Electronic Band Transitions in γ-Ge3N4

Eduard Feldbach, Andreas Zerr, Luc Museur, Mamoru Kitaura, Geeth

Manthilake, Franck Tessier, Veera Krasnenko, Andrei Kanaev

► To cite this version:

Eduard Feldbach, Andreas Zerr, Luc Museur, Mamoru Kitaura, Geeth Manthilake, et al. Electronic Band Transitions in γ-Ge3N4. Electronic Materials Letters, 2021, 17 (4), pp.315-323. 10.1007/s13391- 021-00291-y. hal-03206152

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Distributed under a Creative Commons Attribution 4.0 International License
Electronic band transitions in γ-Ge₃N₄

Eduard Feldbach¹, Andreas Zerr², Luc Museur³, Mamoru Kitaura⁴, Geeth Manthilake⁵, Franck Tessier⁶, Veera Krasnenko¹,⁷, Andrei Kanaev²*

¹Institute of Physics, University of Tartu, 1 W. Ostwald str., Tartu, Estonia

²Laboratoire des Sciences des Procédés et des Matériaux, CNRS, Université Sorbonne Paris Nord, Villetaneuse, France

³Laboratoire de Physique des Lasers, CNRS, Université Sorbonne Paris Nord, Villetaneuse, France

⁴Faculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan

⁵Laboratoire Magmas et Volcans CNRS, IRD, OPGC, Université Clermont Auvergne, 63000 Clermont-Ferrand, France

⁶Institut des Sciences Chimiques de Rennes, CNRS, Université Rennes, Rennes, France

⁷Institute of Solid State Physics, University of Latvia, Riga, Latvia

Received ZZZ, revised ZZZ, accepted ZZZ

Published online ZZZ (Dates will be provided by the publisher.)

**Keywords:** γ-Ge₃N₄, photoluminescence, cathodoluminescence, electronic transitions, exciton

* Corresponding author: andrei.kanaev@lspm.cnrs.fr; Phone: +33 1 49 40 34 30; Fax: +33 1 49 40 34 14

Electronic band structure in germanium nitride having spinel structure, γ-Ge₃N₄, was examined using two spectroscopic techniques, cathodoluminescence and synchrotron-based photoluminescence. The sample purity was confirmed by x-ray diffraction and Raman analyses. The spectroscopic measurements provided first experimental evidence of a large free exciton binding energy $D_c≈0.30$ eV and direct interband transitions in this material. The band gap energy $E_g=3.65±0.05$ eV measured with a higher precision was in agreement with that previously obtained via XES/XANES method. The screened hybrid functional Heyd-Scuseria-

Ernzerhof (HSE06) calculations of the electronic structure supported the experimental results. Based on the experimental data and theoretical calculations, the limiting efficiency of the excitation conversion to light was estimated and compared with that of w-GaN, which is the basic material of commercial light emitting diodes (LEDs). The high conversion efficiency, very high hardness and rigidity combined with a thermal stability in air up to ~700 °C reveal the potential of γ-Ge₃N₄ for robust and efficient photonic emitters.

## 1. Introduction

Germanium(IV) nitride having spinel structure is one of the members of the family of spinel nitrides of the group 14 elements (γ-M₃N₄ where M=Si, Ge, Sn) discovered two decades ago [1-3]. As any high-pressure non-conducting material, γ-Ge₃N₄ exhibits high elastic moduli and hardness, [4]; it is also thermally (meta)stable at atmospheric pressure to ~700 °C [5]. More important is that, soon after their discovery, the group 14 spinel nitrides have been predicted to possesses direct band gaps ($E_g$) spanning the visible wavelength range and extending in the near-IR and UV ranges of light, from 0.6 eV to 5.0 eV [6-10]. In particular, $E_g$ of γ-Ge₃N₄ was predicted to be between 2.07 eV and 3.6 eV depending on the applied calculation approach (LDA, GGA, GW etc.). Direct band gap is one of the basic conditions for a material to be considered as an efficient light emitter while a large exciton binding energy ($D_c>>kT$) is the second necessary condition. The latter parameter has been evaluated theoretically by Boyko et al. [7] for all end-members γ-M₃N₄ and their hypothetical solid solutions using

the hydrogenic-type model. For γ-Ge₃N₄, the approach led to a large value of $D_c=174$ meV while the highest $D_c=333$ meV has been predicted for γ-Si₃N₄ and the lowest $D_c=69$ meV for γ-Sn₃N₄.

