[table]capposition=top

# Excitonic and deep-level emission from N- and Al-polar homoepitaxial AlN
grown by molecular beam epitaxy

L. van Deurzen<EMAIL_ADDRESS>School of Applied and Engineering Physics,
Cornell University, Ithaca, New York 14853, USA  J. Singhal  J. Encomendero
Department of Electrical and Computer Engineering, Cornell University, Ithaca,
New York 14853, USA  N. Pieczulewski  Department of Materials Science and
Engineering, Cornell University, Ithaca, New York 14853, USA  C.S. Chang
School of Applied and Engineering Physics, Cornell University, Ithaca, New
York 14853, USA  Research Laboratory of Electronics, Massachusetts Institute
of Technology, MA 02139, USA  Y. Cho  Department of Electrical and Computer
Engineering, Cornell University, Ithaca, New York 14853, USA  D.A. Muller
School of Applied and Engineering Physics, Cornell University, Ithaca, New
York 14853, USA  Kavli Institute at Cornell for Nanoscale Science, Cornell
University, Ithaca, New York 14853, USA  H.G. Xing  Department of Electrical
and Computer Engineering, Cornell University, Ithaca, New York 14853, USA
Department of Materials Science and Engineering, Cornell University, Ithaca,
New York 14853, USA  Kavli Institute at Cornell for Nanoscale Science, Cornell
University, Ithaca, New York 14853, USA  D. Jena  School of Applied and
Engineering Physics, Cornell University, Ithaca, New York 14853, USA
Department of Electrical and Computer Engineering, Cornell University, Ithaca,
New York 14853, USA  Department of Materials Science and Engineering, Cornell
University, Ithaca, New York 14853, USA  Kavli Institute at Cornell for
Nanoscale Science, Cornell University, Ithaca, New York 14853, USA  O. Brandt
J. Lähnemann  Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut
im Forschungsverbund Berlin e.V., 10117 Berlin, Germany

###### Abstract

Using low-temperature cathodoluminescence spectroscopy, we study the
properties of N- and Al-polar AlN layers grown by molecular beam epitaxy on
bulk AlN{0001}. Compared to the bulk AlN substrate, layers of both polarities
feature a suppression of deep level luminescence, a total absence of the
prevalent donor with an exciton binding energy of 28 meV, and a much increased
intensity of the emission from free excitons. The dominant donor in these
layers is characterized by an associated exciton binding energy of 13 meV. The
observation of excited exciton states up to the exciton continuum allows us to
directly extract the $\Gamma_{5}$ free exciton binding energy of 57 meV.

## Introduction

The ultra-wide gap semiconductor AlN was first synthesized over a century
ago,[1] but it has only recently been recognized that the unique physical
properties of AlN make it of great interest for applications in novel
electronic and optoelectronic devices. Apart from its direct band gap of 6.095
eV corresponding to an emission wavelength of 203 nm, most noteworthy are its
large piezoelectric coefficients (1.5 C/m2 or 5 pm/V),[2] its high breakdown-
field ($>\\!\\!10$ MV/cm) [3, 4], and its high thermal conductivity
($>\\!\\!300$ W/mK) [5]. Moreover, AlN has an exciton binding energy exceeding
twice the value of $kT$ at room temperature,[6] which is notably higher than
in any other III-V semiconductor with the exception of hBN.[7] These
properties make AlN an attractive material for applications in high-power and
millimeter-wave electronics[8, 9, 10] as well as deep-ultraviolet (UV-C) light
emission and lasing[11, 12, 13, 14]. The large exciton binding energy in AlN
is also of high interest for the study of the fundamental science of excitons
and their condensation[15].

This potential of AlN lay dormant until very recently, when researchers
succeeded to synthesize large AlN crystals of high structural perfection with
low dislocation densities ($<10^{4}$–$10^{6}$ cm-2) utilizing either physical
vapor transport (PVT)[16, 17, 18, 19] for bulk growth, or hydride vapor phase
epitaxy (HVPE)[20] for fabricating thick free-standing layers. The
availability of AlN substrates has enabled studies of the fundamental
properties of AlN, particularly regarding its spontaneous emission, yielding
insight into valence-band ordering, spin-orbit and crystal-field splitting,
spin exchange interaction, exciton binding energies, and the detection of
impurities and defects. These investigations have been performed by either
photo- or cathodoluminescence spectroscopy on PVT-grown bulk crystals,[16, 17,
18, 19] HVPE-grown free-standing substrates,[20] or homoepitaxial AlN layers
grown by metal-organic chemical vapor deposition [21, 22, 23, 24, 18, 25].

