8]: K2Hg3(GeS4)2 [K], Ta2I5 [Ta], VSeO4[V] |  |
4.Planar centered-orthorhombic/ monoclinic lattice |  13[7]: Co(SiSe2)2 [Co], Zn(SbO2)4 [Zn] 17[7]: Cu2IO3 [Cu] 20[11]:Li4VF8 [V], Tl2Pt(CN)4 [Pt] 21[11]: ReF4 [Re], VZnF4 [V] |  Each atom has planar 6-fold coordination (elongated hexagonal) |
5.Kagome and breathing-Kagome lattice |  19[10]: P2Pd3S8 [Pd], FeCo3O8 [Co] 18[14]:Nb3SBr7 [Nb], Nb3TeI7 [Nb] |  19: Kagome lattice 18: Breathing Kagome lattice |
6.Planar orthorhombic lattice |  22[7]: LiVF4 [V], LiVCuO4 [Cu] 25[9]: LiAuF4 [Au], MnBiSe2I [Mn] 27[8]: CrPS4 [Cr], Cu(IO3)2 [Cu] 28[14]: Li3Co(NiO3)2 [Co], AlNiF5 [Al] |  |
7.Monolayer square lattice |  24[38]: CuBr [Cu], LaNb2O7 [La] |  |
8.Honeycomb lattice |  41[8]: CrGeTe3 [Cr], TiI3 [Ti] 42[8]:IrBr3 [Ir], CrBr3 [Cr] 43[8]:RhCl3 [Rh],VCl3 [V] |  |
9.Lieb-square lattice |  7[8]: Fe4S5, Os4S5 |  Lattice consists of Lieb-square pattern; each atom is asymmetric 6-fold coordinated |
10.Diamond-chain lattice |  9[7]:ReSe2[Re], ReS2[Re] |  |
11\. Isolated tetrahedra lattice |  37[13]: LiNiPO4 [O], Si(HgO2)2 [O], Cd2SiO4 [O] |  Each atom has 3 fold coordination |
12.Face sharing cuboctahedron |  1[7]: Sr4Mn3(ClO4)2[O], Ba2Cr3O8 [O] |  |
13.Vertex sharing octahedra chain |  2[8]: YFeF5[F], YWF5[F] |  |
14.Edge sharing octahedra-parallel chain |  4[8]:Sc5Cl8[Sc], Sc7Cl10[Sc] |  |
15.Distorted Kagome lattice |  5[15]:MgCl2[Cl], MgF2[F] |  Hexagonal lattice stacked over Kagome to make it distorted |
16.Stacked Kagome-hexagonal lattice |  6[7]:Zr(ReO4)2[O], Mg(AlH4)2[H] |  Two Kagome atomic layers stacked with 2 hexagonal lattice layers at two ends |
17.Dice lattice |  8[18]: HgSe [Se] |  Layers with ABCA.. stacking : atoms from inside layers have 12 fold coordination: form Triangular gyrobicupola ; surface atoms have 9 fold coordination |
18.Stacked square (AB stacking) lattice |  11[7]: InClO[Cl], CaClF[Cl] 12[40]: LiFeF4[F], MoPO5[Mo] 14[10]:BaMgsn [Sn], BaMgPb [Pb] |  Two square lattices stacked with AB stacking |
19.Triangular bipyramid |  29[12]: MoO3[Mo], SbBr3[Br] |  Three layers with middle layer atoms having 6 fold coordination and side atoms having 3 fold coordination. |
20.Planar hexahedra |  33[10]:TaF5[F] SbF5[F] |  |
21\. Vertex sharing planar octahedra |  34[18]: ZrF4[F], SnF4[F] |  |
22.Stacked hexagonal (AB stacking) |  49[20]: Mn3CrO8[O], Mn3CuO8[O] 50[35]: LaBr[La], YCl [Y] |  |
23.Stacked coloring triangle lattice |  48[51]: InBr3[Br], GaBr3[Br] |  |
24.Stacked breathing Kagome |  39[12]: InAs3 [As] |  Can also be called isolated octahedra |
25.Stacked breathing coloring triangle |  35[7]: Ge3Sb2O9 [O] Si3Bi2O9 [O] |  |
26.Stacked orthorhombic-square lattice |  36[14]:HgI2[I] |  The cationic sublattice is a diamond octagon sublattice: But the flat-bands come from the anionic sublattice (stacked orthorhombic-square lattice) |
27.Edge-sharing octahedra chain |  40[11]: Y2Cl3[Y] |  |

