††thanks: These authors contributed equally: Marcin Syperek, Raul Stühler and
Armando Consiglio.††thanks: These authors contributed equally: Marcin Syperek,
Raul Stühler and Armando Consiglio.††thanks: These authors contributed
equally: Marcin Syperek, Raul Stühler and Armando Consiglio.

# Observation of room temperature excitons in an atomically thin topological
insulator

Marcin Syperek  Department of Experimental Physics, Faculty of Fundamental
Problems of Technology, Wrocław University of Science and Technology, Wybrzeże
Wyspiańskiego 27, 50-370 Wrocław, Poland  Raul Stühler  Physikalisches
Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität
Würzburg, 97074 Würzburg, Germany  Armando Consiglio  Institut für
Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence
ct.qmat, Universität Würzburg, 97074 Würzburg, Germany  Paweł Holewa
Department of Experimental Physics, Faculty of Fundamental Problems of
Technology, Wrocław University of Science and Technology, Wybrzeże
Wyspiańskiego 27, 50-370 Wrocław, Poland  Paweł Wyborski  Department of
Experimental Physics, Faculty of Fundamental Problems of Technology, Wrocław
University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370
Wrocław, Poland  Łukasz Dusanowski  Department of Experimental Physics,
Faculty of Fundamental Problems of Technology, Wrocław University of Science
and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Physikalisches Institut and Würzburg-Dresden Cluster of Excellence ct.qmat,
Universität Würzburg, 97074 Würzburg, Germany  Felix Reis  Physikalisches
Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität
Würzburg, 97074 Würzburg, Germany  Sven Höfling  Physikalisches Institut and
Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074
Würzburg, Germany  Ronny Thomale  Institut für Theoretische Physik und
Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität
Würzburg, 97074 Würzburg, Germany  Werner Hanke  Institut für Theoretische
Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat,
Universität Würzburg, 97074 Würzburg, Germany  Ralph Claessen
<EMAIL_ADDRESS>Physikalisches Institut and Würzburg-Dresden
Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany
Domenico Di Sante<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Bologna, 40127 Bologna, Italy  Center for
Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA
Christian Schneider<EMAIL_ADDRESS>Institute of
Physics, University of Oldenburg, 26129 Oldenburg, Germany

###### Abstract

Optical spectroscopy of ultimately thin materials has significantly enhanced
our understanding of collective excitations in low-dimensional semiconductors.
This is particularly reflected by the rich physics of excitons in atomically
thin crystals which uniquely arises from the interplay of strong Coulomb
correlation, spin-orbit coupling (SOC), and lattice geometry. Here we extend
the field by reporting the observation of room temperature excitons in a
material of non-trivial global topology. We study the fundamental optical
excitation spectrum of a single layer of bismuth atoms epitaxially grown on a
SiC substrate (hereafter bismuthene or Bi/SiC) which has been established as a
large-gap, two-dimensional (2D) quantum spin Hall (QSH) insulator. Strongly
developed optical resonances are observed to emerge around the direct gap at
the K and K’ points of the Brillouin zone, indicating the formation of bound
excitons with considerable oscillator strength. These experimental findings
are corroborated, concerning both the character of the excitonic resonances as
well as their energy scale, by ab-initio _GW_ and Bethe-Salpeter equation
calculations, confirming strong Coulomb interaction effects in these optical
excitations. Our observations provide the first evidence of excitons in a 2D
QSH insulator at room temperature, with excitonic and topological physics
deriving from the very same electronic structure.

Optical spectroscopy conducted on atomically thin sheets of transition metal
dichalcogenides (TMDCs) has opened a broad spectrum of fundamental research,
and significantly enhanced our understanding of the physics of elementary
excitations in low-dimensional semiconductors [1, 2]. Indeed, the dramatic
increase of Coulomb correlations in two-dimensional structures has revealed
the emergence of a vast variety of many-body states, including non-hydrogenic
Rydberg series of excitons [3], stable charged- [4, 5], and multi-excitonic
complexes [6, 7].

