# Electronic localization in high-angle twisted boron nitride bilayer

Sylvain Latil  Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette,
France  Van Binh Vu  Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-
Yvette, France  Elisa Serrano Richaud  Université Paris-Saclay, ONERA, CNRS,
Laboratoire d’étude des microstructures (LEM), 92322, Châtillon, France  Hakim
Amara  Université Paris-Saclay, ONERA, CNRS, Laboratoire d’étude des
microstructures (LEM), 92322, Châtillon, France  Université Paris Cité,
Laboratoire Matériaux et Phénomènes Quantiques (MPQ), CNRS-UMR7162, 75013
Paris, France  Lorenzo Sponza  Université Paris-Saclay, ONERA, CNRS,
Laboratoire d’étude des microstructures (LEM), 92322, Châtillon, France
European Theoretical Spectroscopy Facility (ETSF), B-4000 Sart Tilman, Liège,
Belgium

###### Abstract

We investigate the electronic structure and the optical absoprtion onset of
hexagonal boron nitride twisted bilayers at twist angles in the vicinity of
30°. Our investigations are done with ab initio density functional theories
and a purposely designed tight-binding model. Despite its few parameters, we
verify that the model ensures high precision on low-energy excitations. We
demonstrate that approaching 30°, all stacking sequences tend to have the same
band structure which is characterised by a bundle of flat bands laying
slightly above the bottom conduction. The simulated independent-particle
absorption spectra display an intense and well-defined peak at the onset,
stemming solely from transitions to the bundle of flat states. These results
suggest the presence of strong, stable and stacking-independent optical
properties of boron nitride 30°-twisted bilayers. More importantly, the
formation of flat bands and the indistinguishability of the stacking sequence,
are expected to be common characteristics to all homobilayers made hexagonal
monolayers at twist angles close to 30°. In fact, they are a consequence of
the loss of translation symmetry (quasicrystal limit) occurring in all these
systems at twists of exactly 30°.

The wide spectrum of physical properties exhibited by twisted bilayers has led
to significant advances giving rise to the promising field of twistronics Carr
_et al._ (2020); Andrei  _et al._ (2021); Inbar  _et al._ (2023); Du  _et al._
(2023); Liu  _et al._ (2021). By stacking 2D atomic layers to form van der
Waals heterostructures, a geometric moiré superlattice emerges as a result of
a lattice mismatch or a rotational twist Gratias and Quiquandon (2023). The
resulting pattern modulates the potential at the supercell scale and hence
changes the electronic band structure typically through the formation of low
dispersing bands which possibly lead to peculiar transport characteristics, as
unconventional superconductivity Yankowitz  _et al._ (2019); Balents  _et al._
(2020) or insulating behavior driven by correlations Cao  _et al._ (2018).
Typical examples of moiré composites include the pioneering twisted bilayer
graphene Bistritzer and MacDonald (2011), twisted hexagonal boron nitride
(hBN) Xian  _et al._ (2019); Kim  _et al._ (2023); Liu  _et al._ (2021),
hetero- and homobilayers of transition metal dichalcogenides (TMDs) Scuri  _et
al._ (2020) as well as graphene on lattice-mismatched hBN substrates Wang  _et
al._ (2019).

Figure 1: The five hexagonal stackings of BN bilayers clased into the single
and double sublattice coincidence groups. Their stacking sequence takes the
name from the atoms perfectly stacked at high symmetry points (red circles).
The rotation angles are defined as the angles spanned by the B-N bonding at
the coincidence point moving from the bottom layer to the upper layer.

In this context, semiconducting twisted systems, like hBN or TMDs, are
peculiar. In these bilayers, the width of band edge states decreases
continuously with the angle of twist (no magic angle) and the presence of
different atomic species generates several stacking possibilities with
specific electronic properties, hence providing an additional degree of
freedom with respect to graphene bilayers Zhao  _et al._ (2020); Walet and
Guinea (2021); Latil  _et al._ (2023). More specifically, hBN having the
widest band gap in the monolayer (larger than 7 eV Blase  _et al._ (1995);
Wirtz  _et al._ (2006); Ribeiro and Peres (2011); Galvani  _et al._ (2016))
presents outstanding optical properties Sponza  _et al._ (2018); Schué  _et
al._ (2019) with many possible applications Knobloch  _et al._ (2021).

