# Cubic BN optical gap and intragap optically active defects

Anna Tararan  Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR
8502, F-91405, Orsay, France  Stefano di Sabatino  Laboratoire des Solides
Irradiés, École Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128
Palaiseau, France  European Theoretical Spectroscopy Facility (ETSF)  Matteo
Gatti  Laboratoire des Solides Irradiés, École Polytechnique, CNRS, CEA,
Université Paris-Saclay, F-91128 Palaiseau, France  European Theoretical
Spectroscopy Facility (ETSF)  Synchrotron SOLEIL, L’Orme des Merisiers, Saint-
Aubin, BP 48, F-91192 Gif-sur-Yvette, France  Takashi Taniguchi  National
Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
Kenji Watanabe  National Institute for Materials Science, Namiki 1-1, Tsukuba,
Ibaraki 305-0044, Japan  Lucia Reining  Laboratoire des Solides Irradiés,
École Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau,
France  European Theoretical Spectroscopy Facility (ETSF)  Luiz H. G. Tizei
Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR 8502, F-91405,
Orsay, France  Mathieu Kociak  Laboratoire de Physique des Solides, Univ.
Paris-Sud, CNRS UMR 8502, F-91405, Orsay, France  Alberto Zobelli
<EMAIL_ADDRESS>Laboratoire de Physique des Solides, Univ. Paris-
Sud, CNRS UMR 8502, F-91405, Orsay, France

(August 27, 2024)

###### Abstract

We report a comprehensive study on the optical properties of cubic boron
nitride ($c$–BN) and its optically active defects. Using electron energy-loss
spectroscopy (EELS) within a monochromated scanning transmission electron
microscope (STEM) on the highest-quality crystals available, we demonstrate
unequivocally that the optical-gap energy of $c$–BN slightly exceeds 10 eV.
Further theoretical analysis in the framework of the Bethe-Salpeter equation
of many-body perturbation theory supports this result. The spatial
localization of defect-related emissions has been investigated using
nanometric resolved cathodoluminescence (nano-CL) in a STEM. By high-
temperature annealing a $c$–BN powder, we have promoted phase transitions in
nanometric domains which have been detected by the appearance of specific
hexagonal-phase signatures in both EELS and CL spectra. A high number of
intragap optically active centers are known in $c$–BN, but the literature is
rather scattered and hence has been summarized here. For several emission
lines we have obtained nano-CL maps which show emission spot sizes as small as
few tens of nanometers. Finally, by coupling nano-CL to a Hanbury-Brown-Twiss
intensity interferometer, we have addressed individual spots in order to
identify the possible presence of single-photon sources. The observed CL
bunching effect is compatible with a limited set of single-photon emitters and
it permits obtaining emission lifetimes of the order of the nanosecond.

##  I Introduction

Group-III nitrides are wide band-gap semiconductors at the core of many modern
optoelectronic applications. Currently, AlN, GaN, InN and ternary compounds
are for instance fundamental for light-emitting devices. Due to the high
stability of the N2 molecule, these materials are not found in nature. The
synthesis of high-quality crystals has been a challenge for a long time before
these structures could be investigated for technological purposes. Boron
nitride belongs to the same group of semiconductors but its interest as a
potential optical material is still in its infancy. In the last twenty years,
the synthesis of high-quality hexagonal BN ($h$–BN) crystals has permitted
important steps towards the comprehension of the peculiar optical properties
of this wide band-gap layered material. However, studies on other boron-
nitride phases are more incomplete due to the even more limited availability
of high-quality crystals.

Cubic boron nitride ($c$–BN) is the zinc-blend allotropic form of boron
nitride, also known as sphalerite phase. Boron and nitrogen are tetrahedrally
coordinated forming a double-face-centered cubic lattice. The crystal
structure appears thus isoelectronic with diamond, as $h$–BN is with graphite.
The discovery of $c$–BN was inspired by the successful conversion of graphite
into diamond, achieved at General Electrics in 1955.Bundy  _et al._ (1955);
Hall (1961) One year later, in the same research group, Robert H. Wentorf
succeeded in the synthesis of $c$–BN (brand name Borazon) by heating boron and
nitrogen mixtures at high pressure with a nitride catalyst.Wentorf (1957,
1962) $c$–BN is now commonly synthesized by the temperature-gradient method at
high pressure and high temperature (HPHT) in a variety of alkali or alkali-
earth B–N solvents.Mishima and Era (2000) Sintered $c$–BN is currently
extensively used for cutting and abrasive tools, especially in presence of
ferrous metals, which are able to degrade diamond.

Together with a clear interest due to its mechanical properties, the wide band
gap of $c$–BN, by analogy with diamond and other wide-gap nitrides, makes it a
very promising material for potential optical applications. $c$–BN shares
interesting features with its hexagonal counterpart, such as an intense
intrinsic luminescence and bright and stable luminescent crystal defects. The
quality of $c$–BN crystals is generally poor presenting a high density of
point and extended defects, impurities or residual $h$–BN inclusions.Taylor
_et al._ (1994); Matsumoto and Zhang (2001); Yu  _et al._ (2006) In 2001,
high-purity macroscopic crystals, with sizes of the order of 3 mm, were
obtained using fresh barium boron nitride (Ba3B2N4) as the solvent, with
particular care to avoid solvent degradation by air and humidity.Taniguchi and
Yamaoka (2001); Taniguchi and Watanabe (2007) These high-quality samples can
help in enlightening the intrinsic optical properties of $c$–BN and those of
optically active defects.

The $c$–BN band gap is expected to exceed that of $h$–BN, but its precise
value is still debated. Experimentally, a maximum estimation of 6.4 eV has
been deduced from absorption and reflectance measurements.Chrenko (1974);
Mishima and Era (2000) Theoretically, the $c$–BN band structure has been found
to display an indirect band gap. State-of-the-art quasiparticle (QP)
calculations in the GW approximationHedin (1965) (GWA) have determined a value
of the _indirect_ band gap of 6.36-6.66 eV,Klimeš  _et al._ (2014); Jiang and
Blaha (2016) which is only fortuitously in close agreement with experiments.
As a matter of fact, a gap obtained from optical measurements should be
compared more appropriately to the smallest _direct_ gap in the QP band
structure, which in the GWA instead amounts to 11.29–11.66 eV.Klimeš  _et al._
(2014) Moreover, in an optical excitation the electron-hole attraction may
lead to the formation of bound excitons, which in turn reduces the optical gap
with respect to the QP direct gap. However, the solution of the Bethe-Salpeter
equation (BSE), which takes into account excitonic effects accurately,Onida
_et al._ (2002) has resulted into an exciton binding energy of only 0.35
eV,Satta  _et al._ (2004) i.e. way too small to resolve the disagreement with
the measured optical gap.

This difference of $\sim 5$ eV between experiments and theory is much larger
than it was in the case of $h$–BN, where the nature and the value of the band
gap also remained elusive for a long time. Both experimental and theoretical
studies had initially provided underestimated values for the $h$–BN optical
gap, mistaking deep-defect-related emissions with the optical band edge.Zunger
_et al._ (1976); Taylor  _et al._ (1994) Cathodoluminescence (CL) measurements
in the UV range on high-purity crystals have been achieved only in
2004Watanabe  _et al._ (2004) and in 2006 GW-BSE calculations were able to
reconcile theory and experiment by showing that $h$–BN is characterized by an
indirect QP band gap and a large exciton binding energy.Arnaud  _et al._
(2006); Wirtz  _et al._ (2006) Very recently, Cassabois and coworkers, on the
basis of photoluminescence (PL) experiments, have corroborated that
conclusion, highlighting the role of phonons in the high-energy region of the
spectrum and the presence of a weak zero-phonon excitonic line at 5.95
eV.Cassabois  _et al._ (2016a, b)

While the GW-BSE theoretical approach has been proven to be accurate and
predictive in $h$–BN,Arnaud  _et al._ (2006); Fugallo  _et al._ (2015) there
is no reason to expect that it should instead fail for $c$–BN. An indication
in this direction comes from the good agreement between non-resonant inelastic
X-ray scattering (NRIXS) spectra on $c$–BN crystals and GW-BSE calculations of
the dynamic structure factor as a function of momentum transfer
$\mathbf{q}$.Galambosi  _et al._ (2001) However, unfortunately NRIXS
experiments cannot access the optical limit $\mathbf{q}\rightarrow 0$, making
it difficult to determine the optical gap.