The experimental verifications of the theoretical predictions about the direct interband electronic transition, exciton binding energy and band gap energy of these materials remain rather limited. Boyko et al. [6-7] have used soft x-ray emission and x-ray absorption spectroscopy measurements to obtain the band gap energies $E_g$ of γ-M₃N₄ (M=Si, Ge, Sn) and (GeₓSi₁-x)₃N₄ in the range of 1.6-5.0 eV with the relatively low precision of ±0.2 eV. Later on, Museur et al. [11] have conducted studies of the interband and intraband radiative electronic transitions in γ-Si₃N₄ by time- and energy-resolved luminescence method and obtained $E_g=5.05±0.05$ eV and a large free exciton binding energy of ~0.65 eV, which is almost twice larger compared to that theoretically predicted. Today, these papers exhaust available experimental data on the electronic transitions in the spinel nitrides of the group 14 elements, γ-M₃N₄.
Furthermore, compounds with the spinel structure [12] have attracted an additional interest during last decades because of their expected high tolerance to high doses of the irradiation with energetic particles [13-14]. The spinel lattice contains a great number of unoccupied voids, so-called intrinsic structural vacancies: 7/8 of available tetrahedral and 1/2 of octahedral cation sites. For this reason, the spinel lattice can accommodate cations displaced from their regular positions and promote their returning without a significant lattice distortion. Compared with ternary oxides with the general formula $AB_2O_4$ (where A, B are metal cations and O is oxygen anion), the most known example of which is $MgAl_2O_4$, the binary spinel nitrides $\gamma$-M$_3$N$_4$ have an additional advantage that antisite defects are not possible in their lattices. This peculiarity, combined with a high hardness, thermal stability in air, direct band gap and large $D_e$ make $\gamma$-Si$_3$N$_4$ and, potentially, other spinel nitrides interesting as robust and efficient photonic emitters in extreme environments.

In this communication, we report on the first investigation of the electronic band transitions in $\gamma$-Ge$_3$N$_4$ using UV-visible-near-IR time- and energy-resolved spectroscopy. Based on the obtained experimental data and theoretical estimations, we argue about potentially high efficiency of the excitation conversion to light in this material.

## 2. Methods

The polycrystalline sample of $\gamma$-Ge$_3$N$_4$ was synthesized in a multi-anvil press following the procedure reported in Ref. [1] with some modifications. The starting hexagonal phase of germanium nitride, $\beta$-Ge$_3$N$_4$, was prepared via the reaction of $GeO_2$ under ammonia flow in a tubular furnace at 880°C during 12h. The resulting product was annealed under the same conditions for purity purpose and the phase composition was controlled by X-ray diffraction (XRD) to be pure $\beta$-Ge$_3$N$_4$. Elemental analyses, performed using a Leco TC-600 analyzer, showed a low level of oxygen contamination of ~0.3 wt%. No trace of carbon, eventually coming from the synthesis route or from the used vessels, was found. The $\beta$-Ge$_3$N$_4$ powder was compacted in an iron capsule (wall thickness 50 µm), compressed in a multianvil press to ~13 GPa and heated to 1200 °C for 20 min. The 14/8 assembly with a 14 mm edge length $Cr_2O_3$-doped MgO octahedral pressure medium and 8 mm tungsten carbide anvil truncation was used. The recovered product was an almost completely sintered cylinder of ~1.25 mm in diameter and ~1.9 mm in height which split perpendicularly to the cylinder axis in discs of 200-300 µm thickness. One of them, from the sample middle, was used in this study.