Our first observation of excitons in a large-gap QSH insulator at room
temperature opens up a new exciting avenue of combining excitonic physics and
topological electronic properties. Here a first central step was recognizing
that the optical selection rules are governed by the winding number of the
low-energy gapped Dirac bandstructure, which like the Berry curvature, is a
“local” topological quantity in $k$-space [8, 9, 10, 11]. This local physics
is indeed determined by the Berry curvature flux through the small $k$-space
spanned by the relative motion of the $e-h$ pair in the excitonic wavefunction
[12]. This appears, e.g., in a class of materials called gapped chiral fermion
systems, which comprises gapped topological surface states [13], monolayers of
TMDCs [14] and biased bilayers with broken inversion symmetry [15]. Recent
work indeed has connected the strength and required light polarization of an
excitonic transition with the corresponding optical matrix elements’ winding
number [16, 17, 18], and thus successfully described the canonical valley-
contrasting selection rules of K and K’ excitons in TMDCs.

However, despite the fact that optical properties of TMDCs thus are strongly
impacted by the above local $k$-space phenomena [8], the commonly investigated
pristine monolayer TMDCs WS2, WSe2, MoS2, MoSe2, and MoTe2, in their 2H-phase,
belong to the class of topologically trivial insulators. From a fundamental
perspective, the implications of the interlink between optical selection rules
and topological band invariants (a direct link between excitonic physics and
global topology, i.e., the Chern number as $k$-space integral over the Berry
curvature) are expected to trigger a plethora of future experimental and
theoretical investigations. First steps in this direction are discussed in our
conclusions and in the Supplementary Information.

Figure 1: Single-particle electronic properties of Bi/SiC. a Constant current
STM image of bismuthene on SiC(0001). The topographic map displays the high
coverage ($\sim 87\%$) of the epitaxially grown Bi monolayer and the terrace
structure of the underlying SiC substrate.
($T=$4.2\text{\,}\mathrm{K}\text{/}$$ and
$V_{\text{set}}=$2.6\text{\,}\mathrm{V}\text{/}$$,
$I_{\text{set}}=$50\text{\,}\mathrm{pA}\text{/}$$.) Inset: A high-resolution
measurement reveals the bismuthene honeycomb lattice.
($T=$4.2\text{\,}\mathrm{K}\text{/}$$ and
$V_{\text{set}}=$-0.4\text{\,}\mathrm{V}\text{/}$$,
$I_{\text{set}}=$200\text{\,}\mathrm{pA}\text{/}$$.) b Differential tunneling
conductivity $(\mathrm{d}I/\mathrm{d}V)$ spectrum, locally measured in the
center of the bismuthene film ($T=$4.2\text{\,}\mathrm{K}\text{/}$$ and
$V_{\text{set}}=$-0.4\text{\,}\mathrm{V}\text{/}$$,
$I_{\text{set}}=$200\text{\,}\mathrm{pA}\text{/}$$,
$V_{\text{mod}}=$10\text{\,}\mathrm{mV}\text{/}$$). Note that the peak at
$\approx 1.1$ eV is not an intrinsic feature of the bismuthene film, but
originates from a local defect state in the SiC substrate (see Section II of
the Supplementary Information). c $GW$ band structure calculation (blue
curves) for bismuthene. The direct optical transitions at the K-point are
indicated by arrows. Also shown is the DFT band structure obtained with the
HSE hybrid functional (red curves) [19], illustrating the propensity of DFT to
underestimate the band gaps. Characteristic points in the spectrum that
correspond to the valence band onsets ($V_{1}$, $V_{2}$) and the conduction
band onsets ($C_{1}$, $C_{2}$) are marked.

To date, though, any experimental scrutiny of the impact of non-trivial global
band topology on light-matter interaction and exciton physics has so far been
hampered by the lack of suitable topological insulators with sufficiently wide
band gap to allow coupling to high-energy photons. It is only very recently
that such 2D topological insulators have been discovered and synthesized,
namely 1T’-WTe2 monolayers [20] and Bi/SiC, an atomically thin bismuth layer,
which is arranged in a graphene-like unbuckled honeycomb lattice [19]. In
bismuthene, the SOC induced by the heavy Bi atoms and the orbital filtering
resulting from the covalent bonding to the substrate are responsible for the
emergence of a giant topological band-gap [21], which is large enough to be
probed by optical spectroscopy in the near infrared. This is in striking
contrast to any other topological insulator previously known, and opens up
novel experimental opportunities to the study of photo-induced excitonics in
2D topological insulators.