Numerous studies have focused on the excitonic near-bandedge emission of AlN,
resolving transitions due to lowest-energy free excitons with
$\Gamma^{c}_{7}\otimes\Gamma^{v}_{7+}$ symmetry (often called A-excitons)[26]
and several bound exciton transitions [23, 27]. At higher excitation
densities, bi-exciton emission (M-band), inelastic exciton-exciton scattering
bands (P-bands), and electron-hole plasma recombination were observed [22,
17]. Spectra recorded over a wider spectral range revealed the presence of
deep luminescence bands in the 2–4 eV range that frequently dominate over the
near-bandedge emission in terms of integrated intensity. The actual origin of
these ubiquitous deep luminescence bands is unknown, but is generally believed
to be related to complexes of native defects, particularly Al vacancies, with
impurities such as Si and O that form DX centers in AlN. However, also other
native defects and the impurity C have been suspected to play an important
role in the transitions giving rise to these deep bands.[6, 19, 16]

Figure 1: [(a) and (b)] HAADF-STEM images with overlayed wurtzite ball-and-
stick model, [(c) and (d)] $c$-plane atomic force topographs, and [(e) and
(f)] SIMS profiles of samples I (Al-polar AlN, top row) and II (N-polar AlN,
bottom row). The insets in figures (e) and (f) show the respective
$\omega/2\theta$ x-ray diffraction scans across the 002 reflection of the MBE
layers (blue, dotted) and the bare AlN substrate (solid, black).

In general, the incorporation of impurities and the formation of native point
defects is related in a complex way to the particular growth method and
conditions. Surprisingly, there seem to be no detailed studies of the
spontaneous emission of homoepitaxial AlN films grown by molecular beam
epitaxy (MBE). Furthermore, the point defect incorporation and formation often
strongly depends on the bonding configuration on the growth front, i. e., its
crystallographic orientation, as reported, for example, for GaN.[28] To the
best of our knowledge, all of the available studies on the excitonic emission
of AlN have been performed on films grown along the Al-polar ([0001]), semi-
polar, or non-polar ([1$\bar{1}$00] and [1$\bar{2}$10]) directions. However,
N-polar ([000$\bar{1}$]) (In,Ga)N/(Al,Ga)N heterostructures are of particular
interest because they exhibit internal electrostatic fields opposite to those
of their metal-polar counterparts, which is considered favorable for various
advanced device concepts.[29, *akyol_Jpn.J.Appl.Phys._2011,
*dong_Appl.Phys.Lett._2012, *han_Jpn.J.Appl.Phys._2012,
*wong_Semicond.Sci.Technol._2013, *feng_J.Appl.Phys._2015,
*vandeurzenlenEnhancedEfficiencyBottom2021, *choBlueGaLightemitting2019,
*bharadwajEnhancedInjectionEfficiency2020, *leeLightemittingDiodesAlN2020,
*bharadwajBottomTunnelJunction2020]

In the present work, we use cathodoluminescence (CL) spectroscopy in a
scanning electron microscope (SEM) to analyze and compare N- and Al-polar
homoepitaxial strain-free AlN films grown by MBE in terms of their near-
bandedge (around 6 eV) and deep-level (2–4 eV) light emission. With respect to
the bare substrate, we find that both N-polar and Al-polar films feature a
suppression of deep-level luminescence, and the total absence of the donor
dominating the NBE of the substrate. Furthermore, the intense free exciton
emission in the films under investigation allows us to directly measure an
exciton binding energy of 57 meV from the $\Gamma_{5}^{n\rightarrow\infty}$
transition.

## Experiments

The three samples under investigation were grown in an MBE system equipped
with a radio-frequency plasma source for generating active N and solid-source
effusion cells for Al. As substrates, we used bulk AlN wafers grown by PVT
with a dislocation density $<10^{4}$ cm-2. The layers were grown at substrate
temperatures above 1000 ∘C under Al-stable conditions. Details on the surface
preparation and growth parameters for the MBE growth of Al- and N-polar
homoepitaxial AlN layers are given in Refs. 40, 41, 42, 43. Samples I and II
consist of a 700 nm-thick Al-polar and a 900 nm-thick N-polar layer,
respectively. Sample III consists of a 1-$\upmu$m-thick N-polar AlN layer that
was capped with 6.7 nm heteroepitaxial GaN. This GaN cap layer serves as a
passivation layer for the AlN surface and also acts as a charge-spreading
layer during electron beam excitation. As reference and for comparison with
other studies, a freestanding AlN substrate was used.

[capbesideposition=right,center,capbesidewidth=4cm]figure[]

Figure 2: (a) Low-temperature, high-resolution near-bandedge and (b) deep-
level CL spectra for the bare PVT-AlN substrate, as well as samples I and II.
Light was collected with $\mathbf{k}\|\mathbf{c}$. The narrow feature just
above 3 eV in (b) is the second order of the near-bandedge emission.

The samples were investigated by scanning transmission electron microscopy
(STEM) on an aberration-corrected ThermoFisher Spectra 300 CFEG operated at
300 keV to confirm their polarity. STEM samples were prepared using a
ThermoFisher Helios G4 UX Focused Ion Beam system. Carbon and platinum
protective layers were deposited and a final milling step at 5 keV was used to
minimize ion-beam damage. Additionally, atomic force microscopy (AFM) was
performed on an Asylum Cypher AFM microscope in tapping mode for sample areas
of 2 × 2 µm2 and 20 × 20 µm2. Furthermore, the concentrations of O, C, and Si
were measured by time-of-flight secondary ion mass spectrometry (ToF-SIMS)
performed by Evans Analytical Group. Finally, triple-axis $\omega/2\theta$
x-ray diffraction (XRD) scans across the symmetrical 002 wurtzite reflection
were measured using a Panalytical Empyrean x-ray diffractometer equipped with
a PIXcel3D detector and Xe proportional detector. The monochromator consists
of a hybrid two-bounce Ge220 crystal utilizing Cu ${\mathrm{K}\alpha_{1}}$
radiation.

To analyze and compare the optical properties of these samples, low-
temperature emission spectra were recorded by CL spectroscopy in a Zeiss
Ultra55 SEM equipped with a He-cryo-stage allowing sample temperatures down to
10 K and a Gatan MonoCL4 detection system. The spectrometer is operated with
two gratings, one blazed at 300 nm with 2400 grooves/mm (Fig. 2) and the other
blazed at 250 nm with 1200 grooves/mm (Fig. 3), resulting in a spectral
resolution of approximately 0.0035 and 0.007 eV for a slit width of 0.1 mm,
respectively. Note that the line widths of the near-bandedge transitions are
limited by this spectral resolution. The monochromator was calibrated using a
Hg(Ar) spectral calibration lamp. The error of this calibration amounts to
$\pm 2$ Å because of the spectral separation between the highest energy line
of the lamp and the AlN near-bandedge emission. The samples were aligned with
respect to the detection setup such that mostly light propagating parallel to
the $c$-axis, $\mathbf{k}\|\mathbf{c}$, and electric field polarization
($\mathbf{E}$) orthogonal to the $c$-axis, $\mathbf{E}\perp\mathbf{c}$, was
collected. The light was detected monochromatically with a photomultipier tube
(PMT) while stepping through the spectral range and the spectra were converted
to energy scale by a Jacobian transformation, taking into account the
refractive index dispersion of air [44]. For the CL measurements, all of the
AlN samples were sputter coated with 3 nm of Ti to reduce charging effects in
the SEM.

## Results and Discussion

Figure 1 summarizes data pertaining to the structural and morphological
properties of our AlN layers as well as to their purity. The polarity of the
layers is examined by high-angle annular dark-field STEM (HAADF-STEM) as shown
in Fig. 1(a) and 1(b), respectively. The yellow spheres in the overlayed ball-
and-stick model of the wurtzite crystal structure represent Al atoms, and the
white spheres represent N atoms, confirming the nominal polarity of the layers
(growth direction pointing upwards). Figures 1(c) and 1(d) depict $2\times
2~{}\upmu$m2 atomic force topographs of the $c$-plane surface of the as-grown
layers. The trains of well-resolved monolayer steps are characteristic for
step-flow growth, resulting in a root-mean-square roughness as low as 80 and
160 pm, respectively. Figures 1(e) and 1(f) display ToF-SIMS data for the
respective samples revealing comparatively low impurity concentrations for
layers of either polarity. The level of $[\mathrm{Si}]<5\times 10^{16}$ cm-3
is at, and the levels of O and C are close to the detection limit (given in
parentheses): $[\mathrm{O}]\approx 3-4\times 10^{17}$ cm-3 ($1\times 10^{17}$
cm-3) and $[\mathrm{C}]\approx 7-8\times 10^{16}$ cm-3 ($5\times 10^{16}$
cm-3). Hydrogen (not shown) has a concentration below the detection limit for
both samples, $[\mathrm{H}]<3\times 10^{17}$ cm-3. Note that the spike at the
substrate/MBE interface is less pronounced for the Al-polar sample (near 700
nm depth) than for the N-polar sample (near 900 nm depth). This is likely due
to a greater adsorption efficiency of impurities on the N-polar surface [45,
46], as well as the less extensive ex-situ chemical treatment for the N-polar
face, as it is more reactive [40, 42, 43]. Finally, the insets in Figs. 1(e)
and 1(f) depict $\omega/2\theta$ x-ray diffraction scans across the 002
reflection for the Al- and N-polar AlN samples (blue, dotted) compared with
the bare AlN substrates they were grown on (solid, black). The line widths of
approximately 17 arcsec confirm that the MBE-grown AlN layers are free of
strain. The absence of strain is further confirmed by reciprocal space maps
around the asymmetric $\bar{1}05$ reflection for Samples I and II (see Fig. S1
in the Supplementary Material). Symmetric and asymmetric $\omega$ scans
recorded for Samples I and II exhibit widths between 20 and 30 arcsec,
comparable or below to that measured on the corresponding bulk AlN substrates
(see Fig. S2 in the Supplementary Material). For the structural
characterization and SIMS data of sample III, we refer to Figs. S3, S4 and S5
in the Supplementary Material.

Figure 3: (a) Low-temperature, high-resolution near-bandedge CL spectrum for
sample III on a linear and logarithmic (inset) intensity scale. The
integration time was chosen to be $10\times$ longer than for the spectra in
Fig. 2. (b) Expanded view of the excited free exciton transitions, with
$\Gamma_{5}^{n=2}$, $\Gamma_{5}^{n=3}$, and $\Gamma_{5}^{n\rightarrow\infty}$
resolved. The line is a fit of the data with three Gaussians (also shown).

CL spectra of samples I and II are compared with the one of the bare PVT-AlN
substrate (N-polar face) in Fig. 2. An acceleration voltage of 7 keV was used
to excite the samples, ensuring that electron-hole pairs are generated only
within the homoepitaxial layers (first few hundred nm from the AlN/air
interface, see Fig. S7 in the Supplementary Material). Figure 2(a) shows the
low-temperature, high-resolution near-bandedge emission spectra normalized to
their peak intensities. For all three samples, a distinct high-energy line is
observed at $(6.038\pm 0.005)$ eV, with the error due to the uncertainty of
the wavelength calibration. This transition energy is close to the average
energy (6.041 eV) reported for the exciton spin triplet with $\Gamma_{5}$
symmetry in bulk (strain-free) AlN.[21, 17, 22, 25, 23, 27]

For AlN, the negative crystal-field splitting causes the $\Gamma^{v}_{7+}$
band to be the uppermost valence band,[47] resulting in exciton states of
$\Gamma^{c}_{7}\otimes\Gamma^{v}_{7+}$ symmetry (often called A-excitons).[48]
Spin-exchange interaction splits these states into excitons with the
irreducible representation $\Gamma_{1}$ $\oplus$ $\Gamma_{2}$ $\oplus$
$\Gamma_{5}$, with the $\Gamma_{5}$ spin triplet being the only optically
active state that satisfies $\mathbf{E}\perp\mathbf{c}$ and
$\mathbf{k}\parallel\mathbf{c}$, i. e., our measurement geometry. It should be
noted that there currently is no consensus in the literature on the assignment
of the experimentally observed emission lines and thus the ordering of the
$\Gamma_{1}^{n=1}$ and $\Gamma_{5}^{n=1}$ states. In fact, researchers have
reported both negative[49, 22, 25, 17, 23] and positive[18, 20, 50, 51, 52]
values for the spin-exchange splitting $j$. A possible explanation of this
apparent discrepancy is the fact that $j$ depends sensitively on strain.[53,
54, 26] For the case of GaN, Paskov  _et al._ [26] calculated the dependence
of the $\Gamma_{1}^{n=1}-\Gamma_{5}^{n=1}$ splitting (which is given by $2j$)
on biaxial strain, and observed that it changes sign for a biaxial compressive
strain as small as $2\times 10^{-4}$. The scatter of $a$ lattice constants of
bulk AlN amount to a variation of $\pm 2.5\times 10^{-4}$ with respect to the
strain-free value,[55] which, in analogy to GaN, may already be sufficient to
result in a significant change of the spin-exchange splitting in AlN. An
explicit calculation as done by Paskov  _et al._ [26] for GaN is required to
see if this hypothesis can account for the conflicting results in the
literature.

The data in the present work, which are obtained on strain-free AlN, support a
negative value of $j$ and thus a lower energy for the $\Gamma_{1}$ as compared
to the $\Gamma_{5}$ exciton. The geometry ($\mathbf{k\parallel c}$) of the
measurements depicted in Figs. 2 and 3 is compatible with the detection of the
$\Gamma_{5}$, but not the $\Gamma_{1}$ exciton. This fact is confirmed by a
comparison to measurements taken on the cross-section of sample I (see Fig. S6
in the Supplementary Material). In the emission normal to the cross-section
($\mathbf{k\perp c}$), we observe a superposition of the $\Gamma_{5}^{n=1}$
and $\Gamma_{1}^{n=1}$ excitons as expected from the selection rules. The
intense emission of the $\Gamma_{1}^{n=1}$ exciton is situated at 6.031 eV, in
between the $\Gamma_{5}^{n=1}$ and $\mathrm{D^{0}X_{2}}$ lines. The additional
peak observed at higher energy (6.08 eV) for the homoepitaxial layers is
attributed to the first excited state of the free $\Gamma_{5}$ exciton as
discussed in more detail for sample III.

For the PVT substrate, two intense and narrow lines are observed 28 and 13 meV
below the $\Gamma_{5}^{n=1}$ free exciton ground state. The former of these
lines is absent for both samples I and II. These lines have been frequently
observed in previous work, and are attributed to donor-bound exciton
transitions. $\mathrm{D^{0}X_{1}}$, with an exciton binding energy of 28 meV,
has been tentatively assigned to either O [23] or Si [27] as the shallow
donor, while the $\mathrm{D^{0}X_{2}}$ with a binding energy of 13 meV has
been speculated to be related to a native shallow donor.[23] The dominance of
the $\mathrm{D^{0}X_{1}}$ line in the spectra of the substrate, and the total
absence of this transition in our epitaxial layers, implies a significantly
lower concentration of the responsible shallow donor in the layers compared to
the substrate. Accordingly, it is unlikely that this line is related to O, the
concentration of which is even slightly higher in the layers than in the
substrate [cf. Figs. 1(e) and 1(f)]. Si, in contrast, is below the detection
limit for all samples, and it is thus possible that the actual concentration
in the layers is much below that in the substrate. Our experiments are thus
consistent with Si being the donor related to the $\mathrm{D^{0}X_{1}}$ line,
adding support of the assignments made in Refs. 27, 56, 20.

To probe radiative deep levels, we performed CL measurements with lower
resolution in the range of 2–4.2 eV as shown in Fig. 2(b). All spectra in this
figure are normalized to their respective near-bandedge peak intensities. The
integrated intensity of the broad luminescence bands observed in this spectral
range exceeds the near-bandedge intensity by a factor 2.5 for the bulk AlN
substrate, but is lower by factors of 0.5 and 0.8 for samples I and II,
respectively. Apart from the higher intensity, the CL band from the substrate
is much broader and seems to consist of two individual contributions, with the
one at higher energy matching the energy of the band in samples I and II. The
lineshape of samples I and II are almost identical, suggesting that similar
types of defects are incorporated for both polarities.

The overall similarity between the radiative properties of the Al- and N-polar
AlN samples is remarkable. In contrast, studies on N-polar GaN layers,
regardless of whether being grown by MBE or MOCVD, have shown that it is
difficult to obtain layers with a purity and emission characteristics
equivalent to their Ga-polar counterparts. A higher incorporation efficiency
of impurities as well as a lower formation energy for N vacancies in N-polar
GaN growth are discussed as possible reasons for this difference in material
quality [57]. Consequently, N-rich growth seems to be a prerequisite to
suppress the formation of N vacancies for GaN($000\bar{1}$) [28]. For AlN, our
data show that the situation is much more favorable, suggesting that we may
use the alignment of the polarization fields as an additional degree of
freedom for the design of AlN-based light-emitters. Note, however, that
interface recombination may become important in group-III nitride
heterostructures, and may strongly depend on their polarity.[58, 59, 60]

To further elucidate the details of the near-bandedge emission, we turn to
sample III, which exhibits the highest emission intensity of all samples.
Figure 3(a) shows its low-temperature CL spectrum, collected at an
acceleration voltage of 11 keV (still exciting mostly the homoepitaxial layer,
see Fig. S7 in the Supplementary Material). The higher beam energy ensures a
high signal intensity, facilitating the reliable detection of weak
transitions. As seen best in the inset, ten transitions are resolved, the
energies of which are compiled in Table 1. The highest energy transitions
associated to excited states of the free exciton are better visible in the
magnified view displayed in Fig. 3(b). Where available, our values are in good
agreement with previous studies. In particular, the $E(\Gamma_{5}^{n=1})$
exciton and its excited states are resolved at 6.038, 6.079, 6.091, and 6.095
eV, together with their longitudinal optical (LO) phonon replicas at lower
energies. Again, the $\mathrm{D^{0}X_{2}}$ transition and a weak
$\mathrm{D^{0}X_{1}}$ line (originating from the substrate) contribute to the
spectra as well. The prominent transition at 6.079 eV can be readily assigned
to the $E(\Gamma_{5}^{n=2})$ transition, for which energy separations of 37–43
meV with respect to the $E(\Gamma_{5}^{n=1})$ transition have been reported by
other groups,[61, 17, 21, 18, 22, 23, 27] while no values are available for
the higher-order transitions.