The recovered high-pressure product was examined with the powder XRD and Raman spectroscopy. The XRD patterns were obtained using a set-up consisting of a microfocus source delivering Mo $K\alpha$ radiation into a spot of less than 200 µm (1µS high Brilliance-Incoatec Microfocus Source Mo ELM33, Incoatec) and an imaging plate detector

with online readout system (mar345, marXperts). In order to obtain high-quality XRD diffractograms, two-dimensional patterns were collected in the grazing incidence geometry at the angle of ~7° and converted to the standard diffractograms I(20) using the Fit2D software [15]. This set-up permits detection of contaminating phases and compounds (e.g. $GeO_2$) in amounts as low as 0.5 vol.%, not accessible using a standard diffractometer. The Raman spectra were measured with a blue solid-state laser emitting at 473 nm. The spectra were collected by a 100xLWD objective with the lateral resolution of 1 µm using Jobin Yvon HR800 spectrometer equipped with a Pelletier cooled CCD detector; the spectral resolution of measurements was 1 cm⁻¹.

The cathodoluminescence (CL) spectra of samples mounted on a close-cycle He cryostat were recorded from the periphery part of sample area at T=5 K using a setup of the Institute of Physics of Tartu University (for details see Ref. [16]). The reason for avoiding central part of the sample (covered in experiments by Pt foil) was a weak oxygen contamination, as it will be explained below. The CL spectra were recorded with H8259 Hamamatsu photon counting head mounted on ARC SpectraPro 2300i monochromator with 300 l/mm 500 nm blaze grating. The excitation was performed with an electron beam of 10 keV and 0.1 µA. The software for time-resolved CL measurements permitted the collection of emission spectra with different time windows delayed by 5 ns to 2 ms after the excitation pulse of 10 ns duration and rise time of ~1 ns with maximum repetition rate of 5 kHz. The time resolution (~5 ns) of these measurements was limited by Becker&Hickl MSA-300 multiscaler card and photon counting head response.

The photoluminescence (PL) experiments were carried out at the BL3B beamline [17] of the UVSOR facility with synchrotron radiation excitation in the UV-visible spectral range. The samples were mounted on a cold finger of a LHe flow type cryostat maintained during experiments at T=9 K in the vacuum of ~$10^{-7}$ Pa. In order to increase the signal, PL was collected from the total sample area using Spectra Pro 300i monochromator equipped with a CCD detector. The measured photoluminescence excitation (PLE) spectra were corrected on the detector (AXUV100 photo-diode) sensitivity and optical line transmission.

The spectroscopic measurements were accompanied by ab initio numerical calculations using the CASTEP module of the Materials Studio program package [18]. In order to provide more accurate result for Kohn-Sham energy levels, the screened hybrid functional of Heyd-Scuseria-Ernzerhof with optimized screening parameters (HSE06) was used in calculations of the electronic properties on a primitive cell of $\gamma$-Ge$_3$N$_4$, which permits very reasonable band gap energy values for all three spinel nitride structures of Si$_3$N$_4$, Ge$_3$N$_4$ and Sn$_3$N$_4$: the calculated band values of 4.9 eV, 3.73 eV and 1.79 eV were in a very good agreement with experimental data of 5.05±0.05 eV [11], 3.5±0.2 eV [7] and 1.6±0.2 eV [7] for Si$_3$N$_4$, Ge$_3$N$_4$ and Sn$_3$N$_4$, respectively.
The plane-wave basis energy cutoff (with norm con-
serving pseudopotentials) was chosen to be 700 eV. The
Monkhorst-Pack scheme k-points grid sampling was set at
4×4×4 for the Brillouin zone calculation. The convergence
parameters were set as follows: total energy tolerance of
1×10⁻⁵ eV/atom, maximum force tolerance of 3×10⁻³
eV/nm, maximum stress of 0.05 GPa and maximum atomic
displacement of 1×10⁻⁴ nm. The electronic configurations
were 4s²4p² for germanium and 2s²2p³ for nitrogen atoms.

3. Results and Discussion

The XRD pattern shown in Fig. 1 indicated an almost
pure single phase γ-Ge₃N₄ sample with no admixture of the
precursor β-phase. The pattern is dominated by the peaks
of γ-Ge₃N₄ which positions perfectly agree with the cubic
lattice parameter *a*=8.2125 Å, reported earlier [19]. Only a
very weak oxide phase contamination <1 vol.% was ob-
served in the central part of the sample attributed to ger-
manium dioxide with rutile structure, r-GeO₂. The periph-
ery regions of the sample were free of any crystalline con-
tamination. Apparently, the iron capsule served as the
buffer, which reduced oxygen content in the vicinity of the
capsule walls.