Here, we study the electronic as well as the optical band-gap of Bi/SiC
employing photo-modulated reflectivity and scanning tunnelling spectroscopy
(STS) combined with ab-initio $GW$ calculations, accounting for the electronic
Coulomb interactions [22, 23, 24]. Our investigation reveals the emergence of
strong optical resonances within the electronic band-gap which we associate
with the material’s excitonic modes. The experimentally derived excitonic
resonance energies are notably well reproduced by many-body calculations in
the $GW$+Bethe–Salpeter equation framework [25, 26, 27, 28, 29]. Our findings,
which evidence the emergence of excitons at room temperature in a large-gap
QSH insulator, ideally advance present activities to link excitonics and
topological materials research [16, 17, 18, 30, 31, 32, 33, 34, 35, 36, 37,
38].

Figure 2: Optical spectroscopy of Bi/SiC. a The experimental setup for high-
spatial-resolution photo-modulated reflectivity (PR). b The fundamental
excitation spectrum of bismuthene measured in the PR experiment at
$T=$300\text{\,}\mathrm{K}\text{/}$$ (CW excitation,
$E_{\mathrm{laser}}\sim$2.33\text{\,}\mathrm{eV}\text{/}$$,
$P_{\mathrm{ave}}\sim$0.5\text{\,}\mathrm{mW}\text{/}$$). A and B are exciton-
like PR features centred at
$E_{\mathrm{A}}\approx$1.19\text{\,}\mathrm{eV}\text{/}$$ and
$E_{\mathrm{B}}\approx$1.63\text{\,}\mathrm{eV}\text{/}$$, respectively. The
solid red curve represents a numerical fit with the expression from Eq. 1.
Inset: a sketch of the fundamental exciton-like optical transitions at the
K/K’ points in bismuthene.

The bismuthene monolayers used in our optical experiments were grown by
molecular beam epitaxy using a Knudsen cell as Bi source and SiC(0001) as
supporting substrate (for details see Ref. 19 and Methods). The high quality
of the resulting layers is best visualized by scanning tunneling microscopy
(STM). The topographic overview scan in Fig. 1a, obtained by constant-current
STM, shows an excellent Bi coverage of $\sim 87\%$. The inset depicts a
detailed view on the honeycomb arrangement of the Bi atoms, ultimately
confirming successful film growth.

For a detailed discussion of the optical excitations and related exciton
physics it is mandatory to first develop a good understanding of the
underlying single-particle electronic properties of Bi/SiC. Detailed mapping
of the valence bands has already been performed by angle-resolved
photoelectron spectroscopy (ARPES) and found to be in convincing agreement
with density-functional theory (DFT) [19]. However, here we are interested in
both occupied and unoccupied bands and their relative energy separation which
is relevant for the optical transitions. This can be probed by STS which
records the differential tunneling conductivity as a measure of the electronic
local density of states (LDOS).

Fig. 1b displays the STS spectrum measured at the center of the bismuthene
sample where it shows its highest structural quality. The most prominent
feature of the spectrum is the vanishing LDOS between the energy positions
marked by $V_{1}$ and $C_{1}$. This region is identified as the fundamental
band gap, with the spectral onsets $V_{1}$ and $C_{1}$ representing the
valence band maximum (VBM) and conduction band minimum (CBM), respectively.
For a quantitative analysis the onset energy $V_{1}$ is evaluated by
extrapolating the valence band edge to the zero conductance plateau (see
Supplementary Information), an established method to determine gap edges in
STS. The same procedure is applied to the CBM at $C_{1}$. From the energy
separation of $C_{1}$ and $V_{1}$ we infer an experimental single-particle
band-gap $E_{\mathrm{ind}}$ of $$0.80\text{\,}\mathrm{eV}\text{/}$\leq
E_{\mathrm{ind}}\leq$0.96\text{\,}\mathrm{eV}\text{/}$$.

