Local Electronic Structure in AlN Studied by Single-Crystal 27Al and 14N NMR and DFT Calculations

Both the chemical shift and quadrupole coupling tensors for 14N and 27Al in the wurtzite structure of aluminum nitride have been determined to high precision by single-crystal NMR spectroscopy. A homoepitaxially grown AlN single crystal with known morphology was used, which allowed for optical alignment of the crystal on the goniometer axis. From the analysis of the rotation patterns of 14N (I=1) and 27Al (I=5/2), the quadrupolar coupling constants were determined to χ(14N)=(8.19±0.02) kHz, and χ(27Al)=(1.914±0.001) MHz. The chemical shift parameters obtained from the data fit were δiso=−(292.6±0.6) ppm and δΔ=−(1.9±1.1) ppm for 14N, and (after correcting for the second-order quadrupolar shift) δiso=(113.6±0.3) ppm and δΔ=(12.7±0.6) ppm for 27Al. DFT calculations of the NMR parameters for non-optimized crystal geometries of AlN generally did not match the experimental values, whereas optimized geometries came close for 27Al with χ¯calc=(1.791±0.003) MHz, but not for 14N with χ¯calc=−(19.5±3.3) kHz.


Introduction
Aluminum nitride, AlN, is industrially used as a substrate for semiconductor devices such as ultraviolet LEDs, and is also the preferred starting material for the synthesis of chemically inert lightweight ceramics with excellent mechanical properties, such as SiAlONs [1,2]. Ceramic materials are often amorphous or consist of crystalline grains which are embedded in a glassy matrix, and hence characterization of such materials as well as detection and identification of impurities is not always straightforward. Nuclear magnetic resonance (NMR) spectroscopy has proven to be a powerful analytical technique to analyze ceramic structures, because of its ability to selectively probe the local surrounding of the observed nuclides [3][4][5]. For characterization of a multi-component system, it is crucial to know the exact NMR-interaction parameters of the detected nuclei in the various components, in order to correctly assign and distinguish the NMR signals arising from them. The 'gold standard' 2. Single-crystal NMR of 14 N and 27 Al 27 The resonance frequencies of the transitions |mi $ |m + 1i of a nuclide with spin I > 1/2 in an 28 external magnetic field may generally be described by: [12][13][14] 29 n m,m+1 (k) =n 0 + n CS + n (1) m,m+1 (k) + n (2) m,m+1 (k 2 ) with k = m + 1 2 Here, n 0 is the Larmor frequency which solely scales with the magnetic field strength, and n CS 30 is the contribution of the chemical shift (CS). The interaction between the non-symmetric charge 31 distribution of the nucleus and its electronic surroundings further shifts the resonance frequencies 32 by n (1) m,m+1 (k) and n (2) m,m+1 (k), corresponding to the quadrupolar interaction to first and second order, 33 respectively. The factor k is ±0.5 for the two transitions of 14 N with I = 1 and ±0, 1, 2 for the 34 five transitions of 27 Al with spin I = 5/2. The nuclear quadrupole interaction and its orientation 35 dependency is gauged by the quadrupole coupling tensor Q , which may generally be described by a 36 second-rank symmetric and traceless tensor, i.e. Q ij = Q ji and Q xx + Q yy + Q zz = 0. It is generally 37 helpful to define three distinct coordinate systems for NMR spectroscopy of single crystals, i.e. the 38 laboratory frame, where the z axis is defined by the orientation of the external magnetic field, the 39 crystal lattice (CRY) frame and the principal axis system (PAS). For 14 N and 27 Al in AlN, the CRY 40 frame and the PAS are identical and the quadrupole coupling tensor Q for both nuclides is solely 41 defined by the quadrupolar coupling constant C q = Q 33 = c: Single crystal of aluminum nitride, AlN, with the synthesis described in Reference [8].
The crystallographic c axis and the ab plane are indicated by arrows. (b) Wurtzite structure of AlN, according to Reference [9], viewed down the crystallographic [11][12][13][14][15][16][17][18][19][20] direction. The aluminum atoms (blue-grey) and the nitrogen atoms (yellow), both located at Wyckoff position 2b, are tetrahedrally coordinated by each other with one Al-N bond directed parallel to the crystallographic c axis. (c) Individual, tetrahedrally coordinated, aluminum and nitrogen atom in the crystal structure of AlN, in which the three equal, shorter, bonds Al/N-I/II/III with 1.8891(8) Å and the longer bond Al/N-IV with 1.9029(16) Å along the three-fold rotation axis are highlighted. a Drawing generated with the VESTA program [10].