<figure><img src="image_2.png" /><figcaption>Fig. 1. X-ray diffraction pattern of γ-Ge₃N₄ sample.</figcaption></figure>

The Raman spectra of γ-Ge₃N₄ sample shown in Fig. 2
were dominated by two sharp intense bands at 728 cm⁻¹
(T₂g) and 329 cm⁻¹ (T₉) and three other less intense bands
at 854 cm⁻¹ (A₉), 592 cm⁻¹ (T₂g) and 470 cm⁻¹ (E₉), which
have been previously experimentally observed and theoret-
ically validated to be the five Raman-active modes permit-
ted for the ideal spinel structure [2, 20-22]. No extra high-
frequency mode in the range 770-820 cm⁻¹ associated with
nitrogen vacancies [22] were observed, indicating the ex-
pected stoichiometry of the examined sample. The spectra
taken in the central and periphery oxygen-free sample parts
were identical, confirming a negligible influence of oxidic
impurities.

The CL spectra of the γ-Ge₃N₄ sample are shown in Fig. 3. At
room temperature, the CL band at 560 nm (2.20 eV) domi-
nated the spectrum. When cooling sample down to 5 K, its

spectral maximum red shifted to 575 nm (2.15 eV) and a
new UV band appeared as a shoulder at ~390 nm, exhibit-
ing a regular structure with the period of ~0.09 eV, which
corresponds to the strong T2g vibration mode (ħω=90.3
meV). Such sequence of peaks, well-resolved at low tem-
peratures (5 K in our case), can be attributed to phonon rep-
licas of a zero-phonon line [23], appearing in our CL spec-
trum at 3.33 eV, as depicted in Fig. 3. This kind of states
interpreted as large-radius bound excitons usually appears
in solids with Wannier excitons (semiconductors AIIBVI
and many others) and has in most cases relatively small
binding energy below 100 meV. Because CL spectra were
taken from the periphery oxygen-free region of the sample,
identification of the defect with the contaminant r-GeO2
phase can be excluded.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Raman spectra of γ-Ge₃N₄ sample.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. CL spectra (spectral resolution 1 nm) taken from periphery part of γ-Ge₃N₄ sample at T=300 K (a) and 5 K (b, c); intensity of (c) is that of (b) x5.</figcaption></figure>

According to our analysis of another member of the
high-pressure spinel nitrides, γ-Si₃N₄ [11], two different
excited states may contribute to the visible emission: exci-
ton bound to neutral cation X⁻·Si⁺ and metastable paramag-
netic anion radical N⁺ (in the Kröger-Vink notation). Be-
cause of the structural similarity, the both contributions are
expected to appear in γ-Ge₃N₄. In order to resolve these two emissions with strongly different excited-state lifetimes, we applied time-resolved CL measurements with short Δt₁=0-20 ns and long Δt₂=0-200 ns time-windows. These measurements were performed with a much lower spectral resolution, which permitted to increase the signal-to-noise ratio (fine vibronic structure of the UV band was not resolved in these spectra). The results shown in Fig. 4 confirmed the expectations: short- and long-lived spectral components were resolved in these measurements. Accordingly, we assigned the spectrum recorded with the long time-window (CL dashed line) with the maximum at λ=575 nm (hν=2.15 eV) to the anion radical N⁺. In contrast, the broad UV emission, which became dominating in the short time-window spectrum (b), has two contributions from (i) non-resolved lines between 3.33 and 3.0 eV due to the large-radius bound exciton and (ii) continuum with spectral maximum at λ=410 nm (3.0 eV) due to the so-called extrinsic self-trapped or small-radius bound exciton. This interpretation is based on the model introduced by Toyozawa [24] and demonstrated experimentally in numerous studies (see e.g. [25-27]). The main point of this model is that in a perfect lattice the strength of the exciton-phonon interaction is not strong enough for the self-trapping of an exciton and the deformation potential of a defect is also not sufficient for the exciton trapping, but the total short range interaction of an exciton with defect and phonons triggers the trapping process. As a result, large- and small-radius bound excitons coexist. Usually defects responsible for this kind of situation are of neutral type, which in our case is, most probably, neutral structural vacancies Vₐₓ [28]. The zero-phonon line of the localized exciton X⁺-Vₐₓ was observed at 3.33 eV in the spectrally resolved CL (Fig. 3).

<figure><img src="image_5.png" /><figcaption>Fig. 4. Integral (a) and time-gated (b-c) CL spectra (spectral resolution 5 nm) taken from periphery part of γ-Ge₃N₄ sample at T=5 K: Δt₁=0-20 ns (b) and Δt₂=0-200 ns (c). All intensities are normalized for better visibility.</figcaption></figure>

In contrast to CL, PL is state-selective in excitation and can provide more information about the energetic struc-

ture. The PL and PLE spectra of γ-Ge₃N₄ sample are shown in Fig. 5. The PL spectrum consists of a broad visible band and a short-wave shoulder at ~400 nm. The apparent red shift of the principal visible PL band with respect to that observed in CL spectra (Fig. 3b) could be, at least partially, explained by sensitivity of H8259 photomultiplier used in CL measurements, which strongly drops at λ>600 nm. Similarly, a stronger sensitivity of the photomultiplier used in CL measurements with respect to that of the CCD detector of PL setup explains a stronger intensity of the self-trapped exciton in CL spectra (Figs 4 and 5). By analogy with γ-Si₃N₄ [11], we assume two major contributions of the exciton trapped near neutral cation defect X⁺-Geₓ and anion radical N⁺ to the visible PL.

<figure><img src="image_6.png" /><figcaption>Fig. 5: PL (Eexc=3.95 eV) and PLE (EPL=2.05 eV) spectra of γ-Ge₃N₄ sample at T=10 K.</figcaption></figure>

We notice that examination of state-selective PLE spectra for the interband transitions analysis can be preferable in comparison with that of absorbance spectra, because PLE can be free of non-radiative and not relevant (spectrally filtered) radiative transitions of defect/impurity as well as of a contribution of light scattering on grain boundaries in polycrystalline samples. The data examination was performed in agreement with our recent study of γ-Si₃N₄ [11], where visible PL excitation has been shown to fit the interband absorption. We used the Tauc expression for interband transitions in the examined material

$$ \lg(\alpha h \nu) = k \lg(h \nu - E_g) + A \quad (1) $$

where α is absorption coefficient, hv and Eg are photon and band gap energies and A and k are constants. The constant k defines the type of the interband transitions: it is equal to 1/2 and 2 respectively for direct and indirect allowed transitions. In the vicinity of the fundamental absorption onset energy, the normalized PLE intensity I(hv)/I₀ (where I₀ is the maximal PLE intensity) is proportional to the absorption coefficient α. According to Eq. 1, the experimental data in a logarithmic frame should fit a straight line if Eg value is chosen correctly (Fig. 6a); the
slope of this line defines constant k and, consequently, type
of the interband transitions. More precisely, each of the
least-squares fits for a given E_g provided two relevant val-
ues of the coefficient of determination R² and k, which are
plotted as a function of E_g in Fig. 6b. Since the maximum
of R² defines the best fit by Eq. (1), we adopted E_g∈[3.6;
3.7] eV and k≈1/2 as the best solution in framework of the
considered model. Consequently, analysis of the experi-
mental data resulted in the band gap energy E_g=3.65±0.05
eV and evidenced that the interband transitions of γ-Ge₃N₄
are direct allowed. We note that the earlier reported exper-
imental value of E_g=3.5±0.2 eV [7] agrees within the ex-
perimental uncertainties with the present experimental re-
sult.