In theory, the established method for a reliable estimate of the single-
particle gap is the $GW$ approximation, which expresses the electronic self-
energy $\Sigma$ by the electron propagator $G$ and the screened Coulomb
interaction $W({\bf k},\omega)$ via $\Sigma=i{GW}$ [22, 24, 39]. This goes
beyond DFT-type approaches which are designed to yield accurate total (ground-
state) energies but are not devised for extracting single-particle energies
[40] and often tend to underestimate band gaps (see Fig. 1c, red curves for an
example).

The $GW$ band structure for Bi/SiC (see Methods for the details) is shown in
Fig. 1c as solid blue lines. It is characterized by strongly Rashba-split
bands at the K (K’) valleys and an indirect electronic band gap between the
VBM at K and the CBM at $\Gamma$ of
$E_{\mathrm{ind}}=$0.97\text{\,}\mathrm{eV}\text{/}$$, in excellent agreement
with our experimental gap value. The $GW$ band structure accounts also for
some other features in our experimental STS data, as indicated by the dashed
orange lines connecting Figs. 1b and 1c. For instance, the kink in the
measured LDOS at approximately $-0.5\text{\,}\mathrm{eV}\text{/}$ (marked by
$V_{2}$) matches almost perfectly with the maximum of the lower Rashba-split
$GW$ valence band. Likewise, the experimental kink feature $C_{2}$ agrees well
with the K-point minimum of the $GW$ conduction band. Finally, and as of
immediate relevance for the optical transitions, the direct single-particle
gap at the K (K’) valley amounts to
$E_{\mathrm{direct}}=$1.36\text{\,}\mathrm{eV}\text{/}$$ in our $GW$
calculation.

Next, we expand our study to the optical response, i.e., to the two-particle
excitations. Experimentally, bismuthene’s fundamental optical excitation
spectrum is probed via highly spatially-resolved photo-modulated reflectivity
(PR). PR is a very sensitive probe that has been successfully applied to study
the excitonic spectrum of atomically thin crystals [41, 42], since it directly
probes the derivative of the absorption spectrum with respect to energy, in
stark contrast to direct reflectivity techniques.

The experimental setup is sketched in Fig. 2a (see also Methods). All
measurements were performed at room-temperature and with the sample kept under
an inert gas atmosphere to prevent surface oxidation of the Bi monolayers.
ARPES measurements performed before and after exposure to inert gas confirm
that this sample storage method keeps bismuthene chemically intact over weeks
(see also Supplementary Information), also owing to the van der Waals-nature
of the bismuthene surface. The resulting optical excitation spectrum is
displayed in Fig. 2b (see also Supplementary Information). It is dominated by
two absorption-like PR spectral features A and B, located at $\sim
1.19\text{\,}\mathrm{eV}\text{/}$ and $\sim 1.63\text{\,}\mathrm{eV}\text{/}$,
respectively. In analogy to other two-dimensional semiconducting materials
[43] and, in particular, taking into account the aforementioned agreement
between theory and experiment concerning the single-particle properties, it is
most reasonable to assign the captured optical transitions to Coulomb-
interacting electron-hole excitations in the vicinity of K and K’ points of
the Brillouin zone (see inset in Fig. 2b).

Figure 3: Spatial modulation of the optical response of Bi/SiC. a Sketch of
the scanning movement along a vertical and horizontal line on Bi/SiC. b,c
Evolution of the PR resonance of the A-peak along the vertical and horizontal
scanning line, respectively. Solid red lines represent the fitting curves (Eq.
1). d,e,f The energy of the A exciton transition, broadening, and the PR
amplitude, respectively. (Pulsed laser source,
$E_{\mathrm{laser}}\sim$1.65\text{\,}\mathrm{eV}\text{/}$$, average pump
power:$\sim 0.34\text{\,}\mathrm{mW}\text{/}$ , pump fluence: $\sim
0.57\text{\,}\mathrm{mJ}\text{/}{\mathrm{cm}}^{2}$). The PR amplitude in
horizontal direction correlates well with the bismuthene sample quality (see
Supplementary Information).