To identify the origin of the transitions at 6.091 and 6.095 eV, we first
consider the hydrogenic effective-mass approximation, in which the energies of
the excited states as well as the binding energy of the free exciton can be
determined based on the 1S-2S splitting, i. e., the energy difference between
the first excited state and the ground state
$E(\Gamma_{5}^{n=2})-E(\Gamma^{n=1}_{5})$. Specifically, the energy of the
second excited state is predicted to be

$E(\Gamma_{5}^{n=3})=E(\Gamma_{5}^{n=1})+\frac{32}{27}\left[E(\Gamma_{5}^{n=2})-E(\Gamma^{n=1}_{5})\right].$
(1)

Similarly, the energy of the $n\rightarrow\infty$ transition is given by

$\displaystyle E(\Gamma_{5}^{n\rightarrow\infty})$
$\displaystyle=E(\Gamma_{5}^{n=1})+\frac{4}{3}\left[E(\Gamma_{5}^{n=2})-E(\Gamma^{n=1}_{5})\right].$
(2)

These equations yield values of 6.087 and 6.093 eV for the $\Gamma_{5}^{n=3}$
and $\Gamma_{5}^{n\rightarrow\infty}$ transitions, respectively, slightly
lower than the transition energies observed.

Transition | $E$ | $\Delta E(\Gamma^{n=1}_{5})$ | $\Delta E_{\mathrm{lit}}(\Gamma^{n=1}_{5})$
---|---|---|---
| $(\mathrm{eV})$ | $(\mathrm{meV})$ | $(\mathrm{meV})$
$\boldsymbol{\Gamma}^{\mathbf{n}=\mathbf{1}}_{\mathbf{5}}-\mathbf{2}\mathbf{L}\mathbf{O}$ | 5.815 | $-223$ |  $-225$, [22] $-231$ [20]
$\boldsymbol{\Gamma}^{\mathbf{n}\geq\mathbf{2}}_{\mathbf{5}}-\mathbf{2}\mathbf{L}\mathbf{O}$ | $5.849$ | $-189$ |  $-188$[22]
$\boldsymbol{\Gamma}^{\mathbf{n}=\mathbf{1}}_{\mathbf{5}}-\mathbf{L}\mathbf{O}$ | $5.926$ | $-112$ |  $-109$, [22] $-117$ [20]
$\boldsymbol{\Gamma}^{\mathbf{n}\geq\mathbf{2}}_{\mathbf{5}}-\mathbf{LO}$ | $5.965$ | $-73$ |  $-73$[22]
$\mathrm{D^{0}X_{\alpha}}$ |  |  |  $-37$[25]
$\mathbf{D^{\mathbf{0}}\mathbf{X}_{\mathbf{1}}}$ | $6.010$ | $-28$ |  $-28$ [25], $-28.5$[21], $-28.5$[27]
$\mathrm{D^{0}X_{\beta}}$ |  |  |  $-25$[25]
$\mathrm{D^{0}X_{\gamma}}$ |  |  |  $-22.1$ [17], $-22.4$ [23], $-22.5$[27]
$\mathrm{D^{0}X_{\delta}}$ |  |  |  $-19$ [25], $-19.0$ [27]
$\mathbf{D}^{\mathbf{0}}\mathbf{X}_{\mathbf{2}}$ | $6.025$ | $-13$ |  $-13$ [25], $-13.3$[27], $-13.6$ [21]
$\mathrm{D^{0}X_{\epsilon}}$ |  |  |  $-9.5$[21], $-9.5$[27]
$\boldsymbol{\Gamma^{\mathbf{n}=\mathbf{1}}_{1}}$ | 6.031 | $-7$ |  $-8$[25]
$\boldsymbol{\Gamma}^{\mathbf{n}=\mathbf{1}}_{\mathbf{5}}$ | $6.038$ | 0 | 0
$\Gamma^{\mathrm{n=2}}_{1}$ |  |  |  $+31$[25]
$\boldsymbol{\Gamma}^{\mathbf{n}=\mathbf{2}}_{\mathbf{5}}$ | $6.079$ | $+41$ |  $+39.4$[21], $+39.4$[27], $+41$[25]
$\boldsymbol{\Gamma}^{\mathbf{n}=\mathbf{3}}_{\mathbf{5}}$ | $6.091$ | $+53$ |
$\boldsymbol{\Gamma}^{\mathbf{n}\rightarrow\infty}_{\mathbf{5}}$ | $6.095$ | $+57$ |
Table 1: Low-temperature (10 K) near-bandedge transitions in AlN. Transitions
resolved in the homoepitaxial, N-polar AlN layers measured in the present work
are highlighted in bold. All of these values have an experimental uncertainty
of $\pm 1$ meV. We also provide the shift with respect to the $\Delta
E(\Gamma^{n=1}_{5})$ transitions and, where available, corresponding
literature values. An extended version of this table is provided in the
Supplementary Material.

The simple hydrogenic approximation is not expected to be in perfect agreement
with the actual transition energies, since AlN exhibits a notable anisotropy
regarding both the reduced exciton mass $\mu$ and the relative static
permittivity $\varepsilon$. This anisotropy is commonly summarized in the form
of the anisotropy parameter
$\gamma=\varepsilon_{\perp}\mu_{\perp}/(\varepsilon_{\parallel}\mu_{\parallel})$
with the directions relative to that of the $c$ axis.[62, 63, 64, 65] For
values of $\gamma$ other than 1 or 0, exact analytical expressions for the
excitonic levels do not exist. Gil _et al._[64] have provided numerical values
for the $\Gamma_{5}^{n=2}$, $\Gamma_{5}^{n=3}$, and
$\Gamma_{5}^{n\rightarrow\infty}$ transitions, placing them at 6.076, 6.083,
and 6.086 eV for $\gamma\approx 1.8$, the value obtained for AlN using the
material parameters given in Ref. 65. These energies are systematically lower
than those obtained by the simple hydrogen model because $\gamma>1$, which
results in a _reduction_ of the exciton binding energy. In fact, while the
effective Rydberg energy
$\text{Ry}^{*}=\text{Ry(H)}\,\mu_{\perp}/(\varepsilon_{\parallel}\varepsilon_{\perp})$
amounts to 59 meV, where $\text{Ry(H)}=13.6$ eV is the ionization energy of
the hydrogen atom, the binding energy obtained within the framework of an
anisotropic exciton is only 48 meV, as reported previously by Funato _et
al._[21] Even the experimentally observed 1S-2S splitting of 41 meV only
allows for an exciton binding energy of 52 meV, still inconsistent with the
highest energy feature observed in Fig. 3(b).

Indeed, for an accurate determination of the exciton binding energy in AlN,
Ishii _et al._[65] have pointed out that it is crucial to not only consider
crystal anisotropy, but exchange and electron-phonon interactions as well.
They showed the latter to result in a significantly increased exciton binding
energy for the $\Gamma_{5}$ triplet, namely, 64 meV, accompanied by a 1S-2S
splitting of 51 meV. Based on our $\Gamma_{5}$ transition energy of 6.038 eV,
these values would result in transition energies for the $\Gamma_{5}^{n=2}$
and $\Gamma_{5}^{n\rightarrow\infty}$ excitons of 6.089 and 6.102 eV, too
large compared to the experimental values. Clearly, theory has still to be
refined to accurately reproduce the excitonic transition energies observed
experimentally. Currently, we share the observation in the literature that the
hydrogenic approximation still seems to reproduce experimental data best.[49]
Consequently, we assign the emission lines observed at 6.091 and 6.095 eV to
the $n=3$ and $n\rightarrow\infty$ states of the $\Gamma_{5}$ free exciton.
The energy of the $\Gamma^{n\rightarrow\infty}_{5}$ transition corresponds to
the exciton continuum and thus the bandgap of AlN. Therefore, the energy
difference between $\Gamma^{n=1}_{5}$ and $\Gamma^{n\rightarrow\infty}_{5}$
directly provides the binding energy of the $\Gamma_{5}$ exciton, which thus
amounts to 57 meV.

Other experimental values for the free exciton binding energy reported in the
literature range from 47 to 67 meV. Those derived by relying on the 1S-2S
spitting [61, 17, 21, 18, 22, 23, 27] or the energy difference between the
$n=1$ free exciton transition and the exciton-exciton scattering band
$\mathrm{P}^{\infty}$ [17, 20] tend to be close to our value, with the average
being about 54 meV. The large overall variation may be due to strain effects,
sample heating (especially for high-excitation densities), and the large
energy splitting between the $\Gamma_{1}$ and $\Gamma_{5}$ excitons [65].

Besides the zero-phonon exciton emission, we resolve two longitudinal optical
(LO) phonon replicas each for the free exciton state and for the excited
states from this sample. In Fig. 3(a), the LO phonon replicas are denoted as
$\Gamma^{(n=1)}_{5}\mathrm{-}m\mathrm{LO}$ and $\Gamma^{(n\geq
2)}_{5}\mathrm{-}m\mathrm{LO}$ with $m\in\\{1,2\\}$. The energy spacing
between the phonon replicas and the main peaks is a multiple of 112 meV, in
close agreement with the quantized LO phonon energy at the $\Gamma$-point in
AlN [66].

Table 1 summarizes all near-bandedge transition energies measured for our MBE
films. Column $E$ lists the absolute transition energy, and column $\Delta
E(\Gamma^{n=1}_{5})$ the energy shift with respect to the ground state exciton
energy $\Gamma^{n=1}_{5}$. Table 1 also gives values for $\Delta
E_{\mathrm{lit}}(\Gamma^{n=1}_{5})$ of other transitions observed in studies
in the literature but which are not observed in our films. Note that the
actual origin of the $\mathrm{D^{0}X_{\alpha}}$, $\mathrm{D^{0}X_{\beta}}$,
$\mathrm{D^{0}X_{\gamma}}$, $\mathrm{D^{0}X_{\delta}}$, $\mathrm{D^{0}X_{2}}$
and $\mathrm{D^{0}X_{\epsilon}}$ lines is not yet known. These lines are
generally believed to originate from other donor-bound excitons because of
their narrow linewidths, but the contribution of acceptor-bound excitons,
inelastic scattering processes or many-body effects cannot be ruled out. [25,
21, 27] An extended table of all near-bandedge transitions that have been
reported in literature can be found in the Supplementary Material (Table ST1).

Finally, it is interesting to discuss the lack of emission from the $n\geq 3$
free exciton states in any previous study. At the first glance, it may appear
surprising to see recombination from free carriers at a nominal temperature of
10 K, considering the strong Coulomb interaction in AlN. The fraction of
excitons at a given temperature and electron-hole density is governed by
Saha’s law[67] and is predicted to decrease with increasing carrier
temperature and to increase with increasing electron-hole density below the
Mott transition. In the majority of previous works, the samples were excited
with an ArF laser, which delivers intense ns pulses with a low repetition rate
and an energy close to the band edge. Due to the low repetition rate, the
lowest excitation densities are typically around 50 kW/cm2, resulting in a
carrier density on the order of $5\times 10^{17}$ cm-3. On the other hand, the
excess energy delivered per pulse is comparatively low. For the present CL
experiments, the situation is opposite. The continuous-wave excitation by the
electron beam facilitates a comparatively low excitation density. In the
present case, we estimate a carrier density of not more than $4\times 10^{16}$
cm-3 for an acceleration voltage of 11 kV, as detailed in the Supplementary
Material.[68] The carrier temperature, however, is typically rather high,
since the electron beam creates highly energetic secondary electrons and
holes. The cooling of these hot carriers proceeds via the emission of LO
phonons, thus creating a nonequilibrium population of hot LO phonons [69] that
in turn heats the carrier distribution by strongly increasing the probability
of LO phonon absorption. Indeed, from the high-energy slope of the
$\Gamma^{(n=1)}_{5}\mathrm{-}2\mathrm{LO}$ transition,[67, 70] we deduce a
carrier temperature of 120 K. With this carrier temperature and excitation
density, Saha’s law (See figure S8 in the Supplementary Material) predicts an
excitonic fraction of only 0.82, while for the carrier density estimated for
ArF excitation, the fraction increases to 0.95 even for the same carrier
temperature. These estimates provide a clear physical explanation for the
prominence of excited exciton states in our CL experiments.

## Conclusion

In summary and conclusion, we have investigated the properties of AlN layers
grown by plasma-assisted MBE on both the N- and Al-polar faces of bulk AlN
substrates. Regardless of polarity, the layers exhibit atomically smooth
surfaces, high structural perfection and purity, and feature intense free
exciton emission and suppressed emission from bound excitons and deep-level
defects. These results highlight the potential of MBE for the growth of UV
emitters. The ability to grow N-polar samples without compromising the crystal
quality and thus to change the orientation of the polarization fields in AlN
adds an additional degree of freedom that can be exploited in the design of
MBE-grown deep-UV emitters. Finally, we have shown that the unique excitation
conditions in CL spectroscopy facilitates the detection of several excited
states of the $\Gamma_{5}$ exciton, including the excitonic continuum,
directly yielding the exciton binding energy of 57 meV.

## Supplementary Material

See the Supplementary Material for reciprocal space maps as well as symmetric
and asymmetric $\omega$ scans of samples I and II, a detailed characterization
of sample III (AFM, XRD, SIMS), CL spectra measured under different geometries
(sample surface and cross-section), a discussion of the generation volume and
excitation density in CL spectroscopy, the phase diagram of the coupled
exciton-carrier system according to Saha’s law, as well as an extended table
of the near-bandedge transitions observed in AlN.

###### Acknowledgements.

The authors thank Chandrashekhar Savant, Zexuan Zhang, and Shivali Agrawal for
helpful discussions. This work was partially supported by the Cornell Center
for Materials Research with funding from the NSF MRSEC program (Grant No.
DMR-1719875). Further support was granted by the National Science Foundation
with grant No. NNCI-2025233, DMR-1539918, RAISE-TAQS 1839196, MRI 1631282, and
AFOSR Grant No. FA9550-20-1-0148. Finally, LvD acknowledges the "Deutscher
Akademischer Austauschdienst" (DAAD) for funding provided through the Research
Internships in Science and Engineering (RISE) Professional Scholarship.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability

The data that support the findings of this study are available within the
article and its Supplementary Material. The raw data may be obtained from the
corresponding author upon reasonable request.

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