Since the early stages of the research on twist-angle physics, the scientific
community has been investigating the small twist angle limit. In fact, there
are no exhaustive theoretical studies in the literature about large-angle
twisted bilayers. At best of our knowledge, only 30° twisted hBN bilayers
Chernozatonskii and Kochaev (2023) and graphene Yu  _et al._ (2019) have been
considered. Even thought density functional theory (DFT) calculations suggest
that this BN material is a new wide-gap 2D quasicrystal Chernozatonskii and
Kochaev (2023), its electronic and optical properties have never been
addressed specifically. Very recently it has been highlighted in AlN and GaN
binary materials Kang  _et al._ (2023) the emergence of localized states
responsible of unique features. A similar physics is expected to occur also in
large-angle twisted hBN bilayers. Most importantly, a 30°rotation produces
quasicrystals without translational symmetry in all hexagonal bilayers,
encompassing all most popular 2D materials like hBN, graphene, TMDs and their
heterostructures, so all these systems must share some specific
characteristics when twisted at large angles.

prefactor | value (eV) | decay | value
---|---|---|---
$\gamma^{BB}$ | 2.45 | $q_{BB}$ | 3.0
$\gamma^{BN}$ | 0.75 | $q_{BN}$ | 2.0
$\gamma^{NN}$ | 0.32 | $q_{NN}$ | 1.6
Table 1: Set of parameters for the interlayer hopping.

In this Letter, we determine the evolution of the band structure and optical
response as a function of the twist angle in the vicinity of 30° for different
stacking sequences of twisted hBN bilayers. By introducing a tight-binding
(TB) model parametrized on DFT calculations, we show two surprising features.
The first is that as the twist approaches 30°(the asymptotic quasicrystal
limit), the electronic properties of all stackings tend to the same band
structure. This can be understood by the fact that the system slowly
approaches the asymptotic limit where it loses all translational symmetries.
In other words, the system tends to a limit where all local configurations co-
exist and the very notion of stacking sequence has no meaning. The second is
that this common electronic structure is characterised by a strong electron
localization resulting from the emergence of a bundle of low-dispersing states
in the middle of the conduction band. We also show that, despite being in the
middle of the conduction, the bundle is entirely responsible of a very intense
and robust absorption onset which may give rise to spectacular excitonic
properties.

To identify univoquely the structures we investigate, we use the definitions
and nomenclatures that has been introduced in Latil  _et al._ (2023) by some
of us. Let us recall the most important results for the current work. Every BN
bilayer with hexagonal symmetry can be identified univoquely by a stacking
label and a couple of integers $(q,p)$ defining the moiré supercell. There are
only five possible stacking sequences for each $(q,p)$-pair. The five stacking
labels take their name from the pair of atoms placed exactly on top of each
other on the three high-symmetry points of the hexagonal supercell that are
the origin, the point (1/3 1/3) and the point (2/3 2/3). The five stackings
divide into single-coincidence stackings (labelled BB, BN and NN) and double
coincidence stackings (labelled BBNN and BNNB). The $(q,p)$ indeces define two
matrices that, once applied to monolayer unitary vectors, generate the
supercell of the lower layer and the twisted supercell of the top layer. If we
take without loss of generality $p>q$, then the rotation of the top layer with
respect to the bottom layer is given either by an angle $\theta$ (in the
single-coincidence systems) or $-\theta^{\prime}$ (in the double-coincidence
case) with $\theta^{\prime}=\pi/3-\theta$ (cfr. Figure 1. Both twist angles
can be derived from $p$ and $q$ with formulae derived in Latil  _et al._
(2023).

Figure 2: DFT and TB bandstructure of the BB and BN stackings in the (1,3)
supercell, corresponding to a twist angle of $\theta=21.79$°.

In order to study in detail low-energy excitations in these systems, we
developed a tight-binding model purposely designed to describe accurately the
last occupied and the first empty states. We took inspiration from a previous
model developed for graphene moiré bilayers Trambly de Laissardière  _et al._
(2010, 2012) and extended to twisted bilayer MoS2 Venkateswarlu  _et al._
(2020). The basis is constituted of the $p_{z}$ orbitals of B and N and the
Hamiltoinan relies on few parameters: the on-site energy difference
$\Delta=4.90$ eV between B and N species and the first-neighbour in-plane
hopping $t_{\parallel}=-2.65$ eV. The interlayer hopping is included only for
the $\sigma$ components and has a distance-dependent form
$t_{\perp}^{XY}(r)=\gamma^{XY}e^{-q_{XY}(a-r)/a}$ with $a$ being the
interlayer distance 3.22 Å Latil  _et al._ (2023), $r$ being the distance
between two sites belonging to different layers and $XY$ labelling the
pairings $BN$, $BB$ and $NN$. The values of the $\gamma^{XY}$ and $q_{XY}$
parameters are reported in Table 1. The quality of the model can be
appreciated in Figure 2 where we report the top valence and bottom conduction
states of the BB and the BN stackings at twist angle $\theta=27.80$°. These
two stackings have been chosen as paradigmatic because of some specific
characteristic of their band structure that is discussed in more detail below,
and because the corresponding bilayers belong to two different layer symmetry
groups. These have been computed with our TB model (black solid lines) and the
DFT free simulation package Quantum ESPRESSO Giannozzi  _et al._ (2009, 2017)
(red dashed lines).

Figure 3: Independent-particle absorption spectra of the BB(1,3) and BN(1,3)
bilayers computed with ab initio and tight-binding methods.

Our TB model is parametrized to reproduce well some specific features of the
band structure like the general dispersion of the electronic states and the
gapwidth. Particular attention has been paid on the description of the two
bands forming the bottom of the conduction. Indeed, the shape and the flatness
of the bottom conduction bands are correctly reproduced by the model as well
as the ordering. In fact, the DFT predicts the formation of a pretty flat
dispersion in the M-K region in both systems. Actually the two bands avoid
each other in the BB(1,3) electronic structure, even though the splitting is
extremely small, whereas they cross at K in the BN(1,3), consistenlty with
what simulated at smaller angles Latil  _et al._ (2023). Our model catches
very well these features although the splitting in the BB stacking is somewhat
overestimated.

The imaginary part of the independent-particle transverse dielectric function
$\varepsilon(\omega)$ at first order in the coupling with the vector potential
$\textbf{A}(\textbf{r},t)$ is obtained through the formula Grosso and
Parravicini :

$\varepsilon(\omega)=\frac{e^{2}\pi}{m_{0}^{2}\varepsilon_{0}\omega^{2}}\sum_{\textbf{k},m,\mu}\left\lvert\text{v}_{\mathbf{k}\mu
m}(\hat{\bm{e}})\right\rvert^{2}\delta(E_{\mathbf{k}\mu}-E_{\mathbf{k}m}-\hbar\omega)\,,$
(1)

where $e$ is the absolute value of the electron charge, $m_{0}$ is the
electron mass and $\varepsilon_{0}$ is the vacuum permittivity. This
expression describes the absorption of a photon with energy $\hbar\omega$ and
polarization vector $\hat{\textbf{e}}$ through the promotion of an electron
from the valence state $\ket{m,\textbf{k}}$ of energy $E_{\mathbf{k}m}$ to the
conduction state $\ket{\mu,\textbf{k}}$ of energy $E_{\mathbf{k}\mu}$. The
velocity matrix element $\text{v}_{\mathbf{k}\mu
m}=\bra{\mu,\textbf{k}}\hat{\textbf{e}}\cdot\hat{\textbf{v}}\ket{m,\textbf{k}}$
is obtained from the eigenstates of the operator
$\hbar\hat{\textbf{v}}=i[\hat{\textbf{H}},\hat{\textbf{r}}]$. We evaluated
$\varepsilon(\omega)$ in the same systems as before using eigenvalues and
eigenfunctions coming from our DFT calculations, by means of the Yambo Marini
_et al._ (2009) code, and from our TB model. Results are reported in Figure 3.
Both methods predict a well-detached peak at 4.5 eV, corresponding to
transitions towards the bottom conduction states. As expected from the
electronic structure, differences between the two stackings are essentially
negligible, indicating that not only the band structure but also the
corresponding wavefunctions are remarkably similar.