For a more quantitative data analysis we employ an expression for the PR
response of excitons based on an adapted model suggested by Shanabrook et al.
[44] (see Supplementary Information). Within this approach, the PR feature
acquires the form of the first derivative of the dielectric function:

$\frac{\Delta
R}{R}=\mathfrak{R}\left(C_{\mathrm{j}}e^{i\theta}(E-E_{\mathrm{j}}+i\Gamma_{\mathrm{j}})^{-2}\right)\,.$
(1)

Here, $C_{\mathrm{j}}$ and $\theta$ are the resonance amplitude and phase,
$E_{\mathrm{j}}$ and $\Gamma_{\mathrm{j}}$ are the energy and broadening
parameters of the optical transition, respectively. The fitting procedure to
the PR spectrum in Fig. 2b yields $E_{\mathrm{A}}=(1.191\pm
0.004)\leavevmode\nobreak\ $\mathrm{eV}\text{/}$$ and
$E_{\mathrm{B}}=(1.625\pm 0.004)\leavevmode\nobreak\ $\mathrm{eV}\text{/}$$,
$\Gamma_{\mathrm{A}}=(100\pm 4)\leavevmode\nobreak\ $\mathrm{meV}\text{/}$$
and $\Gamma_{\mathrm{B}}=(60\pm 4)\leavevmode\nobreak\
$\mathrm{meV}\text{/}$$. It is worth noting that both transitions show
substantial spectral broadening, which we attribute to effects arising from,
e.g., sample inhomogeneity within the probed area and interaction between
excitons and phonons in these room-temperature measurements [45, 46, 47].

The high spatial resolution of the PR experiment allows us to probe our sample
from the coherent crystal regime in the center of the Bi monolayer to a more
disordered regime at its edges. As schematically depicted in Fig. 3a, we
follow the signature of the A-resonance as the sample is scanned along the
indicated vertical and horizontal lines. The corresponding evolution of the PR
spectrum is presented in Figs. 3b,c. The fitting procedure reveals that the
transition energy (Fig. 3d), broadening (Fig. 3e), and the PR amplitude (Fig.
3f) indeed vary only weakly along the $2\text{\,}\mathrm{mm}\text{/}$-long
scanning line, suggesting the fairly good quality of our bismuthene layer over
a macroscopically large spatial extension. The layer quality deteriorates
close to the bismuthene edges, which is clearly reflected by the diminishing
PR amplitude as well as the increased spectral broadening of the resonance.
This observation is in line with local structure characterization of our
bismuthene films by STM and low-energy electron diffraction (LEED) which
reveal reduced crystalline order towards the sample edges (see Supplementary
Information). The film inhomogeneity from center to edge is most likely
related to a temperature gradient during epitaxial growth. Independent of the
detailed mechanism, the marked correlation between PR signal and local
crystalline quality is a clear indication that the observed PR resonances are
indeed intrinsic to the pristine bismuthene layer.

Turning now to the theoretical picture of the optical response, first we wish
to point out that the many-body effects embedded in the single-particle band
gap versus those contained in the optical excitation spectrum are of different
nature. Even though electronic interactions are already relevant for the self-
energy of the single-particle excitations, as, e.g., captured by our $GW$
approximation above, the optical spectrum is additionally affected by the
Coulomb coupling between electrons and holes generated by the photo-
excitation, lying at the very heart of exciton physics. For a theoretical
description of the two-particle properties and the excitation spectrum
including electron-hole interactions we solve the Bethe-Salpeter equation
(BSE) for the electron-hole amplitude [25, 26, 28]:

$(E_{c{\bf k}}-E_{v{\bf k}})A_{vc{\bf k}}+\Sigma_{v^{\prime}c^{\prime}{\bf
k}^{\prime}}\langle vc{\bf k}|K_{eh}|v^{\prime}c^{\prime}{\bf
k}^{\prime}\rangle A_{v^{\prime}c^{\prime}{\bf k}^{\prime}}=\Omega A_{vc{\bf
k}}\,,$ (2)

where the electronic excitations are given in the basis of electron-hole pairs
with quasi-particle energies $E_{v{\bf k}}$ and $E_{c{\bf k}}$ in the valence
and conduction bands. The $A_{vc{\bf k}}$ are the coefficients of the excitons
in the electron-hole basis and $\Omega$ are the eigen-energies. The kernel
$K_{eh}$ accounts for the screened Coulomb interaction between electrons and
holes, and the exchange interaction, including also local field effects which
are due to the optical excitations of the periodically arranged atomic
orbitals in Bi/SiC [48].