Single-Crystal 14 N and 27 Al NMR
In the solid state, the NMR response of spin I = 1/2 is governed by the chemical shift, and by dipolar (direct) couplings between spins [11]. The dipolar couplings between the nuclear spins in the AlN lattice result in homogeneous line broadening and will not be quantitatively evaluated here. Both 14 N and 27 Al have a spin I > 1/2, and therefore, the quadrupolar coupling between the non-symmetric charge distribution of the nucleus and its electronic surroundings also needs to be considered [12]. For a spin I in an external magnetic field, 2I NMR transitions exist, which are classified according to their magnetic quantum number m. With a particular transition |m → |m + 1 designated by the parameter k = m + 1 2 [13], the resonance frequency ν m,m+1 of this transition may be described by the following general notation: For the two transitions of 14 N with I = 1, the values for k are ± 1 2 . For the five transitions of 27 Al with I = 5/2, the values for k are k = 0 for the central transition, and k = ±1, 2 for the satellite transitions. In Equation (1), ν 0 is the Larmor frequency, ν CS the contribution of the chemical shift (CS), and ν (1) m,m+1 (k) and ν (2) m,m+1 (k) are the effects of the quadrupolar interaction described by perturbation theory to first and second order, respectively. Magnitude and orientation dependency of the quadrupole interaction may be gauged by the quadrupole coupling tensor Q. Similar to the electrical field gradient (EFG) tensor V, to which it is related by Q = (eQ/h)V, this second-rank tensor is symmetric and traceless, i.e., Q ij = Q ji and Q xx + Q yy + Q zz = 0. Generally, for NMR spectroscopy of single crystals, it is useful to define three distinct coordinate systems, i.e., the laboratory frame, where the z axis is defined by the orientation of the external magnetic field, the crystal lattice (CRY) frame and the principal axis system (PAS). In the wurtzite structure of AlN, nitrogen and aluminum are both situated on a three-fold rotation axis parallel to the crystallographic c axis, and therefore the CRY and the PAS frames for 14 N and 27 Al are identical. In their PAS frame, symmetric tensors take diagonal form. This has the consequence that the tensors cannot change when the two formula units are generated by the symmetry elements of Wyckoff position 2b. Therefore, the two 14 N and 27 Al atoms in the AlN unit cell are practically pairwise magnetically equivalent, even though they do not fulfil the strict equivalence criterion of being connected by either inversion or translation. The Q tensor for both nuclides is hence uniaxial (with asymmetry η Q = (Q 11 − Q 22 )/Q 33 = 0), and solely defined by the quadrupolar coupling constant χ = C q = Q 33 : This tensor is conveniently determined from the separations ('splittings') of the symmetric doublet k = ±0.5 for 14 N, and of the satellite transitions (ST's) with k = ±1, 2 for 27 Al, since these are not affected by the chemical shift and the second-order quadrupolar interaction. Thus, the difference ∆ν(k) of the resonance frequencies (where we have dropped the m, m + 1 subscripts used in Equation (1) for brevity) is: The contribution of the quadrupolar interaction to first order for η = 0 is given by [12]: Here, the orientation dependence of ν (1) (k) on the relative orientation of the Q tensor to the external magnetic field is expressed by the Euler angle β, with β being the angle between the eigenvector with the largest eigenvalue, i.e., Q 33 = χ, and the magnetic field vector.
The contribution of the chemical shift ν CS to the resonance frequency is gauged by the chemical shift tensor δ. Taking into account the same symmetry arguments as for the Q tensor above, the chemical shift (CS) tensor for 14 N and 27 Al in AlN is given by: The weighted trace of δ determines the isotropic chemical shift δ iso = 1/3(δ 11 + δ 22 + δ 33 ) and, similar to the Q tensor, the asymmetry parameter for the CS tensor is η CS = (δ 22 − δ 11 )/∆δ = 0. Here, we generally order the tensor components according to the convention |δ 33 − δ iso | ≥ |δ 11 − δ iso | ≥ |δ 22 − δ iso |, and make use of the reduced anisotropy ∆δ = δ 33 − δ iso [14].
To determine the CS tensor of quadrupolar nuclei with half-integer spins, such as 27 Al (I = 5/2), it is customary to trace the orientation dependency of the central transition (CT), i.e., the k = 0 transition [15]. In cases where the CT signal cannot be resolved [16], the variation of the center of the satellite transitions (and for spin I = 1, the center of the doublet with k = ±0.5 in all cases) may be traced instead: For 14 N in AlN, the quadrupolar interaction to second order is negligible, and the CS tensor δ may directly be determined from the doublet centers. The CT of 27 Al in AlN is, however, affected by the quadrupolar interaction to second order, and this contribution has to be subtracted from the CT line position before δ can be determined. This second-order contribution can be written as [17]: After subtracting ν (2) from the observed ν, the change of the CT resonance frequency from the Larmor frequency is solely due to the chemical shift. The line position depends on the relative orientation of the magnetic field vector b 0 to the tensor δ CRY in the crystal frame, which may be compactly expressed by the product [18]: The determination of the actual quadrupole coupling tensors Q N , Q Al and the chemical shift tensors δ N , δ Al for 14 N and 27 Al in aluminum nitride, using the above formalism, is described in the following.

27 Al Quadrupole Coupling Tensor
A single crystal of aluminum nitride with approximate dimensions of 5 × 5 × 4 mm was used for the single-crystal NMR experiments. Since the crystal was grown by a homoepitaxial growth process [8], it is possible to assign the crystal faces to crystallographic planes, as indicated in Figure 1a. It was therefore possible to fix the crystal into in a specific orientation by gluing it with its (10-10) face onto the goniometer axis, which itself is perpendicular to the external magnetic field b 0 . The crystal was then rotated until the [000-1] direction was parallel to b 0 . Both orienting procedures involve small misalignments, which can however be quantified by the data analysis, as described below. Representative 27 Al NMR spectra are shown in Figure 2a, with the full rotation pattern over 180 o shown in Figure 2b, which was obtained by rotating the crystal counterclockwise in steps of 15 • using the goniometer gear. The satellite pairs for k = ±2, in the following denoted as ST(5/2), and k = ±1, in the following denoted as ST(3/2), are symmetrically positioned around the central transition. All 27 Al resonance lines are fairly broad, with a full width at half-maximum fwhm ≈ 9 kHz, caused by hetero-and homonuclear dipolar interactions between aluminum and nitrogen atoms in the structure [19].
The experimentally determined satellite splittings of the ST(5/2) and ST(3/2) doublets in kHz are plotted over the rotation angle ϕ in Figure 3a. The rotation patterns in both Figures 2b and 3a are mirrored at a position very close to 90 • , with the mirror defining the rotation angle for which b 0 is situated in the crystallographic ab plane. The deviation ϕ ∆ of the mirror from 90 • quantifies the original misalignment of the [000-1] direction to b 0 . From the way the crystal is glued on the goniometer axis, we know that the rotation axis must be in the crystallographic ab plane. Also, the above considerations of the effects of crystal symmetry on the tensor structure imply that the eigenvector with the largest eigenvalue (Q 33 = χ) must point along the three-fold rotation axis, i.e., along the crystallographic c axis, which we attempted to align along b 0 for the starting point of our rotation pattern. For this situation, the angle β in Equation (4) can be replaced by β → ϕ − ϕ ∆ , and the magnitude of the satellite splittings (Equation (3)) an be expressed by:      To determine the quadrupole coupling tensor Q Al of 27 Al, the satellite splittings were simultaneously fitted according to Equation  [20]. The full Q Al tensor, with the eigenvalues and corresponding eigenvectors in the PAS frame (Equation (2)), is summarized in Table 1. The quadrupolar asymmetry parameter η Q = 0, and the orientation of the eigenvectors are a consequence of the crystal symmetry, with q 33 aligned exactly along the c axis and q 11 , q 22 placed in the ab plane.

27 Al Chemical Shift Tensor
To determine the chemical shift tensor δ Al of 27 Al, the contribution of the second-order quadrupolar interaction must be subtracted from the central transition (k = 0) line position. In Figure 3b, the 27 Al CT is plotted over ϕ, and the data points clearly show the presence of the quadrupolar-induced shift, which according to Equation (7), contains harmonic terms depending on both cos 4 (β) and cos 2 (β). Using the results obtained from evaluating the splittings (χ = 1.914 MHz and ϕ ∆ = 0.65 • ), this second-order quadrupole shift can be calculated for each crystal orientation according to Equation (7) with β = ϕ − ϕ ