††thanks: Corresponding author<EMAIL_ADDRESS>
# Spectroscopic evidence of flat bands in breathing kagome semiconductor
${\mathrm{\textbf{Nb}}}_{3}{\mathrm{\textbf{I}}}_{8}$

Sabin Regmi  Department of Physics, University of Central Florida, Orlando,
Florida 32816, USA  Tharindu Fernando  Department of Physics, University of
Washington, Seattle, Washington 98195, USA  Yuzhou Zhao  Department of
Physics, University of Washington, Seattle, Washington 98195, USA  Anup
Pradhan Sakhya  Department of Physics, University of Central Florida, Orlando,
Florida 32816, USA  Gyanendra Dhakal  Department of Physics, University of
Central Florida, Orlando, Florida 32816, USA  Iftakhar Bin Elius  Department
of Physics, University of Central Florida, Orlando, Florida 32816, USA  Hector
Vazquez  Department of Physics, University of Central Florida, Orlando,
Florida 32816, USA  Jonathan D Denlinger  Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Jihui Yang  Department of
Materials Science and Engineering, University of Washington, Seattle,
Washington 98195, USA  Jiun-Haw Chu  Department of Physics, University of
Washington, Seattle, Washington 98195, USA  Xiaodong Xu  Department of
Physics, University of Washington, Seattle, Washington 98195, USA  Ting Cao
Department of Materials Science and Engineering, University of Washington,
Seattle, Washington 98195, USA  Madhab Neupane  Department of Physics,
University of Central Florida, Orlando, Florida 32816, USA

###### Abstract

Abstract - Kagome materials have become solid grounds to study the interplay
among geometry, topology, correlation, and magnetism. Recently, niobium halide
semiconductors
$\mathrm{Nb}_{3}X_{8}(X~{}=~{}\mathrm{Cl},~{}\mathrm{Br},~{}\mathrm{I})$ have
been predicted to be two-dimensional magnets and these materials are also
interesting for their breathing kagome geometry. However, experimental
electronic structure studies of these promising materials are still lacking.
Here, we report the spectroscopic evidence of flat and weakly dispersing bands
in breathing-kagome semiconductor $\mathrm{Nb}_{3}\mathrm{I}_{8}$ around 500
meV binding energy, which is well supported by our first-principles
calculations. These bands originate from the breathing kagome lattice of
niobium atoms and have niobium d-orbital character. They are found to be
sensitive to polarization of the incident photon beam. Our study provides
insight into the electronic structure and flat band topology in an exfoliable
kagome semiconductor, thereby providing an important platform to understand
the interaction of geometry and electron correlations in two-dimensional
materials.

Introduction -  Owing to the possibility of exploring the interplay among some
or all of parameters such as geometry, topology, electronic correlations and
magnetism, quantum materials with kagome lattice have recently been attracting
a bulk of research studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Comprised of
corner sharing triangles forming a hexagon within, a kagome lattice may
feature relativistic Dirac band crossing at the corner of the Brillouin zone
(BZ) [11]. Because of the structural geometry, the electrons are trapped
within the hexagon. This geometry-driven self-localization of the electrons
means that flat bands may be present in the electronic structure of the kagome
materials and give rise to strong electronic correlations [4, 12, 13, 14, 15].
By introducing spin-orbit coupling and magnetism that breaks the time-reversal
symmetry, the Dirac band crossing at the corner of the BZ can be gapped out
leading to intrinsic Chern quantum phases [16, 17, 8, 18]. The band gap may be
opened also when the alternating triangles in the kagome lattice have
different bond lengths resulting a different geometry than the conventional
kagome (in which all the triangles are identical) - called the breathing
kagome [19]\- leading to the possibility of higher order topological phase
[20]. The intrinsically topological flat bands, however, may be robust to this
geometrical perturbation [19]. The difference in the size of the alternate
triangles in the breathing kagome geometry may result in local electric dipole
moment leading to ferroelectric order [21].

Figure 1: Crystal structure and bulk band calculation of
${\mathrm{\textbf{Nb}}}_{3}{\mathrm{\textbf{I}}}_{8}$. a Crystal structure of
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$, where the green and magenta spheres
represent Nb and I atoms, respectively. b Nb atoms forming a breathing kagome
lattice. An schematic of breathing kagome lattice is presented in c. The
alternating traingles are not identical giving rise to breathing lattice. d
Projected hexagonal BZ on the (001) surface showing the high-symmtery points
and directions. e Calculated band structure for bulk
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$.

Angle-resolved photoemission spectroscopy (ARPES) [22, 23, 24] has served as a
valuable technique allowing direct visualization of the band structure in
quantum materials. In recent times, it has been successful in revealing the
electronic structure of various kagome materials. Topological Dirac bands
and/or flat bands in the electronic structures of $\mathrm{Fe_{3}Sn_{2}}$ [4,
18, 25], $\mathrm{FeSn}$ [14], $\mathrm{CoSn}$ [12, 13],
$\mathrm{\textit{R}\textit{T}_{6}Sn_{6}}$($\mathrm{\textit{R}=~{}Tb,Y,~{}Er,~{}Ho,~{}Gd}$
and $\mathrm{\textit{T}~{}=~{}Mn,~{}V)}$ [8, 15, 26, 27, 28],
$\mathrm{CsV_{3}Sb_{5}}$ [29] have been observed by utilizing ARPES. A point
to note is that most of these materials are metals with conventional kagome
lattice, $\mathrm{Fe_{3}Sn_{2}}$ with breathing kagome lattice reported in a
recent ARPES study suggests a possibility of magnetic Weyl semimetallic phase
in this material [25]. ARPES study of materials with the breathing kagome
lattice is still limited. The ${\mathrm{Nb}}_{3}{\mathrm{\textit{X}}}_{8}$
family of materials, where $\mathrm{\textit{X}~{}=~{}Cl,~{}Br,~{}I}$ [30, 31],
provides material platforms to study the electronic structure arising from the
breathing kagome geometry. Importantly, these materials have two-dimensional
(2D) crystalline structure with very weak interlayer van der Waals interaction
and hence can be thinned down to 2D limit via mechanical exfoliation [32, 33,
34]. They are semiconducting with a moderate band gap [33], which is suitable
for electronic applications. In the monolayer limit, they are predicted to
host ferromagnetic (FM) order [31, 35, 36], allowing a great platform to study
the interplay among geometry, electronic correlations, and magnetism in 2D.
Monolayer ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ is predicted to exhibit large
spontaneous valley polarization making it suitable for valleytronics
applications [37]. Systematic studies of the electronic structure of these
semiconducting van der Waals breathing kagome materials are worthwhile,
however, are lacking.

In this article, via ARPES measurements, we report a systematic electronic
structure study of breathing kagome semiconductor
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ assisted by density-functional theory
(DFT) computations. Our experimental data is consistent with the
semiconducting nature of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ and reveals the
presence of flat and weakly dispersing bands, that arise from the Nb-breathing
kagome, in the electronic band structure of this material. These bands are
observed to be sensitive to the polarization of the incident photon beam. The
experimental observations are well reproduced by theoretical calculations. Our
study provides a layered 2D material platform with flat bands thereby giving
an opportunity to study the interaction among symmetry, geometry, electronic
correlations and potentially magnetism given the prediction of FM monolayer.

Figure 2: ARPES measured energy contours (ECs) and band dispersions. a-c ECs
at the binding energies of 300 meV, 1.05 eV, and 1.6 eV, respectively. Red
hexagons represent the BZ with high-symmetry points marked and labeled. d-e
Band dispersions along different high-symmetry directions as shown in the
plots. f Second derivative plot of e. Data were collected at the ALS beamline
4.0.3 at a temperature of 260 K using photon source of mono-energy 90 eV.

Methods -  High-quality single crystals of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$
were synthesized by utilizing chemical vapor transport method (modified from a
previously reported method [32]) using Iodine as a transport agent (see
Supplementary note 1). The crystal structure was examined using X-ray
diffraction and the chemical compositions of the obtained crystals were
verified using scanning electron microscopy and energy dispersive X-ray
spectroscopy analysis (See Supplementary Fig. 2).

Synchrotron based ARPES measurements were performed at the Advanced Light
Source (ALS) beamline 4.0.3 equipped with R8000 hemispherical analyzer. In
order to obtain high-quality and fresh surfaces required for ARPES
experiments, single crystals were cleaved in situ under ultra high vacuum with
pressure maintained in the order of $\mathrm{10^{-11}}$ Torr. The measurements
were carried out at a temperature of 260 K to evade charging effects. This
leads to thermal broadening of the bands in the ARPES spectra. LH and LV
polarized photon sources in the range $\mathrm{30-124~{}eV}$ were used during
the measurements. For experimental geometry, see Supplementary Note 3 and
Supplementary Fig. 6.