Figure 4: Ab-initio $GW$+BSE results. a The solid line displays the absorption
spectrum of Bi/SiC inferred from the imaginary part of the dielectric function
from the BSE calculation. The dashed line displays the joint density of states
from the $GW$ calculation, where the electron-hole interaction is neglected. b
Convergence of the A and B exciton peaks, as a function of the inverse of the
number of $k$-points. The intersection of the gray dashed line with the
$y$-axis gives the result ideally obtained from an infinitely dense $k$-mesh.
c Lateral and top views of the A exciton wave-function. The radius of the
exciton is $\sim$3\text{\,}\mathrm{nm}\text{/}$$.

The absorption spectra calculated with and without electron-hole interactions
are displayed in Fig. 4a. From these calculations, we can extract an excitonic
binding energy of $\sim 0.15\text{\,}\mathrm{eV}\text{/}$ for the lowest
excitonic peak A, estimated as the energy difference between the independent
particle $GW$ absorption gap $E_{\mathrm{direct}}$ indicated by the arrow and
the energy position of peak A. The second evident peak B is shifted by an
energy comparable to the Rashba splitting of the valence states at the K and
K’ valleys. This results in a significant separation between the A and B
excitonic peaks, and the eventual merging of the latter into the continuum of
electron-hole excitations (forming a Fano resonance depending on coupling
strength between exciton B and continuum).

We find that the position of the theoretical A and B peaks is highly sensitive
to the number of $k$-points employed in the BSE calculation. Fig. 4b shows the
convergence for several $k$-meshes, highlighting a linear scaling towards
$E_{\mathrm{A}}\approx$1.23\text{\,}\mathrm{eV}\text{/}$$ for an infinitely
dense sampling, in excellent agreement with its measured value
($1.19\text{\,}\mathrm{eV}\text{/}$). For exciton peak B, the extrapolation
yields a theoretical value of
$E_{\mathrm{B}}\approx$1.73\text{\,}\mathrm{eV}\text{/}$$, still satisfyingly
close to the experimental result ($1.63\text{\,}\mathrm{eV}\text{/}$).

The aforementioned convergence issue arises from the extended nature of the
Wannier-Mott type excitons in Bi/SiC. In Fig. 4c, we give side and top views
of the A peak excitonic wave-function obtained from our calculations. The
exciton radius of $\sim 3\text{\,}\mathrm{nm}\text{/}$ extends over several
lattice constants, resulting from the strong localization in reciprocal space
around the K and K’ valleys. This, in turn, requires a dense $k$-mesh to
reduce the spurious Coulomb repulsion between periodic excitons. Remarkably,
Fig. 4c shows also how the exciton wave-function is not strictly localized
inside the Bi monolayer, but rather extends for at least two atomic layers
into the SiC substrate. This further evidences the pivotal role played by the
substrate in bismuthene at the single- and two-particle levels: SiC is not
only responsible for the orbital filtering mechanism and the huge topological
gap [19, 21], but also provides an important screening channel for the Coulomb
interaction.

In conclusion, we report the observation of excitons in a large-gap,
atomically-thin QSH insulator at room temperature. We capture two prominent
transitions in the many-particle response of bismuthene which we
quantitatively associate with excitonic transitions from the Rashba-split
valence bands. It is important to emphasize that the many-particle response of
bismuthene differs fundamentally from what is known from the vastly studied
2H-TMDCs. First, whereas in the latter materials the low-energy interband
transitions are captured by two decoupled gapped chiral Dirac fermion models
[16, 17] with the same winding number in each valley (with opposite sign
between K/K’-valley), in bismuthene, low-energy interband transitions are
described by chiral models with different winding numbers in each valley.
Therefore, in contrast to the strict valley-locked optical selection rules
that govern 2H-TMDCs, bismuthene is sensitive to controlling the coupling to
$\sigma^{+}$\- and $\sigma^{-}$-polarized light by reordering electronic bands
with external electric and magnetic fields [18, 49] (see also Supplementary
Information). The latter provides a clear technological advantage over TMDCs
for engineering electro-optical devices based on atomically thin materials.
Second, in bismuthene the excitonic _and_ the topological physics originate
from the very same electronic states, i.e., they establish a direct link
between excitonic physics and topology, which has only been proposed
theoretically so far (see for example, Ref. [50]). We foresee that our
findings trigger a plethora of future experimental investigations, for example
related to the helical excitonic transport in topological edge modes [35],
finding and assessing “global” topological properties of the QSH phase via
optical band-to-band transitions and selection rules (see also Supplementary
Information), topological polaritonics [51], and other exotic many-body
quantum phases.

Methods
Sample preparation – The bismuthene sample was grown by molecular beam epitaxy
using a Knudsen cell as Bi source and SiC(0001) as supporting substrate.
During growth, the temperature of the SiC substrates was controlled via direct
current resistive heating. The characteristic of this heating method is a
temperature gradient of $\sim
20\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ along the longitudinal
extent of the sample (the horizontal direction). As bismuthene growth is
sensitive to such a narrow temperature window, we observe a strong correlation
of bismuthene film coverage with the spatial position in the horizontal
direction. Based on low energy electron diffraction (LEED), the bismuthene
film quality can be macroscopically checked. High intensity of the
$\sqrt{3}\times\sqrt{3}$ spots associated with the bismuthene reconstruction
on SiC(0001) and low diffuse background intensity provide a qualitative
measure of film quality (see Supplementary Information).

Optical setup – In the PR experiment (Fig. 2a and Fig. 3a,c), the sample is
kept under a N2 inert gas atmosphere ($<0.1$ ppm H2O; $<0.1$ ppm O2) and
mounted on an x-y-z mechanical stage, allowing for selecting the probed
sample’s area with a $\sim 100\text{\,}\mathrm{nm}\text{/}$ accuracy in each
spatial direction. The infinity-corrected microscope objective ($0.42$
numerical aperture and $20\text{\,}\mathrm{mm}\text{/}$ working distance) used
in the PR experiment operates in the spectral range
$0.41.8\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ with $\times 20$
magnification. The focal plane of the modulation laser beam is inspected
through the imaging setup with a CCD camera showing the sample surface, the
white light spot, and the laser spot. Since the modulation laser spot is
roughly $1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ in diameter, it
defines the total area of a sample tested in a single experimental run. White
light comes from the Halogen lamp. A mechanical chopper modulates the laser
beam with a $50\%$:$50\%$ duty cycle at an $f_{\mathrm{m}}$ frequency
depending on the utilized photon detector: (i) the
$f_{\mathrm{m}}=$10\text{\,}\mathrm{Hz}\text{/}$$ for the Si diode, sensitive
in the $0.31.0\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ spectral range,
or (ii) $f_{\mathrm{m}}=$270\text{\,}\mathrm{Hz}\text{/}$$ for the InGaAs
photo-diode, covering the spectral range of
$0.91.5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. Several types of laser
sources are employed: (a) the continuous-wave (CW) neodymium-doped yttrium
aluminum garnet laser emitting at $532\text{\,}\mathrm{nm}\text{/}$ photon
wavelength; (b) the ultrafast Ti: Sapphire oscillator, providing a train of
$\sim$140\text{\,}\mathrm{fs}\text{/}$$-long-pulses at 76 MHz repetition
frequency, spectrally tunable in the range of
$0.70.95\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ or
$0.350.475\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ (second harmonic
generation), and (c) the synchronously-pumped optical parametric oscillator
providing a train of $\sim$200\text{\,}\mathrm{fs}\text{/}$$-long-pulses
spectrally tunable in the range of $0.50.76\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ or $0.971.1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. A
spectral light analysis is provided by a spectrometer consisting of a
$0.3\text{\,}\mathrm{m}\text{/}$-focal-length grating monochromator
($0.2\text{\,}\mathrm{nm}\text{/}$ ultimate spectral resolution) and two
detectors mentioned above. The spectral resolution is limited to an
arbitrarily chosen monochromator scanning step in the PR experiment, which is
$\sim 4\text{\,}\mathrm{meV}\text{/}$. The lock-in amplifier reads the
detector signal at the $f_{\mathrm{m}}$ carrier frequency.

Theoretical framework – The approach to determine and quantify the importance
of the electron-hole interaction, and corresponding excitonic effects, in
Bi/SiC employed the following methods (in this hierarchical order):
$DFT\rightarrow GW\rightarrow BSE$. Density Functional Theory (DFT) is used to
understand the ground-state properties, with the interacting many-electrons
problem being mapped onto a non-interacting electrons problem. Spectroscopic
properties, however, involve excited particles above the ground state and a
correct step in this direction can be done by considering the many-body
electron-electron interaction in the framework of $GW$ approximation. Starting
from DFT single-particle energies, one can compute the Quasi-Particle (QP)
energies by taking into account the self-energy $\Sigma$ expressed through the
single-particle Green’s function $G$ and the screened Coulomb interaction $W$
(here computed within the random phase approximation RPA including local field
effects). This approximation is known to be correct at about 0.1 eV [52],
which makes our $GW$ results overall consistent with all our STS spectra shown
in the manuscript. Finally, on top of the $GW$ results, the effect of the
electron-hole interactions are included using the Bethe-Salpeter Equation
(BSE) and the excitonic two-body wave-function is written via a product of
occupied and unoccupied orbitals. The results here presented have been
obtained using the simulation packages Quantum Espresso [53] and YAMBO [54,
55]. Norm-conserving pseudo-potentials are used to quantify the electron-ion
interactions and the Perdew-Burke-Ernzerhof (PBE) functional [56] is used for
the exchange-correlation potential. Wave-functions are expanded in plane-waves
with an energy cut-off of 40 Ry, large enough to guarantee converged results.
The Brillouin zone is then sampled with a 12$\times$12$\times$1 grid for DFT
ionic relaxation and non self-consistent calculations, but it has been
increased up to 24$\times$24$\times$1 $k$-points for many-body calculations.
Spin-orbit coupling is included self-consistently. QP energies are then
computed using a single shot $G_{0}W_{0}$ approximation.

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Acknowledgements
The authors acknowledge insightful discussions with B. Rosenow, B. Scharf and
C. Franchini. The research leading to these results has received funding from
the European Union’s Horizon 2020 research and innovation program under the
Marie Skłodowska-Curie Grant Agreement No. 897276. We gratefully acknowledge
the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu) for
funding this project by providing computing time on the GCS Supercomputer
SuperMUC-NG at Leibniz Supercomputing Centre (https://www.lrz.de). We are
grateful for funding support from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy through the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter ct.qmat (EXC 2147, Project ID 390858490) as well as through the
Collaborative Research Center SFB 1170 ToCoTronics (Project ID 258499086). P.
H. acknowledges support from Polish National Science Center within the Etiuda
8 scholarship (Grant No. 2020/36/T/ST5/00511). M. S. acknowledges support from
the Polish National Science Center within the Opus 18 Grant
No.2019/35/B/ST5/04308 and the Polish National Agency for Academic Exchange
(PPI/APM/2018/1/00031/U/001). The Flatiron Institute is a division of the
Simons Foundation.

Author contributions
C.S. and R.C. conceived the project. M.S., P.H, P.W. and Ł.D. performed the
photo-modulated reflectivity experiments and analyzed the resulting data with
help from S.H. and C.S.. R.S. has grown the sample and conducted the STM/STS
and ARPES characterization. R.S. and F.R. conducted the XPS characterization.
A.C. and D.D.S. performed the ab-initio $GW$+BSE calculations, with input from
R.T. and W.H.. C.S., R.C., W.H. and D.D.S. wrote the manuscript with input
from all coauthors.

Additional information
Extended data is available for this paper at https://xxxx
Supplementary information The online version contains supplementary material
available at https://xxx
Correspondence and requests for materials should be addressed to Christian
Schneider, Ralph Claessen and Domenico Di Sante.

Competing financial interests
The authors declare no competing financial interests.