The large discrepancy between optical spectroscopy and theory in $c$–BN, on
the one hand, could be explained by the fact that the gap is most likely too
wide to be accessed by standard optical excitation techniques, such as optical
absorption and PL. Even using CL, where higher excitations can be in principle
addressed, the limitation comes from an efficient detection of the far UV
radiation. On the other hand, the generally poor quality of the investigated
crystals makes one wonder whether the existing optical spectra are influenced
by possible $h$–BN inclusions Satta  _et al._ (2004); Cappellini  _et al._
(2001); Widmayer  _et al._ (1999) and whether they are even largely dominated
by defect-related signatures.Tkachev  _et al._ (1985); Shipilo  _et al._
(1986); Zaitsev  _et al._ (1986); Shipilo  _et al._ (1988); Shishonok and
Steeds (2002); Manfredotti  _et al._ (2006)

This highly unsatisfactory situation thus calls for a new joint experimental
and theoretical effort in order to obtain a definite answer to the issue of
the $c$–BN gap. Furthermore, by analogy with $h$–BN, interesting optical
properties may be expected from $c$–BN luminescent defects themselves. Indeed,
in the last years it has been demonstrated that several emissions observed
within the $h$–BN optical gap may act as very stable and bright single-photon
sources (SPS) in the visible and UV spectral range.Tran  _et al._ (2016a, b);
Shotan  _et al._ (2016); Jungwirth  _et al._ (2016); Chejanovsky  _et al._
(2016); Martínez  _et al._ (2016); Exarhos  _et al._ (2017); Bourrellier  _et
al._ (2016) Similar sources could be present also in $c$–BN also in virtue of
its structural and electronic analogies with diamond and silicon carbide,
where SPS have been extensively documented.Aharonovich  _et al._ (2016)

To address the open questions on the optical properties of $c$–BN it is
necessary to investigate absorption and emission spectra in the far UV
spectral range. On the one hand, the relatively high density of defects, even
in the highest-quality samples, imposes a high spatial resolution to the
spectroscopy techniques employed. Conventional optical methods are diffraction
limited and even in a confocal microscope a spatial resolution below the
hundred of nanometers can hardly be achieved. On the other hand, high-
frequency lasers or doubling-frequency techniques are required to excite high-
energy levels. Both these limitations can be overcome by using high-energy
electrons as the exciting radiation. Indeed, in a scanning transmission
electron microscope (STEM) electron beams can be focused into sub-nanometric
or even sub-Ångstrom probes providing the spatial resolution necessary for the
study of highly heterogeneous materials. The response function of $c$–BN can
then be accessed in an extended energy range through low-loss electron energy-
loss spectroscopy (EELS). Furthermore, the recent development of nano-
cathodoluminescence techniques (nano-CL) within a STEM allows one to
investigate the luminescence of a material with a nanometric spatial
resolution.Zagonel  _et al._ (2010); Kociak and Zagonel (2017) Unlikely
photoluminescence, high-energy electrons permit an easy access to an energy
range whose upper limit is solely defined by the light collection-analysis
optical chain and not by the energy of the exciting radiation.Ponce  _et al._
(1996); Bourrellier  _et al._ (2014) Finally, nano-CL provides the capability
to address individual color centers and this signal can then be driven to a
light-intensity Hanbury-Brown-Twiss (HBT) interferometer.Hanburry Brown and
Twiss (1956) The potential nonclassical nature of highly localized emissions
can be demonstrated by the appearance of an anti-bunching signature in the
second-order correlation function, $g^{(2)}(\tau)$. Previously, this original
experimental setup has been successfully applied in the study of NV centers in
nanodiamonds and defect centers in h-BN.Tizei and Kociak (2013); Tizei  _et
al._ (2013); Bourrellier  _et al._ (2014); Meuret  _et al._ (2015)

In this work we have investigated the optical properties of $c$–BN crystals
using an experimental and theoretical approach combining electron energy-loss
spectroscopy, nano-cathodoluminescence and many-body perturbation theory. In
order to probe the intrinsic optical properties of $c$–BN and to separate
defect-related optical features, we have applied these high spatially resolved
techniques to the highest-quality samples available.

The paper is organized as follows. In Sec. II we provide an overview of the
theoretical and experimental methods employed. In Sec. III.1 we address the
question of the optical gap of $c$–BN by EELS. The use of a highly
monochromated electron beam permits also the observation of optical phonon
modes in the infrared-energy domain of the spectra. These experimental results
are then interpreted on the basis of numerical simulations and excitonic and
plasmonic features in the loss spectra are identified. In Sec. III.2 we show
modifications in EELS and nano-CL spectra occurring after high-temperature
annealing which promotes local phase changes within the $c$–BN crystals. In
Sec. III.3 we discuss defect-related emission in $c$–BN. After an overview of
the literature in the field (Sec. III.3.1), we present nanometric resolved
hyperspectral emission maps of defect emissions (Sec. III.3.2). Finally,
intensity interferometry experiments have been conducted for the higher-energy
features in order to establish the lifetime of the associated excited state
and their possible behavior as single-photon emitters (SPE).

##  II Methods

###  II.1 Samples preparation

Colorless high-quality $c$–BN single crystals have been synthesized by
spontaneous nucleation at high pressure and high temperature using the
temperature gradient method in Ba3B2N4 solvent.Taniguchi and Yamaoka (2001);
Taniguchi and Watanabe (2007) The crystals have been washed in acetone and
ethanol and successively crushed in a mortar to obtain a high-quality
submicrometric powder. TEM grids have been prepared by drop casting the
crystal dispersed in isopropanol. Lower-quality commercially available
micrometric powders (Element Six) have been employed in CL experiments of
defect centers. The latter samples have been used also for the study of
temperature-induced phase transformations.

###  II.2 Electron energy-loss spectroscopy and cathodoluminescence

Monochromated EELS experiments were performed on the ChromaTEM microscope
(modified Nion Hermes 200) operated at 60 keV with the sample at room
temperature. The electron-beam convergence half angle was 25 mrad and the EEL
spectrometer collection half angle was set to about 30 mrad. The electron beam
was monochromated to 20 meV full width at half maximum (FWHM) with an
electron-beam current of the order of a few pA. The spectrometer dispersion
was set to 1.9 meV/pixel, 9.6 meV/pixel and 53.3 meV/pixel dependently upon
the spectral range investigated. Plural scattering contributions were removed
using the Fourier-log method.

All nano-CL experiments and the EELS experiments correlated with nano-CL were
performed in a dedicated VG-HB501 STEM operating at 100 keV. The VG microscope
was provided with a liquid-nitrogen cooling system for the sample stage (150
K). Electron energy-loss hyperspectral images have been collected using a
spectrometer dispersion of 0.1 eV. The spectra have been deconvoluted using a
Richardson-Lucy routineGloter  _et al._ (2003) after removal of plural
scattering. CL signals were collected using an Attolight Mönch 4107 STEM-CL
system fitted with an optical spectrometer with a 300-groove diffraction
grating blazed at 2.5 eV or 4.1 eV dependently from the signal analyzed,
assuring an energy dispersion of 0.34 nm and 0.16 nm, respectively. The
sampling of the spectrometer CCD was 0.17 nm/pixel. Hyperspectral images were
obtained by sequentially recording one full CL spectrum per pixel while
scanning the sample with a nanometric step size.