Computational results were obtained from first-principles calculations
performed within the framework of DFT [38, 39] as implemented in the Quantum
ESPRESSO package [40] and Vienna ab initio simulation package (VASP) [41].
Norm-conserving scalar relativistic ONCVPSP (Optimized Norm-Conserving
Vanderbilt Pseudopotential) [42, 43] and Perdew-Burke-Ernzerhof-type exchange-
correlational functional [44] were used with van der Waals correction included
in the D2 formalism [45]. For more information, see Supplementary Note 2.

Results -  ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ crystallizes in trigonal
crystal system with rhombohedral space group $\mathrm{R\overline{3}m}$ (number
166) with lattice parameters ${a}$ = ${b}$ = $\mathrm{7.60~{}\AA}$ and ${c}$ =
$\mathrm{41.715~{}\AA}$ [30]. A three-dimensional bulk crystal structure with
six layers within a unit cell is shown in Fig. 1a where each layers along the
crystallographic $\mathrm{c}$-direction are connected through a weak van der
Waals bond and within each layer, there is strong covalent bonding. The Nb
atoms form a breathing kagome lattice with two distinct Nb-Nb distances in
alternate Nb triangles and are sandwiched between the top and the bottom
I-atom sheets, each containing two atomic layers (I1, I3 on the top sheet and
I2, I4 on the bottom sheet) (see Supplementary Note 1 and Supplementary Fig.
1). The breathing kagome formed by the Nb-atoms can be seen in Fig. 1b, where
each Nb atom has two long inter-triangular distances and two short intra-
triangular distances with its four nearest neighbor Nb-atoms within the kagome
plane. A schematic of the breathing lattice is presented in Fig. 1c, where the
alternate triangles, unlike in a conventional kagome lattice, are not
identical. Because of the presence of this special geometry in
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$, flat bands are expected in its electronic
structure. In Fig. 1d, a projected hexagonal BZ on (001) surface is presented
with high-symmetry points and directions. In Fig. 1e, we present the
calculated band structures for bulk ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ at
$k_{z}$ = 0. According to the calculation, ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$
is semiconducting with a Kohn-Sham band gap (direct) of
$\sim\mathrm{520~{}meV}$ at the $\mathrm{K}$ point. Flat bands and weakly
dispersing bands can be seen within 500 meV above and below the mid-gap. Such
bands are present around 1 eV below the Fermi level as well. The overall
frontier band structures of the bulk seems to be similar to that of monolayer,
indicating that the interlayer electronic couplings are weak (Supplementary
Fig. 3 for monolayer calculation).

Figure 3: Observation of flat and weakly dispersing bands. a,b ARPES measured
band dispersions along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
measured using photon energy of 90 eV and Integrated EDC (Voigt fit, also see
Supplementary Note 6 and Supplemenatry Fig. 9) within an momentum window of
($\mathrm{-0.4~{}/\AA}$ \- $\mathrm{0~{}/\AA}$) in a. c,d Second derivative
plot of dispersion map in a and respective EDC curve taken within momentum
window of ($\mathrm{-0.5~{}/\AA}$ \- $\mathrm{0~{}/\AA}$), respectively. e
Experimental band dispersion along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
direction. f Calculated bands along $\mathrm{M}-\mathrm{\Gamma}-\mathrm{M}$. g
Experimental band dispersion along
$\overline{\mathrm{\Gamma}}-\overline{\mathrm{K}}-\overline{\mathrm{M}}-\overline{\mathrm{K}}$
direction. h Calculated bands along
$\mathrm{\Gamma}-\mathrm{K}-\mathrm{M}-\mathrm{K}$. ARPES data were collected
at the ALS beamline 4.0.3 at a temperature of 260 K.

In Figs. 2 \- 4, we present the electronic structure of
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ measured using ARPES.
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ is a semiconductor and the Fermi level
lies within the gap. Therefore, no photoemission signal was observed at the
Fermi level. ARPES signal is observed only about 300 meV below the Fermi level
[see Fig. 2a]. Several pockets can be seen below 1 eV from the Fermi level, as
shown in Fig. 2b. At a binding energy of about 1.6 eV, a nice hexagonal
symmetry is observed in the energy contour [see Fig. 2c]. Next, in order to
reveal the electronic structure of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$, we
present the dispersion maps along various high-symmetry directions in Figs.
2d-e. The dispersion maps along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$ and
$\overline{\mathrm{\Gamma}}-\overline{\mathrm{K}}-\overline{\mathrm{M}}-\overline{\mathrm{K}}$
show the presence of bands in accordance to the observations in the energy
contours. The photoemission intensity is not uniform and specially at around
500 meV, is strongly suppressed out of a single BZ (also seen in Fig. 2a),
possibly because of matrix element effects. This can be seen in the second
derivative plot of the dispersion along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$,
presented in Fig. 2f.

Next, we proceed to analyze the details of the bands observed along the high-
symmetry directions along with a careful comparison with theoretically
calculated bands using the DFT approach in Fig. 3. Figure 3a shows the
dispersion map along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
obtained from ARPES measurement using 90 eV photon energy. Around 500 meV
below the Fermi level, there are two bands, one of which is almost flat (named
B) and the other is weakly dispersing (named A), which correspond to the peaks
in the integrated energy distribution curve (EDC) (Voigt fit) in Fig. 3b [also
see Supplementary Fig. 9]. It has been predicted that flat and weakly
dispersing bands in breathing kagome lattices might be intrinsically robust
[19]. Another nearly flat band (named C) can also be observed below 1.2 eV
binding energy. In Fig. 3c, we present the second derivative plot of the
dispersion map in Fig. 3a, where the presence of two bands - one weakly
dispersing (A) and the other almost flat (B) - can be clearly seen. Two minima
peaks in the integrated EDC taken within ($\mathrm{-0.5~{}/\AA}$ \-
$\mathrm{0~{}/\AA}$) of the second derivative plot (Fig. 3d) further confirm
the presence of two bands. The minima peaks exactly coincide the maxima peaks
in the integrated EDC in Fig. 3b. In Figs. 3e and 3f, we present the
experimental band structure along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$ taken
with 110 eV photon energy and calculated band structure using DFT approach,
respectively, where we can see an reasonable matching between the two. Within
500 meV of the mid-gap, the calculated data shows a set of almost flat bands
(labeled as A and B) that have weak dispersions persisting down to the
monolayer limit. Another set of almost flat bands (labeled C) appears around 1
eV below the Fermi level. Below C, DFT calculations show a continuum of band
states consistent with the finite intensity observed in the experimental APRES
spectra. The flat bands do not seem to vary their dispersion with the change
in photon energy regardless of the variation in intensity, which possibly
comes from the photoemission matrix element effect. This shows that these
bands are of 2D nature, which is expected for 2D materials like
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ [For more variation in photon energy, see
Supplementary Note 4 and Supplementary Fig. 7]. The dispersive bands below 1.5
eV binding energy, however, seem to strongly depend on the energy of the
incident photon beam indicative of their 3D nature. Figures 3g and 3h
represent the experimental and calculated band structures along
$\overline{\mathrm{\Gamma}}-\overline{\mathrm{K}}-\overline{\mathrm{M}}-\overline{\mathrm{K}}$,
respectively. The calculated result well reproduces the experimental
observation.

Figure 4: Polarization dependent band dispersion. a-c ARPES measured band
dispersions along
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
measured using photon energies of 120 eV, 70 eV, and 36 eV, respectively using
LH or LV polarization as noted on each plot. d Fits of integrated EDCs (Voigt
fits) taken within momentum window of ($\mathrm{-0.2~{}/\AA}$ \-
$\mathrm{0~{}/\AA}$) for 120 eV LH polarized (bottom) and LV polarized (top)
data in a [also see Supplementary Fig. 9]. e Calculated orbital-resolved
dispersion along ${\mathrm{M}}-{\mathrm{\Gamma}}-{\mathrm{M}}$. ARPES data
were collected at the ALS beamline 4.0.3 at a temperature of 260 K.

In Fig. 4, we present the polarization dependence of the bands observed along
the $\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
direction. Polarization dependent measurements have been performed at several
photon energies including 120 eV, 70 eV, and 36 eV and presented in Figs. 4a-c
respectively. It can be seen that the bands around 500 meV binding energy have
strong intensity in the linear horizontal (LH) polarization [see bottom panel
of Fig. 4d]. These bands are strongly suppressed in intensity when the
incident photon beam is linear vertical (LV) polarized. With LV polarization,
the intensity of the bands below 1.2 eV becomes strong [see top panel of Fig.
4d]. A weakly dispersing band can also be seen with LV polarized light [shown
by arrow in the right panels of Fig. 4a and 4c], supported by two-peak feature
in Fig. 4d top panel. In Fig. 4e, we present the contribution of $\mathrm{Nb}$
d and $\mathrm{I}$ p in the electronic structure of
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ along the
$\overline{\mathrm{M}}-\overline{\mathrm{\Gamma}}-\overline{\mathrm{M}}$
direction. It can be seen that the bands within 500 meV and just below 1 eV
binding energy have strong contribution from Nb d and the highly dispersive
bands near the center of the BZ and below 1.5 eV are strongly dominated by I
p. The flat bands observed in our experiments, therefore, originate from the
breathing kagome lattice of the Nb atoms. We further perform orbital-resolved
calculations to study the contribution of each d orbitals, namely $d_{zx}$,
$d_{x^{2}-y^{2}}$, $d_{z^{2}}$, $d_{yz}$, and $d_{xy}$ [see Supplementary Note
5 and Supplementary Fig. 8 for orbital resolved calculations]. The band sets A
and B have strong contribution from $d_{z^{2}}$ orbitals. On the other hand,
the band sets C have significant contribution from all orbitals with majority
coming from the $d_{xy}$ orbitals.

Discussion -  Because of the breathing kagome geometry of the Nb atoms in
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$, the Dirac fermion is not preserved in
this system as in the case of the conventional kagome geometry. This might
have led to the semiconducting nature of this material as seen in our
theoretical calculations and backed by our experimental observation. However,
the flat and weakly dispersing bands may remain robust despite the geometric
perturbation from kagome to breathing kagome [19]. In our experimental
measurements, we observe multiple flat and weakly dispersing bands at various
binding energies. Our calculations also show that the frontier flat and weakly
dispersing bands of the bulk is similar to those of the monolayer. This makes
the choice of kz not so significant in the measurements and can be validly
compared with the DFT calculations (also see calculations for kz = 0 and kz =
$\pi$ in supplementary Fig. 5, where no significant differences can be seen).
Our analysis shows that no band inversion occurs for the bands around
$\mathrm{500~{}meV}$ binding energy indicating non-topological character and
quantized non-zero Chern numbers do not exist for the lower bands by rigidly
shifting the Fermi level as they cannot be completely isolated in energy (see
Supplementary Note 2 and Supplementary Fig. 4). Moreover, we can see in the
experimental results that some bands, which are far below the Fermi level,
depend on the choice of the photon energy (kz value). This is expected because
although ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ is a 2D material, the
experimental sample is technically a three-dimensional bulk material.
Importantly, our calculations show that the flat and weakly dispersing bands
arise from the 2D breathing kagome plane of the Nb atoms.

In summary, we carried out ARPES measurements on a semiconducting kagome
material ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ that can be exfoliated down to 2D
limit (see Supplementary Note 7 and Supplementary Fig. 10) and discover flat
as well as weakly dispersing bands that arise from the breathing kagome
lattice of the $\mathrm{Nb}$ atoms. Computational results using DFT support
our observation. These bands have 2D nature as expected for a 2D material
based on our photon-energy-dependent measurements and show a strong dependence
on the polarization of the photon beam. Our study points to
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ being an important ground to study the
interplay among geometry and electronic correlations in 2D materials. Given
that the monolayer of this compound is predicted to be FM and the exfoliation
to stable few-layers to monolayer limit already reported to be achieved, it
could throw magnetism as well in the mix.

Note: During the preparation of the manuscript, the authors became aware of
similar work on a similar compound ${\mathrm{Nb}}_{3}{\mathrm{Cl}}_{8}$ [46].

Acknowledgments -  M. N. acknowledges the support from the National Science
Foundation (NSF) CAREER award DMR-1847962, the Air Force Office of Scientific
Research MURI Grant No. FA9550-20-1-0322, and NSF Partnerships for Research
and Education in Materials (PREM) Grant DMR-2121953. X. X., T. C., J.-H. C., &
J. Y. acknowledge the support from UW Molecular Engineering Materials Center,
a NSF Materials Research Science and Engineering Center (Grant No.
DMR-1719797). H.V. was supported by NSF PREM Grant DMR-2121953. This research
used resources of the ALS, which is a DOE Office of Science User Facility
under Contract No. DE-AC02-05CH11231.

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Supplemental Information

Supplementary Note 1. Crystal structure and sample characterization
$\mathrm{Nb}_{3}\mathrm{I}_{8}$ has a layered crystal structure (rhombohedral
space group no. 166) with intralayer covalent bonding and interlayer van der
Waal bonding along the crystallographic $\mathrm{c}$-axis. A side view of the
bulk crystal structure is shown in Supplementary Fig. 1a where a unit cell
consists of sixtuple layers. Nb atoms are sandwiched in between two sheets of
I atoms with I1, I3 atoms forming the top sheet and I2, I4 forming the bottom
sheet. Top view of the crystal structure is presented in Supplementary Fig.
1b. Nb atoms form a breathing kagome lattice [see Supplementary Fig. 1c]. An
schematic of breathing kagome lattice is shown in Supplementary Fig. 1e, where
the alternating triangles are of unequal size unlike in conventional kagome
lattice in which all the triangles are of equal size [Supplementary Fig.
1(d)].

Supplementary Fig. 1: Crystal structure and kagome geometry in
${\mathrm{\textbf{Nb}}}_{3}{\mathrm{\textbf{I}}}_{8}$. a Side view and b top
view of the crystal structure. c Breathing kagome plane formed by Nb atoms.
d-e Schematic of conventional kagome and breathing kagome lattices,
respectively.

As mentioned in the method section of the main text, high-quality single
crystals of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ used in this study were grown
by employing chemical vapor transport method with Iodine used as a transport
agent. 0.2303 g High purity $\mathrm{Nb}$ powder (99.99%, 325mesh from Alfa
Aesar) and 0.8021 g $\mathrm{Iodine}$ (99.999% from Aldrich) were sealed
inside a quartz tube (12.75 mm OD $\times$ 10.5 mm ID $\times$ 12 cm) under
vacuum ($<$ 20 mtorr). The tube was loaded into a dual zone furnace and ramped
for 20 hours to reach 652 $\degree$C on the precursor side and 650 $\degree$C
on the growth side. After 70 hours growth, the tube and the furnace were
allowed to naturally cool down to obtain flat and shiny crystals. The crystal
structure was examined using X-ray diffraction (XRD). XRD on a
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ crystal grown in the same batch as the
ARPES measurements is presented in Supplementary Fig. 2a. Sharp
$\mathrm{(000l)}$ peaks are observed indicating that the crystal surface is
oriented in the $\mathrm{ab}$-plane. The compositional homogeneity of the
crystals were verified using scanning electron microscopy (SEM) and energy
dispersive X-ray spectroscopy analysis (EDS) performed on
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ flakes exfoliated on a
$\mathrm{Si}\mathrm{O}_{2}/\mathrm{Si}$ substrate [see Supplementary Fig. 2b].
The quantitative elemental analysis shows a $\mathrm{Nb}:\mathrm{I}$ ratio of
8.24:22.31 = 2.96:8 [see table 1].

Supplementary Fig. 2: Sample characterization. a X-ray Diffraction (XRD) on powdered (cyan) and single crystal (red) ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$. b Energy-dispersive X-ray spectroscopy (EDS) of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ flakes exfoliated on a $\mathrm{Si}\mathrm{O}_{2}/\mathrm{Si}$ substrate. Supplementary Table 1: EDS analysis of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$. Element | Atomic Number | Series | Wt% | Wt% Sigma | Atomic%
---|---|---|---|---|---
Silicon | 14 | K Series | 29.86 | 1.26 | 54.20
Iodine | 53 | L Series | 55.53 | 1.64 | 22.31
Oxygen | 8 | K Series | 4.78 | 0.8 | 15.24
Niobium | 41 | L Series | 15.02 | 0.56 | 8.24

Supplementary Note 2. First-principle calculations
First-principles calculations were performed, for the bulk stable state
(Antiferromagnetic, interlayer spin configuration with alternate layers having
opposite spins) and monolayer stable state (FM), utilizing the density
functional theory approach [1, 2] as implemented in the Quantum ESPRESSO
package [3] by using norm-conserving scalar relativistic ONCVPSP (Optimized
Norm-Conserving Vanderbilt Pseudopotential) [4, 5] and Perdew-Burke-Ernzerhof-
type exchange-correlational functional [6] and with van der Waals correction
included in the D2 formalism [7]. A conventional unit cell with six layers
with $a=\mathrm{7.600~{}\AA}$ and $c=\mathrm{41.715~{}\AA}$ [8] was used for
the bulk calculations, and a single ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ layer
with $c=\mathrm{52.500~{}\AA}$ was considered for monolayer calculations.
Interlayer AFM was considered with each consecutive layers in six layers
having alternate spins. Hubbard potential $U=\mathrm{2~{}eV}$ on
$\mathrm{Nb}~{}d$ orbitals was used to account for the on-site interaction
[9]. Convergence tests were run to justify a kinetic energy cutoff,
$ecutwfc=\mathrm{85~{}Ry}$, and an 8 x 8 x 1 Monkhorst-Pack k-point grid. We
used a convergence threshold (for self-consistency) of $\mathrm{1\times
10^{-8}~{}Ry}$ .

In Supplementary Fig. 3, DFT calculations for bulk as well as monolayer
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ are presented. For comparison with
experimental data, we set the mid-band gap to be the zero-energy level. The
overall band structures of bulk and monolayer look similar. Direct Kohn-Sham
band gap at the $\mathrm{K}$ point for bulk is $\sim\mathrm{520~{}meV}$ and
that of monolayer is $\sim\mathrm{490~{}meV}$ indicating semiconducting nature
of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ in both three- and two-dimensions. We
calculated the mirror operator eigenvalues of -1, 1, 1, 1, 1, -1, 1, and -1
for the bands c3, c2, c1, v1, v2, v3, v4, and v5 bands (see Supplementary Fig.
4 for monolayer Kohn-Sham bands), respectively. With the inclusion of spin-
orbit coupling (SOC), the band degeneracy of these bands at the
$\mathrm{\Gamma}$ point is broken, indicative of potential topological
character. For c1 and v1, no band inversion occurs, therefore these bands are
non-topological. For bands c2, c3, v2, v3, v4, and v5, since any of these SOC-
gapped bands cannot be completely isolated in energy by shifting the Fermi
level (unlike band v1), quantized non-zero Chern numbers do not exist in this
band structure by rigidly shifting the Fermi level alone.

Supplementary Fig. 3: Bulk and monolayer band calculations. a DFT band
structure calculation for bulk ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$. For bulk,
majority- and minority-spin bands are degenerate. b DFT band structure
calculation for monolayer ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$. Blue and red
denote majority- and minority-spin bands, respectively. Supplementary Fig. 4:
Kohn-Sham bands of monolayer
$\mathrm{\textbf{Nb}}_{3}\mathrm{\textbf{I}}_{8}$. a Calculated Kohn-Sham
bands without considering SOC. b Kohn-Sham bands with SOC included.

In Supplementary Fig. 5, we present the calculations for bulk
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ at different $k_{z}$ planes (0 and $\pi$).
Note that these calculations have been implemented in VASP [VASP]. The overall
band structure remains similar to the one obtained by implementing in Quantum
ESPRESSO. Also, the band structure within the bands of interest (flat and
weakly dispersing bands that lie well within $\mathrm{1.5~{}eV}$) does not
have significant difference for $k_{z}=0$ and $k_{z}=\pi$ planes.

Supplementary Fig. 5: Bulk bands calculated for different $k_{z}$ planes. DFT
band calculations for bulk ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ for a $k_{z}=0$
and b $k_{z}=\pi$ planes. The calculations are performed using VASP software
[10].

Supplementary Note 3. Experimental details
Experimental measurements using the angle-resolved photoemission spectroscopy
(ARPES) technique were carried out at the Advanced Light Source Beamline 4.0.3
equipped with Scienta R8000 hemispherical electron analyzer in the Berkeley
National Laboratory. Single crystals of ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$
were cleaved in-situ in ultra-high vacuum conditions to achieve clean and
fresh surface. Measurements were done at a temperature of 260 K in order to
dodge the charging effects. For the whole measurement period, the sample
surface was stable and showed no sign of degradation. Energy and angular
resolutions were set to be better than 20 meV and 0.2$\degree$, respectively.
Photon beam with monoenergy in the range 30-124 eV was used with linear
horizontal (LH) and linear vertical polarization (LV).

Supplementary Fig. 6: Experimental geometry used in the ARPES measurements.

An experimental geometry for the ARPES measurements is shown in Supplementary
Fig. 6. At normal emission, the beam is incident at a 25$\degree$ grazing
angle to the sample, i.e. the angle between the x-rays and Scienta lens axis
is fixed at 65$\degree$. The Scienta electron analyzer is mounted with a
vertical slit and a polar map rotates the sample about a vertical axis. LH
(LV) polarization oscillates the electrons in the undulator in the horizontal
(vertical) plane and so the x-ray polarization vector is in the horizontal
(vertical) plane of the incident x-ray propagation direction. When the sample
tilt angle is zero, then LV polarization gives purely s-polarization
(polarization vector in the plane of the sample surface). LH polarization has
both s- and p-polarization components, which depend on the polar angle
($\theta$) as: $e_{p}=cos(\theta+25\degree)$ and
$e_{s}=sin(\theta+25\degree)$. For our polarization dependent cuts, the value
of $\theta$ is -4$\degree$.

Supplementary Note 4. Photon energy dependent dispersion maps
In order to show the two-dimensional (2D) nature of the flat bands, we show
the photon energy dependent dispersion maps along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
direction in Supplementary Fig. 7. Similar to what was observed with the
photon energy variation in the main text, the flat bands do not change their
dispersion with changing photon energy (only the photoemission intensity is
changing). This confirms that these bands are indeed of 2D nature. On the
other hand, the dispersing bands seem to be dependent on photon energy showing
their three-dimensional bulk origination.

Supplementary Fig. 7: Photon energy-dependent dispersion maps. a-h Dispersion
maps along the
$\mathrm{\overline{M}}-\mathrm{\overline{\Gamma}}-\mathrm{\overline{M}}$
direction for different value of photon mono-energy. The energy values are
noted on each plots.

Supplementary Note 5. Orbital resolved band structure calculations

Supplementary Fig. 8: Contribution from individual $\mathrm{\textbf{Nb}}~{}d$
orbitals. a Dispersion map along the $\mathrm{M}-\mathrm{\Gamma}-\mathrm{M}$
direction with total contribution from Nb d orbitals. b-f Band structure
showing individual contributions from $d_{zx}$, $d_{x^{2}-y^{2}}$,
$d_{z^{2}}$, $d_{yz}$, and $d_{xy}$ orbitals, respectively.

Supplementary Fig. 8a shows the total contribution from Noibium d in the
electronic structure along $\mathrm{M}-\mathrm{\Gamma}-\mathrm{M}$. We can see
strong contribution of Nb d to the bands A - C. Subsequent plots in
Supplementary Figs. 8b-f present the individual contribution from $d_{zx}$,
$d_{x^{2}-y^{2}}$, $d_{z^{2}}$, $d_{yz}$, and $d_{xy}$ orbitals, respectively.
It can be seen that the bands A and B have strong dominance from $d_{z^{2}}$
orbitals. The bandset C, on the other hand, seem to have significant
contribution coming from all orbitals with majority contribution coming from
$d_{xy}$ orbitals.

Supplementary Note 6. Energy distribution curves (EDCs)
In Supplementary Fig. 9, we present the EDCs (both raw and fits using voigt
function) for the plots used in main text Figs. 3a and 4d. We can observe that
the intensity of the flat and weakly dispersing bands coming from Nb breathing
kagome plane depend upon the polarization of the incident photon beam. The
bands around 500 meV (A and B), which have dominant $d_{z^{2}}$ orbital
character appear strong when the incident beam is LH polarized. On the other
hand, the bands below 1.2 eV (bandset C) with major $d_{xy}$ orbital
contribution are enhanced when the incident photon beam is LV polarization.

Supplementary Fig. 9: Energy distribution curves and Voigt fits. EDCs and
respective fits using voigt function for a 90 eV linear horizontal polarized
light, b 120 eV linear horizontal polarized light, and c 120 eV linear
vertical polarized light.

Supplementary Note 7. Optical and atomic force microscopy images
In Supplementary Fig. 10, we present the images of the exfoliated
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ samples. Panel (a) shows the optical image
of the crystal that is exfoliated and the panels (b) and (c) show the optical
image and atomic force microscopy images after exfoliation onto
$\mathrm{SiO}_{2}/\mathrm{Si}$ substrate. Our images reveal that the bulk
${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ crystals can be exfoliated into thinner
flakes, thereby indicating the potential utilization in two-dimensional
heterostructure studies and applications.

Supplementary Fig. 10: Exfoliation of
${\mathrm{\textbf{Nb}}}_{3}{\mathrm{\textbf{I}}}_{8}$. a Optical image of
crystal before exfoliation. b,c Optical image and atomic force microscopy
image of exfoliated ${\mathrm{Nb}}_{3}{\mathrm{I}}_{8}$ samples.

## Supplementary References

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