### Supplementary Note I: Finer search for recurrent lattice structures

We also use a finer clustering solution by HDBSCAN which identifies more flat
band sublattices as compared to those observed with MS = 7 and SS = 6 in Fig.
5 and Table LABEL:tab_7_6. As demonstrated in Fig. 4a-b, the clustering
solution for MS = SS = 3 yields the best DBCV and S-Dbw indices. However, this
also means a much larger number of clusters (230), inducing extra complexity
in identifying major groups of structure fingerprints. Instead, we choose MS =
4, SS = 3 to obtain a total of 145 clusters while maintaining a good number of
unclassified materials (1071), comparing to other combinations like MS = 3, SS
= 4 or MS = SS = 4, which lead to a larger number of uncategorized examples.
Comparing to the case of MS = 7, SS = 6, we now have smaller clusters, many of
which were identified as outliers before. Results of HDBSCAN clustering with
MS = 4 and SS = 3 are shown as a condensed tree plot in Fig. S1a and as a 2D
t-SNE visualization in Fig. S1b. All the recognizable groups have been
identified as clusters, and the number of unclassified examples around defined
clusters is reduced.

We identified all the exemplar lattice structures found in the 145 HDBSCAN
clusters shown in Fig. S1. The coordination patterns from the t-SNE
representation in Fig. 5b could guide the identification/nomenclature of these
structures. We again use t-SNE to combine the nearly identical exemplar
candidates and obtain the total list of sublattice structures. Most sublattice
structures of this list were identified with MS=7, SS=6 from Fig. 5 and are
already listed in Table LABEL:tab_7_6. New structures which were elusive with
MS=7, SS=6 are described in detail in Tables LABEL:mono-LABEL:multilayer. We
observe that most of the clusters are homogeneous as a common sublattice is
present in all the materials within one cluster. However, 24 clusters were
found to have either inhomogeneous sublattice structures or structures which
are too complex to be categorized; these clusters lie near the diffused region
in the middle of t-SNE plot, Fig. S1b.