Figure 4: From top to bottom: Bundle states in the conduction bands of the
(3,8), the (4,11) and the (11,30) supercells of all the five stacking
sequences (in different colors).

Now that we have validated the TB model, we can extend our investigation to
twist angles closer to 30°and systems that can hardly be addressed with DFT.
In Figure 4 we report the bottom conduction states of the five stackings in
the supercells (3,8), (4,11) and (11,30) corresponding to twist angles ranging
from 29.41° to 29.96° in the single coincidence stackings (BB, BN and NN) and
from 30.59° to 30.04° in the double coincidence ones (BBNN and BNNB). The
tendency observed already in the (1,3) supercells is here confirmed and
strengthen: all stackings present basically the same band structure at fixed
supercell.

More interestingly, many low-dispersing states forming a bundle of flat bands
are present above the bottom conduction and concentrated in an interval
$\Delta E$ centered around 4.40 eV and pretty narrow (about 100 meV). Quite
interestingly, this interval is at about the same energy where the M-K flat
region was encountered in the (1,3) stackings (cfr. Figure 2). Such behaviour
is in contrast to small-angles twisted hBN bilayers where one or more single
states are formed directly in the gap and clearly separated in energy by about
0.1 eV Xian  _et al._ (2019); Zhao  _et al._ (2020); Liu  _et al._ (2021).
Furthermore, BN being recognized as a cardinal compound in 2D material
research for its exceptional optical features, it is therefore particularly
worthwhile to understand the impact of these emerging states on absorption
properties.

To this aim we computed with TB the independent-particle absorption spectra of
the BB(3,8), the BB(4,11), the BN(3,8) and the BN(4,11) bilayers and we report
them in Figure 5 as red solid and black dashed lines. As in the case of the
(1,3) stackings, the spectral onset is dominated by a very intense and well
detached peak at an energy compatible with the bundle states. As a
confirmation of its origin, we computed the spectra including only transitions
toward the bundle states (shaded patterns). We recover the same peak at
essentially the same intensity indicating that, despite being in the middle of
the conduction band, the bundle states are responsible of the absorption
onset, the bottom conduction contributing only at higher energies.

Figure 5: Normalised independent-particle absorption spectra at the onset. BB
stacking in the left panel, BN stacking in the right panel. Supercells (3,8)
and (4,11) in black dashed and solid red respectively. Patterned areas stand
for spectra obtained including allowing $\mu$ of equation (1) to run only on
the bundle state index.

To conclude, we have investigated the electronic and optical properties of hBN
bilayers at twist angles close to 30°. To this aim, we have developed a simple
and versatile tight-binding model which accurately describes low-energy
excitations in all stacking sequences and twist angles. Its parametrisation
has been validated against ab initio calculations of band structure and
optical absorption. With our model, we have demonstrated that at twist angles
close to 30° all hBN bilayers develop the same electronic properties,
irrespective to the stacking sequence. This common electronic structure is
characterised by the emergence of a bundle of low-dispersing states right
above the bottom of the conduction band which are responsible of an intense
and robust peak at the onset of the absorption spectrum. Our results suggest
that 30°-twisted BN bilayers may host extremely strong excitonic phenomena
independent on the stacking sequence. More generally, the tendency of
developing the same electronic structure is a consequence of the fact that any
hBN bilayer becomes a quasicrystal in the 30° limit, i.e. a structure with no
translation symmetry where all the possible local configurations co-exist.
Since this very characteristic is actually common to the majority of
homobilayers, we predict that a similar trend can be found in the 30° limit in
all most common 2D materials including graphene, transition metal
dichalcogenides, buckled graphene structures (antimonene, silicene),
transition metal monochalcogenides, and all homostructures formed of hexagonal
single-layers.

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