###  II.3 Hanbury Brown and Twiss interferometry

In our experimental nano-CL setup, the CL signal collected by the parabolic
mirror has been driven to a HBT interferometer using an optical multi-mode
fiber ($600\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter core). Experiments
have been performed in the visible-UV region above 2.59 eV, selected by the
convergent lens at the entrance of the optical path (no additional filter has
been employed). Photon detection has been performed by two photomultiplier
(PMT) modules. Time-delay histograms (which are proportional to the second-
order correlation function) have been acquired using a Time Harp correlation
electronics from Picoquant. The typical room background noise varied from 200
to 500 count/s per PMT. Time-delay histograms have been normalized to one for
$\tau\gg 0$.

###  II.4 Theoretical framework and computational details

Calculations have been carried out within the framework of many-body
perturbation theory.111For an extended introduction to the theoretical
formalism see Ref. 103. In the $G_{0}W_{0}$ scheme,Hybertsen and Louie (1986);
Godby  _et al._ (1988) the GW self-energy $\Sigma_{xc}$ is obtained as a
convolution in frequency space of the Kohn-Sham (KS) Green’s function $G_{0}$
and the dynamically screened Coulomb interaction $W_{0}$ evaluated in the
random-phase approximation (RPA). The QP energies $E_{n\mathbf{k}}$ that form
the QP band structure are calculated evaluating the first-order perturbative
corrections with respect to the KS eigenvalues $\varepsilon_{n\mathbf{k}}$:

$E_{n\mathbf{k}}=\varepsilon_{n\mathbf{k}}+Z_{n\mathbf{k}}[\text{Re}\Sigma_{n\mathbf{k}}^{xc}(\varepsilon_{n\mathbf{k}})-V^{xc}_{n\mathbf{k}}]$
(1)

with $V_{xc}$ the exchange-correlation KS potential and the renormalization
factor $Z$ defined as:

$Z_{n\mathbf{k}}^{-1}=1-\left.\frac{\partial\text{Re}\Sigma_{n\mathbf{k}}^{xc}(\omega)}{\partial\omega}\right|_{\omega=\varepsilon_{n\mathbf{k}}}.$
(2)

In order to calculate the loss function, the BSE two-particle Hamiltonian
$H_{\textrm{exc}}$ has to be inverted, which can be formulated as an
eigenvalue problem. The direct electron-hole interaction derived from the GWA
is $W_{0}$. Here we use a static approximation to $W_{0}$. In the one-particle
transition basis
$|n\rangle=|\mathbf{k}vc\rangle=\phi_{v\mathbf{k}}(\mathbf{r})\phi_{c\mathbf{k}}(\mathbf{r})$
[where $\mathbf{k}$ is in the first Brillouin zone and in the Tamm-Dancoff
approximation $v$ ($c$) runs over valence (conduction) bands]
$H_{\textrm{exc}}$ reads:Albrecht  _et al._ (1998); Benedict  _et al._ (1998)

$\langle
n|H_{\textrm{exc}}|n^{\prime}\rangle=(E_{c{\mathbf{k}}}-E_{v{\mathbf{k}}})\delta_{n,n^{\prime}}+\langle
n|\bar{v}_{c}-W_{0}|n^{\prime}\rangle.$ (3)

Here $\bar{v}_{c}$ is a modified Coulomb interaction in which the long-range
component $\bar{v}_{c}(\mathbf{G}=0)$ is set to 0 in reciprocal space. From
the eigenvalues $E_{\lambda}$ and eigenvectors $A_{\lambda}$ of the excitonic
Hamiltonian (3), one obtains the macroscopic dielectric function
$\epsilon_{M}$, which in the dipole limit $\mathbf{q}\rightarrow 0$ is:

$\epsilon_{M}(\omega)=1-\lim_{\mathbf{q}\rightarrow
0}\frac{8\pi}{q^{2}}\sum_{\lambda}\Big{|}\sum_{{\mathbf{k}}vc}A_{\lambda}^{{\mathbf{k}}vc}\langle\phi_{v\mathbf{k}}|e^{-i\mathbf{q}\mathbf{r}}|\phi_{c\mathbf{k}+\mathbf{q}}\rangle\Big{|}^{2}\\\
\times\left[\frac{1}{\omega-
E_{\lambda}+i\eta}-\frac{1}{\omega+E_{\lambda}+i\eta}\right].$ (4)

Here both the resonant and antiresonant contributions (the first and second
term in the square bracket, respectively) are explicitly taken into account,
which is important for EELS spectra calculations.Olevano and Reining (2001)
The effect of the electron-hole interactions $\bar{v}_{c}$ and $W_{0}$ in Eq.
(3) is the mixing of the amplitudes through the coefficients $A_{\lambda}$ in
the first line of (4) and the change of the excitation energies $E_{\lambda}$
in the denominators of the second line, which in absence of excitonic effects
would just correspond to energy differences between QP bands:
$E_{c{\mathbf{k}}}-E_{v{\mathbf{k}}}$.

Finally, while $\text{Im}\epsilon_{M}(\omega)$ yields the absorption spectrum,
the loss function $-\text{Im}\epsilon_{M}^{-1}(\omega)$ measured by EELS (for
$\mathbf{q}\rightarrow 0$) can be conveniently expressed as:

$-\text{Im}\epsilon_{M}^{-1}(\omega)=\frac{\text{Im}\epsilon_{M}(\omega)}{[\text{Re}\epsilon_{M}(\omega)]^{2}+[\text{Im}\epsilon_{M}(\omega)]^{2}}.$
(5)

Peaks in the loss function derive either from $\text{Re}\epsilon_{M}=0$ with
$\text{Im}\epsilon_{M}$ not too large, which corresponds to plasmon
resonances, or from peaks of $\text{Im}\epsilon_{M}$, which corresponds to
(screened) interband transitions.

In this work we have employed norm-conserving Troullier-Martins
pseudopotentialsTroullier and Martins (1991) in a plane-wave approach.Gonze
_et al._ (2005) KS eigenenergies and eigenfunction have been obtained in the
local-density approximation (LDA) with a cutoff of 30 Hartree, adopting the
experimental lattice parameter $a=3.615$ Å.Wyckoff (2014) The Brillouin zone
has been sampled with a 4-times shifted 8$\times$8$\times$8 Monkhorst-Pack
grid of $\mathbf{k}$ points (which for the calculation of spectra were shifted
to avoid high-symmetry points). In the $G_{0}W_{0}$ calculations we have used
the Godby-Needs plasmon-pole approximation,Godby and Needs (1989) including
300 bands in the evaluation of the self-energy. The resulting GW corrections
to the LDA were taken into account in the BSE by applying a scissor operator
of 1.86 eV to open the LDA band gap and by stretching valence and conduction
bands by 6.7 % and 4.5 %, respectively. The BSE spectra converged with 30
bands. In the reciprocal space the dielectric function was represented as a
matrix of size 89 $\mathbf{G}$ vectors. To compare the calculated loss
function with the experiments we introduced a 0.5 eV gaussian broadening. For
BSE calculations we used the EXC code.exc

##  III Results and discussion

###  III.1 Electron energy-loss spectroscopy

Low-loss EELS is a very powerful technique to investigate material spectral
response in the wide energy-range relevant for large band-gap materials. The
few EELS spectra in the UV range published on poor quality $c$–BN crystals
presented a very poor energy resolution and a sole broad electronic excitation
centered at about 30 eV could be identified.Schmid (1995); Weißmantel  _et
al._ (2002)