Figure S1: Clustering results from HDBSCAN for MS=4, SS=3. (a) Condensed tree plot showing hierarchical information among clusters with numbers representing cluster IDs. (b) t-SNE plot showing adjacency of materials in the embedded space. Numbers represent cluster IDs. Table S2: New monolayer flatband sublattices identified from Fig S1 Lattice type |  Cluster ID# [cluster size]:Example [sublattice] |  Remarks |  Figure
---|---|---|---
1.Square chain lattice (Cruetz lattice) |  12[4]:KFe2Se3[Fe], RbFe2Te3 [Fe] |  Spaced rows of squares |
2.Vertex shared square chain lattice |  3[6]:Be2Te7Cl6[Te], Tl2Te5 [Te] |  Spaced rows of squares |
3.Linear chain |  90[5]: NbS3 [Nb] |  |
4.Isolated quadrilaterals |  41[7]: MoSe3 [Mo] |  |
5.$(3^{2},4,3,4)$ Archimedean lattice |  38[10]:TiP [P] |  |
Table S3: New bilayer flatband sublattices identified from Fig S1 Lattice type |  Cluster ID# [cluster size]:Example [sublattice] |  Remarks |  Figure
---|---|---|---
1.Stacked Kagome lattice (AA stacking) |  6[4]: TaIr3[Ir] |  Two stacked Kagome lattice (each has out of plane distortions) |
2.Stacked centered orthorhombic lattice |  23[4]:SrLaBr5[Br], BaLaCl5[Cl] |  Elongated hexagonal lattices stacked such that each atom has 6 fold in plane coordination and 2 fold out of plane coordination |
3.Stacked rectangles (AB stacking) lattice |  52[7]: Ba2P [P] |  Two rectangle lattices stacked with AB stacking. |
4.Stacked hexagonal (AA stacking) |  59[5]:TiO2[O] 118[10]: Rb3Nb2Br9 [Nb], CuAsO3 [Cu] |  Only one nearest neighbour because of stacking |
5.Linked sawtooth lattice |  88[9]: TaI4 [Ta] |  2-fold OOP and 2-fold IP coordination |
6.Stacked centered-orthorhombic-square chain |  115[6]:SmF3 [F] 121[8]: CaThBr6 [Br], TbCl3 [Cl] |  Type of Archimedean lattice: Lattice consists of alternate chains of cenetered orthorhombic and square shapes |
7.Reverse sawtooth lattice |  119[4]: TaCoTe2 [Co], LiVF5 [V] 125 [16]: PdCl2 [Pd] |  |
Table S4: New multilayer flatband sublattices identified from Fig S1 Lattice type |  Cluster ID# [cluster size]:Example [sublattice] |  Remarks |  Figure
---|---|---|---
1.Isolated twisted-octahedra |  15[4]:ReSeCl[Re], MoCl2[Mo] |  Alternate rows of twisted octahedras |
2.Stacked square-tetrahedra chain |  18[6]:Sr2TiCu2O7[O], Sr2Cu2SbO7[O] |  |
3.Distorted stacked hexagonal lattice |  25[4]: BeBr2[Br] 82[5]: Fe3S4 [Fe] |  Each atomic layer is hexagonal (monoclinic distortion present) |
4.Stacked colouring triangle-hexagonal lattice |  29[4]:Li2Fe(PO4)2[O], Ni(ClO4)2[O] |  Two colouring triangle lattices stacked with two hexagonal lattices on ends |
5.Stacked $(3^{2},4,3,4)$ Archimedean lattice |  91[5]: CoBr4 [Br] |  2 layers of Archimedean lattice; The Archimedean lattice is distorted out-of-plane |
6.Vertex-sharing triangular pyramid chain |  92[4]:V2CuO6[O], V2CoO6[O] |  |
7.Connected sawtooth chains |  94[5]: In4Se3 [Se], Tl4S3 [S] |  |
8.Edge sharing octahedra chain (zigzag) |  97[14]:ZnMoO4[O], ZnWO4[O] |  |
9.Vertex-edge sharing planar octahedra |  104[4]: CaTlCl3[Cl], TlPbI3 [I] |  |

### Supplementary Note II: Visualization of flatband clusters using UMAP

Figure S2: Optimized UMAP visualization (Nearest$\\_$neighbor = 30) for
HDBSCAN output of MS=7 and SS=6.

UMAP 51 is another nonlinear dimensionality reduction algorithm, similar to
t-SNE, that we have used to present our results. We visualized HDBSCAN output
of MS=7, SS=6 using UMAP in 2 dimensions to assess its effectiveness in
comparison to that of t-SNE. For UMAP, we optimized the 3 following
parameters, nearest$\\_$neighbors, minimum$\\_$distance, and distance metric.
Nearest$\\_$neighbors and minimum$\\_$distance denote the number of neighbor
distances that are retained while reducing the dimensions and minimum distance
within a cluster in the embedded space respectively. Here we show clustering
results with Nearest$\\_$neighbors=30 and minimum$\\_$distance=0.1, which is
considered to be the best visualization. The resulting embedded space is shown
in Fig. S2.Despite the seemingly different presentation, it shares many
similar features with that of our t-SNE clustering results. For example,
clusters #41-43 are well isolated from the rest of the clustering. Meanwhile,
the transition from hexagonal to rectangular sublattices (mostly yellow and
green in color) which have very few outlier points in between, as well as the
dispersed distribution of structural fingerprints on the opposite site, are
all in good agreement with the t-SNE plot in Fig. 5b. The clustering solution
visualized by UMAP method, therefore, does not offer a significantly better
solution than the t-SNE method used in the main text.

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