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Modified Tauc plot, according to Eq. (1), obtained from PLE data for several indicated E_g values (a) and R² and k values derived from the least-squares fits of experimental data in the range of E_g between 3.5 and 3.8 eV (b). The adopted solution E_g=3.65±0.05 eV and k=1/2 corresponds to the maximum of R² (b). Solid lines in (b) are to guide the eye.</figcaption></figure>

The obtained experimental data also permitted an im-
portant conclusion about the value of the free-exciton bind-
ing energy D<sub>e</sub> of &gamma;-Ge<sub>3</sub>N<sub>4</sub>. In fact, a comparison of CL
spectra in Fig. 3 indicated the disappearance of the local-
ized exciton (b) at room temperature (a). This signifies that
the exciton detached from the weakly bound state with the
dissociation energy &delta;E<0 meV and quenched in the host
matrix. Taking into account the position of zero-phonon
line at E&deg;=3.33 eV, one can derive the exciton binding en-
ergy: D<sub>e</sub>=E<sub>g</sub>-E&deg;-&delta;E&asymp;0.30 eV. We notice that a significantly
smaller free exciton binding energy of 174 meV has been
calculated for the hydrogen-like exciton [7]. This tendency
to the underestimation of the free exciton binding energy
has been previously observed for another member of the
spinel nitrides family, &gamma;-Si<sub>3</sub>N<sub>4</sub>, where the experimental
value of 0.65 eV [11] was significantly larger than the cal-
culated one of 0.333 eV [7].

The direct allowed transitions in defect-free crystals
generally show up in an intense luminescence. We evaluat-
ed quantum efficiency of γ-Ge₃N₄ luminescence by a com-
parison with the reference sample (Lu,Y)₂Si₅:Ce (LYSO),
which is a well-known efficient scintillator. The compari-
son showed that the intensity of our sample was about two
orders of magnitude lower than that of the reference mate-
rial. The quantum efficiency of ~0.4% was estimated in
our γ-Ge₃N₄. This rather low emission efficiency is charac-
teristic of a non-perfect solid, which is what our polycrys-
talline sample has to be. Summing up, the synthesis proce-
dure has to be further improved in order to obtain efficient
photonic emitters of this material.

The results of the evaluation of the electronic band
structure and density of states in γ-Ge₃N₄ using HSE06
calculations with optimized screening parameters [29] are
shown in Fig. 7.

<figure><img src="image_8.png" /><figcaption>**Figure 7:** Electronic band structure (a) and density of states (b) of γ-Ge₃N₄ obtained from DFT calculations using the HSE06 functional.</figcaption></figure>

Firstly, our calculations confirmed the direct allowed
character of the interband transitions in γ-Ge₃N₄. Secondly,
the calculated bandgap value of E_g=3.73 eV is in a good
agreement with our experimental value of E_g=3.65±0.05
eV, which is the most accurate one reported so far for γ-
Ge₃N₄. According to these calculations, Ge₃N₄ has the di-
rect gap with minimum of the conduction band and maxi-
mum of the valence band at Γ point of the Brillouin zone
center, as zoom image in Fig. 8a shows. The similar result
for γ-Ge₃N₄ has been reported by Gao et al. (E_g=3.46 eV)
[30], though Jayatunga et al. [31] have found that the band
gap is indirect (E_g=3.48 eV). These both cited authors have
used an accurate but very time-consuming GW method of
calculations. The obtained in the present work effective
masses of electron m_e=0.58m_0 and hole m_h=5.90m_0 (m_0 is
the mass of free electron) and dielectric constant ε=2.88
were used in estimations of the emission efficiency (see
below). The calculated density of states (DOS) diagram
(Fig. 7b) helps to understand the origin of the electronic
bands. The valence band in the range from -11 to 0 eV is
composed mainly of N 2p states and in the range from -15
to 20 eV of N 2s states, with a small Ge 4s and Ge 4p con-
tributions. The contours of the Ge 4p and N 2p curves in
the valence band below the Fermi level are similar and
overlap indicating the covalent character of the bonds formed between Ge and N atoms. This meets the conclusions by Liu et al. [32]. The conduction band with the width of about 8 eV is formed mainly by a superposition of Ge-p, Ge-s and N-p states; the upper part of the conduction band shows an increasing weight of the Ge-p states.

Our calculations showed that most of the valence electrons are localized on N atoms. Figure 8b shows the electron density difference in a plane with normal along (0, $\sqrt{2}/2$, $\sqrt{2}/2$) direction of the crystalline lattice and half bond shifted from the atomic plane. This density difference is affected by nitrogen and germanium atoms on the octahedral and tetrahedral sites: Ge-N bond between N-p and Ge-p in the tetrahedral site is slightly more covalent than between N-p and Ge-p in octahedral site, since it exhibits more elongated shape. The results confirm formation of weak covalent bonds between Ge and N atoms in this structure (see also [32]).

<figure><img src="image_9.png" /><figcaption>**Figure 8:** Magnified/zoomed representation of the top of the valence band of γ-Ge<sub>3</sub>N<sub>4</sub> confirming direct nature of band gap (a) and electron density difference in cross-section oriented along (0, $\sqrt{2}/2$, $\sqrt{2}/2$) direction and half bond shifted (in electrons/Å<sup>3</sup>).</figcaption></figure>

The calculated Mulliken charges [33] for γ-Ge<sub>3</sub>N<sub>4</sub> compound are collected in Table 1. We notice slightly different charges of Ge cations in the octahedral and tetrahedral sites.

<table><caption>**Table 1:** Effective Mulliken charges of γ-Ge<sub>3</sub>N<sub>4</sub> (in units of proton charge).</caption><thead><tr><th rowspan="2">Species</th><th colspan="2">Electron configuration</th><th rowspan="2">Charge (e)</th></tr><tr><th>s</th><th>p</th></tr></thead><tbody><tr><td>N</td><td>1.69</td><td>4.49</td><td>-1.18</td></tr><tr><td>Ge (octahedral)</td><td>0.83</td><td>1.60</td><td>1.57</td></tr><tr><td>Ge(tetrahedral)</td><td>0.86</td><td>1.55</td><td>1.59</td></tr></tbody></table>

The results obtained here and published earlier permitted an estimation of the limiting efficiency η of the exciton energy conversion to luminescence in an ideal defect-free crystal, which can be expressed as:

$$ \eta = \frac{r_r}{r_r + r_d} \qquad (2) $$

where $r_r = 1/\tau_r$ is the rate of exciton radiative transitions ($\tau_r$ is the exciton radiative lifetime) and $r_d$ is the exciton dissociation rate estimated from the Onsager-Braun model [34]

$$ r_d = \nu \cdot \exp(-D_e / kT) \qquad (4) $$

with the frequency factor

$$ \nu = 3e(\mu_e + \mu_h) / 4\pi\epsilon a^3 \qquad (5) $$

where $e$, $\mu_e$, $\mu_h$, $\epsilon$, and $a$ are, respectively, electron charge, electron mobility, hole mobility, dielectric constant and the exciton Bohr radius. In the following, we used our experimental value $D_e \approx 300$ meV for the free-exciton binding energy in γ-Ge<sub>3</sub>N<sub>4</sub>. Because the physical parameters in Eq. (5) are not measured for γ-Ge<sub>3</sub>N<sub>4</sub>, we used effective masses of electron from our calculations and estimated the exciton Bohr radius $a=0.29$ nm. Due to the absence of any experimental or theoretical data on the charge carrier mobility in γ-Ge<sub>3</sub>N<sub>4</sub>, we used the electron mobility $\mu_e \approx 1$ cm<sup>2</sup>/V/s measured in γ-Sn<sub>3</sub>N<sub>4</sub> [35], which we consider as representative for the spinel nitrides. We also neglected $\mu_h$ in Eq. (5) due to the large effective hole mass. As a result, the frequency factor $\nu = 4.9 \cdot 10^{14}$ s<sup>-1</sup> was obtained from Eq. (5) for γ-Ge<sub>3</sub>N<sub>4</sub> and Eq. (4) led to the exciton dissociation rate $r_d = 3.0 \cdot 10^9$ s<sup>-1</sup> at room temperature ($kT=25$ meV). Taking the radiative free-exciton lifetime $\tau_r \le 10$ ns, as estimated from our time-gated CL experiments (Fig. 4), we obtained for γ-Ge<sub>3</sub>N<sub>4</sub> the exciton emission factor $\eta \ge 3.2\%$ at room temperature. For a comparison, we calculated efficiency $\eta$ for gallium nitride having wurtzite structure, w-GaN, which is the basic material of commercial light emitting diodes (LEDs). This direct band semiconductor emits in the UVA spectral range ($\lambda \approx 360$ nm) and has a comparable band gap energy ($E_g \approx 3.47$ eV at 0 K) but significantly lower exciton binding energy of $D_e = 26$ meV [36]. We used the following experimental parameters measured at 300 K: dielectric constant $\epsilon=5.35$, the effective masses of electrons and holes of $m_e=0.2m_0$ and $m_h=0.8m_0$, respectively [37]. Applying the exciton Bohr radius $a=3.29$ nm [38] and the mobilities $\mu_e=1000$ cm<sup>2</sup>/V/s and $\mu_h \le 200$ cm<sup>2</sup>/V/s [37], we obtained for w-GaN, at room temperature, $\nu = 2.2 \cdot 10^{14}$ s<sup>-1</sup> and the exciton dissociation rate of $r_d = 7.6 \cdot 10^{13}$ s<sup>-1</sup>. Accordingly, the exciton emission factor of w-GaN at room temperature was found to be $\eta \approx 0.004\%$ if the free-exciton radiative lifetime $\tau_r = 295$ ps is adopted [39]. Summing up, significantly higher conversion efficiency of the excitation to light could be expected in LEDs based on γ-Ge<sub>3</sub>N<sub>4</sub>.

The above estimations have shown that high exciton-emission efficiency can be expected in γ-Ge<sub>3</sub>N<sub>4</sub>, which contains sufficiently low defect concentration. The low luminescence quantum efficiency of ~0.4% measured in our polycrystalline sample is characteristic of a strongly non-perfect solid, which is not surprizing on this stage of the research. Accordingly, further efforts are needed to improve the synthesis procedure in order to meet requirements for an efficient photonic emitter based on γ-Ge<sub>3</sub>N<sub>4</sub>. Furthermore, the earlier and recent reports on a reproduci-
ble deposition of γ-Sn₃N₄ [35, 40-41] show a promising method for large-scale synthesis of the spinel nitrides including thin films of γ-Ge₃N₄.

## Conclusions

In this work, spectroscopic examination of the electronic interband transitions in γ-Ge₃N₄, synthesized at high pressures and temperatures in a multianvil press, was performed using cathodoluminescence and synchrotron based photoluminescence spectroscopy. The observed luminescence was assigned to (i) coexisting large-radius (δE≤25 meV, E⁻=3.33 eV) and small-radius bound (hνmax=3.0 eV) excitons trapped near neutral cation defect V<sub>Ge</sub><sup>x</sup> and (ii) anion radical N⁺ (hν=2.15 eV). The time-gated CL measurements conducted at cryogenic and room temperatures allowed first experimental estimation of the free exciton binding energy: D<sub>e</sub>≈0.30 eV, which suggests exciton stability at room and elevated temperatures. The measurements provided an evidence of the direct allowed interband transitions in γ-Ge₃N₄, which bandgap energy E<sub>g</sub>=3.65±0.05 eV was measured with a higher precision compared to that previously obtained in XES/XANES experiments. The screened hybrid functional Heyd-Scuseria-Ernzerhof (HSE06) calculations with optimized screening parameters supported the experimental conclusions. Based on these results, limiting efficiency of the exciton energy conversion to luminescence in defect-free γ-Ge₃N₄ crystal was estimated and compared with that of w-GaN. The significantly higher conversion efficiency together with advanced mechanical properties reveal the potential of γ-Ge₃N₄ for development of robust photonic emitters, suitable for use in extreme environments.

## Acknowledgements

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Support from Estonian Research Council grant PUT PRG 619 is gratefully acknowledged. The multi-anvil experiments at LMV were supported by the French Government Laboratory of Excellence initiative n°ANR-10-LABX-0006, the Région Auvergne and the European Regional Development Fund (ClerVolc contribution number 478).

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