In Fig. 1 we present three low-loss EELS spectra acquired from the same area
of a 80 nm thick $c$–BN crystallite derived by crashing a high-quality
macroscopic crystal. The thickness $t$ has been estimated from the Poisson
statistics as $t/{\lambda}=\ln(I_{t}/I_{0})$, where $I_{t}$ is the total area
of the spectrum, $I_{0}$ is the area under the zero-loss peak (ZLP) and
$\lambda$ is the mean free path for inelastic scattering, derived from
acquisition parameters and the effective atomic number of the specimen.Egerton
(2011) Thinner regions of the crystals were not considered to minimize
surface-loss contributions to the spectra. In Fig. 1(a) a well defined peak
appears at 143 meV which corresponds to optical phonon modes of $c$–BN, in
accordance with previous EELS experiments.Nicholls  _et al._ (2015, 2018);
Hage  _et al._ (2018) As the experiments are performed on the bulk of a cBN
crystal and are not resolved in momentum, we expect in an EELS experiment to
preferentially measure the longitudinal-optical mode (LO) with respect to the
transverse-optical (TO). In Fig. 1(b) the onset can be clearly identified with
the rise of the loss function around 10.1 eV and the appearance of a prominent
peak at 11.1 eV. The small constant background measured within the forbidden
optical region which progressively decreases at 6 eV can be tentatively
attributed to Cherenkov or Transition radiations (CR or TR) effects. CR happen
when the index of the medium is larger than the speed of light to electron
speed ratio. In our case, the calculated static index ($\sim 2.16$) is
slightly smaller than the light to electron speed ratio ($\sim 2.24$) which
suggests that the small plateau most probably arises from TR. The low-loss
spectrum recorded in a wider energy range is presented in Fig. 1(c) where a
broad and structured peak centered at 33 eV is visible. These results are
consistent with the NRIXS spectraGalambosi  _et al._ (2001) even though the
present EELS experiment is carried out at the optical limit, i.e. with a
momentum transfer $\mathbf{q}\rightarrow 0$, while NRIXS experiments were
performed at much higher $\mathbf{q}$.

Figure 1: EELS low-loss full spectrum of a thin region (about 80 nm thick) of
a $c$–BN high purity crystal. (a) Infrared spectral region showing a prominent
peak corresponding to $c$–BN optical phonon modes. (b) Spectral region
focusing on the onset of the response function. The weak signal in the lower
energy region corresponds to the signature of Cherenkov effects. (c) Complete
loss function presented after removal of multiple scattering. Figure 2: (a)
The loss function, (b) the real and (c) the imaginary part of the dielectric
function calculated from the BSE (violet solid lines) and neglecting the
electron-hole interactions in the GW approximation (green dashed lines). In
(a) the calculated loss functions are compared with the measured spectrum
where the zero loss peak has been removed (black dashed line).

In order to interpret these experimental findings, in Fig. 2(a) we compare the
measured EELS spectrum where the zero-loss peak has been removed (black lines)
with the loss function calculated from the BSE (violet solid lines). The
overall good agreement allows us to analyse the origin of the different
spectral features on the basis of the calculations. The main broad peak P is
the valence plasmon resonance: it corresponds to a zero of
$\text{Re}\epsilon_{M}$, see Fig. 2(b). The onset of the calculated loss
function is characterized by the two smaller structures I1 and I2. They are
interband transitions that stem from the peaks in the absorption spectrum
$\text{Im}\epsilon_{M}$, see Fig. 2(c), and are only slightly blueshifted by
the contribution of $\text{Re}\epsilon_{M}$ to Eq. (5). The onset of the
calculated spectra has an excitonic origin. This can be easily understood by
comparing the BSE spectra (violet solid lines) with the GW spectra (green
dashed lines), where the electron-hole interactions in Eq. (3) have been
neglected: besides a small redshift of all the spectra, as a result of
excitonic effects both the real and imaginary parts of the dielectric function
get strongly enhanced towards lower energies. In the GW absorption spectrum
the smallest-energy transitions take place between the threefold-degenerate
top-valence and bottom-conduction bands at the $\Gamma$ point of the band
structure, see Fig. 3. The minimum direct gap at $\Gamma$ is 8.83 eV in LDA
(blue lines in Fig. 3) and becomes 11.10 eV in GW (black lines in Fig. 3). The
effect of the GW corrections in the band structure is mainly a constant
opening of the band gaps along the different $\mathbf{k}$ points, justifying
the use of a scissor operator and a small stretching of the bands. The
electron-hole interaction in the BSE gives rise to nine degenerate bound
excitons in the absorption spectrum, with a binding energy of $\sim 0.3$ eV.
We note that the binding energy of the excitons in $c$–BN is smaller than in
$h$–BN, where it is 0.7 eV.Arnaud  _et al._ (2006) Indeed, while in $h$–BN the
bound excitons derive from transitions between parallel flat bands, supporting
an interpretation in terms of a tightly bound Frenkel exciton,Arnaud  _et al._
(2006) in the case of $c$–BN the bands at the origin of the bound excitons are
parabolic, leading to a Wannier-exciton picture. Satta  _et al._ (2004) The
calculated BSE optical gap in the absorption spectrum thus amounts to $\sim
10.7$ eV, i.e. much larger than the $\Gamma$X indirect band gap, which is 6.31
eV in GW (4.45 in LDA). Excitonic effects are important also for the rest of
the spectrum, shifting the spectral weight towards lower energies and
improving considerably the agreement with the experimental spectrum. Notably,
the shoulder S on the low-energy side of the main plasmon peak P in the loss
function is quite weak in the GW spectrum and gets strongly enhanced, in
agreement with experiment, when electron-hole interactions are taken into
account in the BSE.

Figure 3: Band structure of c–BN from KS eigenvalues within the LDA (blue
lines) and from the GWA in the $G_{0}W_{0}$ approach (black lines). The zero
of the energy axis has been set to the top-valence maximum.

The values of the $G_{0}W_{0}$ quasiparticle gaps in the band structure and
the optical gap in the BSE absorption spectrum agree well with the
corresponding results from recent calculations.Klimeš  _et al._ (2014); Jiang
and Blaha (2016); Satta  _et al._ (2004) By using a more accurate partially
self-consistent GW approach instead of the perturbative $G_{0}W_{0}$ scheme
employed here, it has been shown that the direct gap in the quasiparticle band
structure would be even larger by 0.3 eV.Klimeš  _et al._ (2014)

###  III.2 Detecting phase transformations in annealed $c$–BN

Boron nitride exists in two low-density $sp^{2}$ phases, the hexagonal and
rhombohedral ($r$–BN) allotropes, where weakly interacting planes of
hexagonally arranged B and N atoms are stacked respectively in a AA′ and ABC
order. The $sp^{3}$ coordination gives rise to two further high-density
phases, the cubic and wurtzite ($w$–BN) allotropes. The latter structure
corresponds to a hexagonal lattice isoelectronic with the carbon allotrope
lonsdaleite. The stacking similarities between the different BN structures
suggest that the conversion of $h$–BN into $w$–BN and $r$–BN into $c$–BN can
be driven by simple compression, through the formation of new out-of-plane
covalent bonds.Mirkarimi  _et al._ (1997); Nistor  _et al._ (2005) On the
contrary, the $h$–BN to $c$–BN transition requires bond breaking and atom
diffusion to transform the AA′ stacked $h$–BN atomic planes into the ABC
ordered puckered lattice planes in the $\left\langle 111\right\rangle$
direction of c-BN face-centered cubic lattice. HRTEM studies have reported
evidences against a direct $h$–BN to $c$–BN transformation, in favor of the
formation of transient phases mediating the conversion.Horiuchi  _et al._
(1996); Nistor  _et al._ (2005) Single and multiwalled nanoarches formed by
joining dangling bonds at $h$–BN sheet edges have been observed in the
starting $h$–BN material.Nistor  _et al._ (2005) At the beginning of the high-
pressure/high-temperature conversion, compression and local folding of $h$–BN
planes occur, inducing a transition to a monoclinic crystal structure.Horiuchi
_et al._ (1996); Nistor  _et al._ (2005) Moreover, additional nanoarches form
and are believed to act as nucleation sites for c-BN crystallites [the
nanoarches planes becoming the (111) planes of the cubic lattice].Nistor  _et
al._ (2005) The coalescence of pairs of planes promoting the stacking order
change is associated with an $r$–BN arrangement. In small $c$–BN grains,
$w$–BN areas have been observed Horiuchi  _et al._ (1996).

These findings demonstrate the relative abundance in $c$–BN of defects
consisting in inclusions of hexagonal-related phases. Besides these native
defects, it is possible to deliberately induce the coexistence of several
phases in higher-quality $c$–BN crystals by promoting inverse transformation.
The phase-change temperature from $c$–BN to $h$–BN at ambient pressure has
been experimentally measured at $1320^{\circ}\text{C}\pm
380^{\circ}\text{C}$,Will  _et al._ (2000) in good agreement with the $h$–BN
$\rightleftharpoons$$c$–BN equilibrium line proposed in the theoretical phase
diagram of BN.Solozhenko  _et al._ (1999) By annealing $c$–BN at sufficiently
high temperature, the first stages of this structural transformation
correspond to the nucleation of tiny hexagonal domains. This local phase
change could in principle be optically detected using spatially resolved
techniques such as EELS and CL.

To prove this, commercially available micrometric powders have been heated in
a tubular furnace at 1200 ∘C for 36 hours in an inert atmosphere. In Fig. 4(a)
we present high-angle annular dark-field (HAADF) images of a particle obtained
after crashing the treated powder in a mortar and in Fig. 4(b) a magnified
image of the region indicated by the rectangle in Fig. 4(a). The morphology of
the particle does not indicate any significant difference compared to the
untreated system. An EELS hyperspectral image has been acquired in the area
marked by the small dashed rectangle in Fig. 4(b). The region has a rather
constant thickness as deduced by the homogeneity of the HAADF image intensity.
However, the intensity of the signal integrated after the ZLP subtraction in a
0.7 eV window centered at 8.2 eV presents strong spatial variations [Fig.
4(d)]. Indeed, EELS spectra are fundamentally identical to those of untreated
$c$–BN powders almost everywhere in the particle [red spectrum in Fig. 4(c)]
but for two pronounced features (6.6 eV and 8.2 eV) which appear within the
$c$–BN band gap [blue spectrum in Fig. 4(c)] at specific positions in the
particle. Furthermore, the higher-energy region appears modified, with a
dominant contribution at about 26.0 eV. These additional components match the
energy of the main plasmon peaks of the h-BN bulk spectrum.Arenal  _et al._
(2005); Tarrio and Schnatterly (1989)

Figure 4: (a),(b) HAADF images of a micrometric particle from the thermally
treated $c$–BN powder. The smaller rectangle in (b) indicates the region
investigated by EELS hyperspectral imaging, the larger rectangle the one
investigated by CL hyperspectral imaging. (c) EELS spectra extracted from
regions 1 and 2 indicated in the intensity map in (d). The color scale in (d)
represents the intensity of the 8.2 eV feature. (e) Typical CL signals
occurring at the thermal treated $c$–BN powder. CL maps obtained integrating
the luminescent signal in the 4.8–6.0 eV energy window.

A further confirmation of a local phase transformation can be derived by an
analysis of the cathodoluminescent signal. In Fig. 4(e) we present spectra
extracted from a CL hyperspectral image acquired in a wider area of the
crystal overlapping the previous EELS analysis zone [corresponding to the
solid rectangle in Fig. 4(b)]. A larger analysis region is now possible since
cathodoluminescence does not require the thin-sample constrain as for EELS.
Compared to the luminescence of the pristine sample, we observe the appearance
of peaks at 5.20 eV, 5.30 eV and 5.37 eV. The map of the 5.30 eV emission peak
[Fig. 4(f)] presents very localized emission spots with a FWHM size of just
few tens of nanometers, i.e. of the same order of magnitude of defect
emissions later discussed in Sec. III.3.2. Notably, this emission spot occurs
in the area previously analyzed by EELS (dashed rectangle) and it is spatially
correlated with the additional features observed by EELS.

These spectroscopic observations are strong indications for a local phase
transformations within the particles. EELS spectra suggest the presence of
$h$–BN domains but the cathodoluminescence spectra do not show the
characteristic 5.75 eV emission of the pure $h$–BN phase. Previous studies
have demonstrated that the CL spectrum of $h$–BN, unlike EELS spectra, is
strongly affected by changes in the stacking order.Bourrellier  _et al._
(2014) One may guess that these domains present a hexagonal phase which does
not respect the AA′ stacking order of bulk $h$–BN. As we discussed at the
beginning of this section, $c$–BN can be obtained by a direct transformation
from the $r$–BN phase and only indirectly from the $h$–BN phase. Therefore,
the inverse transformation would drive to the $r$–BN phase having an ABC
stacking. From another side, it has been demonstrated that $c$–BN presents
stacking faults that can be described as very local $w$–BN phase
domains.Horiuchi  _et al._ (1996) In that case, temperature treatments could
induce the relaxation of these native local domains to the more stable $h$–BN.
Finally, the non-correspondence on CL spectra could be due to confinement
effects. These aspects of the observed phase transformation are open to
further investigations and certainly reference spectra of pure $r$–BN and
$w$–BN would provide an important contribution. However, this spectroscopic
information is currently unavailable due to the lack of good-quality crystals
for these metastable phases.

###  III.3 Optically active defects

The very large band gap of $c$–BN makes it unlikely to be employed as an
optical functional material, but its technological potential might reside in
the presence of bright and stable optical defects. During the last 30 years,
many luminescence features have been observed by PL and CL in a spectral range
going from the infrared to the far UV. Studies have been conducted on a
variety of $c$–BN samples such as crystal grown in B–rich or N–rich
conditions, annealed, electron irradiated, ion bombarded or compressed. Many
of these emissions present a rich vibrational structure with several phonon
replica which can be related with main $c$–BN phonon modes.Shishonok and
Steeds (2003, 2007) Whereas complete direct measurements of $c$–BN phonon band
structure have not been reported yet, theoretical values obtained in the
framework of DFT can be confidently employed to identify well-defined phonon
modes.Kern  _et al._ (1999); Karch and Bechstedt (1997); Bechstedt  _et al._
(2000); Bağcı  _et al._ (2006) Despite a rich literature in the field, no
complete characterization (concerning lifetime, spatial localization,
polarization, etc.) of these emission lines has been reported yet.
Furthermore, $c$–BN might contain single-photon sources as in diamond or SiC,
but this hypothesis, which received also a recent theoretical support,Abtew
_et al._ (2014) has not been explored experimentally. This situation motivates
further experimental studies using modern characterization techniques. In this
section, after an overview on previous works, we present new results on defect
luminescence in $c$–BN obtained by nano-CL showing the high spatial
localization of its defect-related emissions and, for some of them, their
lifetime.

####  III.3.1 A literature survey

A rather fragmented picture emerges from the rich literature on defect
luminescence in $c$–BN with a broad and sometimes confusing nomenclature. Here
we survey earlier studies on the luminescence of $c$–BN with the aim to
provide a unified vision of the field. Key information has been reported in
Table 1. Tentative attributions of different emission lines to intrinsic or
extrinsic defects have been discussed most of times on the sole correlation
between growth conditions or post-growth treatments and luminescence
observations. Whereas these hypotheses are often very plausible, the lack of
more complete experimental characterizations does not permit an unequivocal
identification of the observed defects and therefore generic references to
vacancies, interstitial or impurities have been often employed.

Table 1: Summary of the major emission lines observed on $c$–BN with indication of their lifetime, main phonon replica and tentative attribution indicated using the Kröger-Vink notation where possible. Series | Nomenclature | Energies | Lifetime | Main phonon |  Tentative |  References
---|---|---|---|---|---|---
|  | (eV) | (ns) | replica (meV) |  attribution |
General cubic | GC-1 | $1.76$ |  | $64$ |  N vacancies complexes |  27; 28
| GC-2 | $1.63$ |  | $56$ |  B vacancies complexes, $O_{N}$–$V_{B}$ |  27; 28,this work
| GC-3 | $1.55$ |  |  |  |  27; 28
| GC-4 | $1.44$ |  | $56$ |  |  28
| M1111This emission has been named R center in Ref. 83. | $1.65$ |  |  |  $C_{B}$ |  82; 83
| M2 | $1.64$ |  |  |  $Si_{B}$ |  82
| $\Gamma$ | $2.99$ |  | $33$ |  Stoichiometry changes |  82
Irradiation induced | RC-1 | $2.27$ |  | $64$ |  Vacancies |  29; 84; 83; 85,this work
defects | RC-2 | $2.15$ |  | $64$ |  Vacancies |  29; 84; 83; 31; 85,this work
| RC-3 | $1.99$ | $3.7$ | $64$ |  Vacancies |  29; 84; 83; 31; 85
| RC-4 | $1.86$ |  | $64$ |  Vacancies |  84; 83; 31
| BN-1 | $3.3$ |  | $161$ |  $B_{i}$ |  31; 76
Pressure cubic | PC–1 | $2.84$ |  | $100$ |  Interstitial atoms |  30; 86
| PC–2 | $2.33$ |  | $100$ |  Interstitial atoms |  30; 86
| PC–3222This center might be the same as the center named G in Ref. 87. | $1.79$ |  |  |  Interstitial atoms, transition metals impurities |  30; 86; 87; 31
| PF–1333In several works the first phonon replica has been named also PF–2. The full band, observed without resolving the phonon structure, has also been named UCL, US–1. | $3.57$ | $0.55$ | $140$ |  $V_{N}^{-}$–$B_{i}^{+}$, $B_{N}$ |  88; 31; 5; 89; 75; 32,this work
| GB–1444This emission has been named O center in Ref. 82. | $3.21$ |  | $25$ |  Impurities |  90; 82
| T | $4.93$ | $1.4$ | $145$ |  Si impurities |  82; 91; 92,this work
|  | $4.09$ |  | $175$ |  $C_{N}$ at $h$–BN inclusions |  90

  * •

The first series of color centers at energies GC–1=1.76 eV, GC–2=1.63 eV,
GC–3=1.55 eV and GC–4=1.44 eV (GC for General Cubic), was identified in 1985
by Tkachev et al. by CL on polycrystalline $c$–BN materials.Tkachev  _et al._
(1985) These centers present phonon replica with well identified energies, 64
meV for the CG-1 center and 56 meV for the GC–2 and GC–4 centers. Later
studies conducted under high-temperature annealing conditions and nitrogen
implantation permitted to ascribe the GC–1 center to defects containing
nitrogen vacancies and the GC–2 center to defects containing boron
vacancies.Tkachev  _et al._ (1985); Shipilo  _et al._ (1986) A more recent
theoretical work proposed the GC–2 center to be related to substitutional
oxygen at a nitrogen site adjacent to a boron vacancyAbtew  _et al._ (2014)
($O_{N}$–$V_{B}$), which would be an analogous in $c$–BN to the well known
N–V- center in diamond.

  * •

In the presence of carbon or silicon doping two centers appear at the energies
M–1=1.65 eV and M–2=1.636 eV.Shipilo  _et al._ (1991) The intensity of these
centers increases when the growth occurs in a nitrogen-excess atmosphere which
might promote the formation of boron vacancies. Therefore, the M–1 center has
been ascribed to C or Si substitutionals at B sites ($C_{B}$) while the M–2
center to Si substitutionals at B sites ($Si_{B}$). The centers can be seen as
an evolution of the GC–2 line in the presence of dopant atoms. When the growth
occurs in a boron excess, carbon or silicon doping leads to the appearence of
the $\Gamma$ center at 2.99 eV presenting a phonon side band at 33 meV.Shipilo
_et al._ (1991)

  * •

A series of emissions named Radiation Cubic (RC) has been observed by CL at
the energies RC–1=2.27 eV, RC–2=2.15 eV, RC–3=1.99 eV and RC–4=1.86 eV on
$c$–BN crystals bombarded with electrons having an energy in the order of the
MeV or in the range 150–300 keVZaitsev  _et al._ (1986); Shishonok and Steeds
(2002, 2004); Erasmus and Comins (2004) or by ion irradiation.Zaitsev  _et
al._ (1987) More recent investigations reported the generation of these
centers by femtosecond laser pulses.Buividas  _et al._ (2015) In the same
work, a PL decay close to a single exponential with time constant 3.7 ns has
been also reported for the RC–3 line. All emissions present phonon replica
spaced by 64 meV. These features have been ascribed to vacancy-related defects
since they disappear after annealing at temperatures around 1000 ∘C.Zaitsev
_et al._ (1986); Shishonok and Steeds (2002) Zero-phonon lines of the RC–1 and
RC–2 centers present finer structures, depending upon the particle irradiation
energy, which have been associated to local symmetry breakings at the defect
centers.Shishonok and Steeds (2004); Erasmus and Comins (2004) Another
electron-irradiation induced emission, named BN–1 center, was identified at
3.29 eV accompanied by a very rich phonon structure.Shishonok and Steeds
(2002, 2007) By analogy with a similar center in diamond, the BN–1 center was
thought to include most probably a boron interstitial ($B_{i}$).Shishonok and
Steeds (2002)

  * •

Emissions observed by CL after high-pressure treatments at almost 10 GPa have
been classified as Pressure Cubic (PC) PC–1=2.84 eV, PC–2=2.33 eV and
PC–3=1.79 eV. These centers present local phonon replica of 100 meV,Shipilo
_et al._ (1988); Shishonok and Steeds (2002) a higher energy than in the GC or
the RC series. This suggested the possible presence of interstitial atoms. An
influence of transition metal impurities has been suggested for the PC–3
line.Shishonok  _et al._ (1989)

  * •

A broad emission band centered at about 3.12 eV has been reported by several
works and different notations such as UCL or US–1 have been used for
it.Mishima and Era (2000); Zhang  _et al._ (2002); Manfredotti  _et al._
(2006) Other works reported the presence of two sharp peaks at the high-energy
side of the band, indicated as PF–1=3.57 eV and PF-2=3.41 eV.Shipilo  _et al._
(1990); Shishonok and Steeds (2002) Low-temperature photoluminescence and
cathodoluminescence measurements demonstrated a very rich phonon structure for
the PF–1 center, the PF–2 center being then the first intense phonon replica.
Shipilo  _et al._ (1990); Shishonok and Steeds (2003) The broad bands
previously discussed in the literature correspond to the unresolved phonon
replica. This center has been interpreted as related to donor-acceptor Frenkel
pairs, $V_{N}^{-}$–$B_{i}^{+}$, by comparing CL of B-implanted, electron-
bombarded and annealed crystals.Shipilo  _et al._ (1990); Shishonok and Steeds
(2002) Manfredotti et al. proposed a different assignment with a boron
substitutional at a nitrogen site ($B_{N}$).Manfredotti  _et al._ (2006)

  * •

A series of five sharp lines spaced by 20-25 meV has been observed in the
energy range 3.23 and 3.13 eV, with intense peaks at 3.18 eV and 3.23 eV. This
series has been called GB–1 center but also O center.Shishonok (2007); Shipilo
_et al._ (1991) Shipilo et al. reported that these emissions are encountered
in Si or carbon doped crystals only grown in an excess of nitrogen.Shipilo
_et al._ (1991) Shishonok et al. proposed that this center might be associated
with heavy atom impurities due to the low energy of the phonon replica which
does not correspond to any typical vibration mode of $c$–BN.Shishonok (2007)

  * •

A zero-phonon line at 4.93 eV followed by several phonon replica separated by
152 meV, originally indicated as T center, has been observed using
cathodoluminescenceShipilo  _et al._ (1991); Kanda  _et al._ (1997) and
synchrotron radiation stimulated luminescence.Evans  _et al._ (2008) This
emission has been firstly detected in deliberately highly Si doped crystals
and interpreted as the interaction between Si impurities with other defective
sites.Shipilo  _et al._ (1991)

  * •

In lower-quality $c$–BN crystals, a 4.09 eV emission followed by phonon
replica spaced by 175 meV has been observed.Shishonok (2007) These features
are typical of $h$–BN, which is a common inclusion in $c$–BN crystals, and
they have been discussed as being associated to carbon substitutionary atoms
at nitrogen sites.Katzir  _et al._ (1975); Taniguchi and Watanabe (2007);
Bourrellier  _et al._ (2016)

We have limited this summary to intrinsic defects or extrinsic defects related
to common impurities that might be introduced involuntarily at the synthesis.
These defects are therefore the most commonly encountered ones when analyzing
usual samples. However, the wide band gap of $c$–BN represents a solid
platform to host optically active defects through deliberate doping. For this
reason, there exists an active research in the field with most of works
dedicated to rare earth which promote the appearance of very bright additional
emission lines.Nakayama  _et al._ (2005); Shishonok  _et al._ (2005, 2007,
2010); Zhigunov  _et al._ (2016)

####  III.3.2 Nano-cathodoluminescence

The summary given here provides a global picture of the variety of optical
active centers identified in $c$–BN. This richness is shared with other wide
band-gap semiconductors such as diamond, SiC or GaN, whose optical response
has been the object of a much larger number of studies. One of the most
promising application for optical defects in bulk materials is their use as
single-photon sources which are at the core of future quantum information
technologies. During the last years, $h$–BN has appeared as a very promising
new platform for single-photon emission (SPE). Bright, stable and room-
temperature active SPEs have been identified in the visibleTran  _et al._
(2016a, b); Shotan  _et al._ (2016); Jungwirth  _et al._ (2016); Chejanovsky
_et al._ (2016); Martínez  _et al._ (2016); Exarhos  _et al._ (2017) and
UVBourrellier  _et al._ (2014) energy region. However, the most studied
emissions present a large energy distribution (1.7–2.3 eV) dependently on the
defect environment. A reason for this behavior might be the layered structure
of $h$–BN which allows very low energy, and thus frequent structural
deformations such as plane glides, bends or dislocations. $c$–BN could be also
a potential host for SPEs with the advantage over $h$–BN of presenting a
smaller set of emission energies, as it was summarized in the previous
paragraph. However, $c$–BN is a rather defective material and individual
defects can hardly be isolated using conventional optical techniques. This
problem has hindered up to now more in-depth studies and for instance
lifetimes and quantum nature of the optical centers have not been investigated
yet. Here we partially overcome this problem by employing nano-CL and we
provide the first maps of the spatial localization of the defects and insights
on their quantum nature.

Figure 5: (a),(d),(g) Examples of typical spectra associated with defects in
$c$–BN, (b),(e),(h) HAADF images of the particles from which they have been
extracted. (c),(f),(i) Corresponding CL emission intensity maps obtained by
integrating the signal in the 10 meV window indicated by yellow regions in the
spectra. For the spectra (a) and (g), the integration window correspond to
zero-phonon lines. For the spectrum (d) a lower-energy phonon replica has been
chosen in order to limit the overlap with the broad band centered at 2.5 eV.

By analyzing micro-particles of both high-quality and commercial $c$–BN
crystals, several emission features have been clearly identified. In Fig.
5(a,d,g) we present some example spectra, in Fig. 5(b,e,h) HAADF images from
the particles from which they have been extracted and in Fig. 5(c,f,i) CL
emission intensity maps. The zero-phonon line (ZPL) observed at 1.63 eV
(possibly a double peak) with a 56 meV phonon replica [Fig. 5(a)] can be
unequivocally identified as the GC–2 center. Two ZPL at 2.31 eV [Fig. 5(d)]
and 2.17 eV [Fig. 5(g)] can also be identified with multiple-phonon replica
spaced by about 70 meV. Despite a minor energy shift, these defects can be
reasonably associated to the RC–1 and RC–2 centers which have been observed in
high-energy electron irradiated samples.Zaitsev  _et al._ (1986); Shishonok
and Steeds (2002) In our experiments, these defects do not appear after
deliberate prolongated irradiation with the microscope electron beam
suggesting that the irradiation energy threshold should be above our 60 keV
operating energy and that these defects might occur also as native defects. In
Fig. 5(d,g) two broad bands centered at about 2.5 eV and 3.12 eV are clearly
visible. The first one resembles to the unidentified broad bands discussed in
Ref. 30. The second one corresponds to the unresolved phononic structure of
the PF–1 defect.

Intensity maps have been derived by integrating the hyperspectral signal
within energy windows of about 10 meV [Fig 5(c,f,j)]. For the 1.63 eV and 2.31
eV emissions, integration windows were centered at the energy of the ZPL peaks
while for the 2.17 eV emission we chose to center the window at a lower-energy
phonon replica in order to limit the overlap with the broad bands. All
emissions arise from highly spatially localized spots with the 2.17 eV and
2.31 eV centers having a FWHM below 100 nm, which is a value of the order of
magnitude of excitation diffusion length in $h$–BN Bourrellier  _et al._
(2016) or diamond.Tizei and Kociak (2012, 2013) Therefore, it can be
reasonably proposed that, by analogy with the case of other wide band-gap
semiconductors, these $c$–BN optically active centers could be related to
point defects and ultimately act as single-photon sources. The relatively low
quantum efficiency of these centers, whose spectra also often overlap,
presents a strong limit to investigate this hypothesis using our nano-CL setup
coupled with a HBT interferometer. High-energy emissions are more adapted to
these studies since they present stronger luminescences and their spectra can
be clearly isolated, which grants a higher signal-to-noise and a higher
signal-to-background ratio.

In Fig. 6(a) we present the emission spectrum of the PF–1 series with a ZPL
clearly visible at 3.57 eV and six strong phonon replicas. The phonon
structure has been fitted with a multi-Gaussian function set, giving an
average line separation of 141 $\pm$ 6 meV. By comparison with the calculated
phonon density of states,Kern  _et al._ (1999); Karch and Bechstedt (1997);
Bechstedt  _et al._ (2000); Bağcı  _et al._ (2006) this spacing corresponds to
the LO phonon mode. It is also in agreement with the value measured in the
EELS spectrum [Fig. 1(a)]. Occasionally a slightly higher energy emission at
3.62 eV can be observed with phonon replicas having a little higher energy
[see insert in Fig. 6(a)]. Both emissions probably arise from the same type of
defect, and the energy shift may be due to a different local environment.
Similar shifts were also observed in Ref. 88. The deexcitation process can be
characterized by the Huang-Rhys factor $S$ which represents the average number
of phonons involved in the emission. The energy
$\hbar\omega_{\text{ZPL}}-S\hbar\omega_{\text{phon}}$, where
$\hbar\omega_{\text{ZPL}}$ is the energy of the ZPL and
$\hbar\omega_{\text{phon}}$ is the phonon energy, should indicate the maximum
of the emission band. From our fit we obtain a Huang-Rhys factor of 4.07 and
hence a maximum of intensity expected at 2.99 eV in very good agreement with
the experiments. A similar characterization had been reported in previous
works Manfredotti  _et al._ (2006); Shishonok and Steeds (2003) obtaining
slighty higher values (4.9 and 5.26), but it should be kept in mind that the
Huang-Rhys factor varies as a function of the local environment and therefore
it is not an intrinsic parameter of the defect emission.

In Fig. 6(b) we present a HAADF image of the particle from which the spectrum
in Fig. 6(a) has been extracted. In Fig. 6(c) we report the emission intensity
map obtained by integrating the signal in an energy window about 100 meV wide
and centered at the emission band maximum (3.00 eV). Whereas a faint signal is
present all over the particle, once more it is possible to identify a
localized emission spot about 500 nm wide indicating a limited number of
emission centers with the possibility of isolated centers.

This emission has been further investigated by driving the CL signal to a
unique home-made HBT interferometer. Fig. 6(d) reports a typical intensity
autocorrelation function acquired from these centers. The curve shows an
evident zero-delay bunching character. This behavior occurs in CL due to the
synchronized emission of several defect centers excited by the same electron
through the deexcitation of a bulk plasmon into few electron-hole pairs. This
effect has been previously observed for point defects in diamond and $h$–BN
Meuret  _et al._ (2015) and more recently in InGaN quantum wells.Meuret  _et
al._ (2017) The bunching is characteristic of emissions by packets of photons:
although anti-bunching is the irrefutable evidence of a SPE, in the particular
case of CL, bunching is obtained from a set of independent but spatially close
SPEs. Our experiments have not permitted yet to isolate a single defect
associated with the 3.57 eV luminescence even in high-quality crystals. The
relatively high spatial extension observed in the emission maps indicates the
clustering of several defect centers within the typical plasmon-excitation
volume. Finally, an analysis of the bunching decay provides the lifetime of
the optical center.Meuret  _et al._ (2016) By fitting the experimental curve
with exponential functions we estimate the lifetime of the 3.57 eV emission to
be $1.4\pm 0.1$ ns.

A similar analysis has been conducted for the T series having a ZPL at 4.94 eV
(Fig. 7). In the spectrum [Fig. 7(a)], eight phonon replicas can be
distinguished with an average line separation of $145\pm 5$ eV, once more
compatible with LO phonons. The calculated Huang-Rhys factor is 4.99 from
which one derives an energy for the maximum of the emission band of 4.22 eV,
in good agreement with the experimental value. Fig. 7(b,c) present a
microscopy image of a whole $c$–BN particle and the associated intensity map
for the 4.94 eV emission, derived by signal integration in a 100 meV energy
window centered at the band maximum. At least three intensity maxima can be
distinguished in the map, indicating a possible clustering of several defect
centers. HBT experiments have been performed and a typical autocorrelation
function is presented in Fig. 7(d). Once more we obtain a bunching function
indicating the possible presence of multiple close SPEs. From the exponential
fit we estimate the lifetime of the center as $0.55\pm 0.09$ ns.

Nano-CL experiments have also been conducted on Tb and Eu earth doped $c$–BN
crystals (not presented here). We observe a strong catholuminescence related
to the rare earth atoms analogous to previously published results.Nakayama
_et al._ (2005) However related nano-CL maps show a luminescence delocalized
all over the particles indicating a high doping level which does not allow to
isolate individual emitters.

Figure 6: (a) The PF–1 zero-phonon line at 3.57 eV and associated phonon
replica. Dashed curves are individual gaussian components used to fit the
phonon replica, the red curve is the resulting global fit. In the insert we
report a 3.62 eV emission which most likely arises from the same type of
defect in a different local environment. (b) HAADF image of the $c$–BN
particle from which the spectrum has been extracted and (c) CL intensity map
of the this emission. (d) Autocorrelation function derived from HBT
interferometry. An emission lifetime of $0.55\pm 0.09$ ns is obtained from the
fit of the bunching peak. Figure 7: (a) The T defect zero-phonon line at 4.94
eV and associated phonon replica. Dashed curves are indivual gaussian
components used to fit the phonon replica, the red curve is the resulting
global fit. (b) HAADF image of the $c$–BN particle from which the spectrum has
been extracted and (c) CL intensity map of this emission from which several
separated spots can be distinguished. (d) Autocorrelation function derived
from HBT interferometry. An emission lifetime of $1.4\pm 0.1$ ns is obtained
from the fit of the bunching peak.

##  IV Conclusions

In this work we investigated the optical properties of $c$–BN tackling both
the problem of the optical gap and the luminescence of intragap defects. By
using EELS with a monochromated electron beam on high-quality $c$–BN crystals
we demonstrated unequivocally that the optical gap slightly exceeds 10 eV. A
further theoretical analysis confirms this result permitting to interpret the
nature of the spectral features observed. The main peak corresponds to a
plasmon excitation while the onset of the optical response is identified as a
Wannier-type exciton. Excitonic effects modulate the intensity of the spectra
and they can be associated to several features observed in experiments.
Previous estimation of the optical gap, corresponding to much lower values,
should be understood as misinterpretations of optically active defects,
possibly related to inclusions of hexagonal phases. By high-temperature
annealing a $c$–BN powder we have deliberately promoted phase transitions. In
very local domains of these treated samples, we observed in correlated EELS
and CL experiments the appearance of spectroscopic signatures characteristic
of the hexagonal phase.

A high number of optically active centers are known in $c$–BN but the
literature is rather scattered and hard to follow. Here we summarized the main
known results with the aim to provide an unified picture of defect
luminescence in $c$–BN. Using our unique nano-CL experimental setup integrated
within a STEM, we have detected a number of these centers in the visible and
UV spectral range. Due to the large band gap of $c$–BN, higher-energy
optically active defects might also be present. These transitions can in
principle be excited by the microscope’s high energy electron beam but they
cannot be observed by our CL system which has an upper detection limit of
about 6.2 eV. Finally, we have investigated by nano-CL the spatial
localization of several emission centers showing small spots, compatible with
a limited number of optically active centers. For the brightest defect types,
lying in the UV spectral range, we have performed Hanbury Brown and Twiss
interferometry experiments in order to identify the possible presence of
single-photon sources. The bunching effect observed is compatible with a very
limited set of SPEs synchronously excited but no anti-bunching has been
detected yet. We hope that these promising results will motivate additional
investigations which might succeed in isolating individual centers. Moreover,
SPEs might be associated with emissions in the visible spectral range which
present also a high spatial localization. In this work, we have not tested
these centers by HBT interferometry due to their relatively low CL-
luminescence and a spectral overlap with the tail of higher energy emissions
inevitably excited by CL with fast electrons. Photoluminescence in a confocal
microscope might an adapted technique whereas a too high density of defects in
currently available samples might still hinder these studies.

###### Acknowledgements.

AT, MK, TZ and AZ acknowledge support from the Agence Nationale de la
Recherche (ANR), program of future investment TEMPOS-CHROMATEM (No.
ANR-10-EQPX-50). The work has also received funding from the European Union in
Seventh Framework Programme (No. FP7/2007 -2013) under Grant Agreement No.
n312483 (ESTEEM2), from a Marie Curie FP7 Integration Grant within the 7th
European Union Framework Programme and from the European Research Council (ERC
Grant Agreement n. 320971). Computational time was granted by GENCI (Project
No. 544).

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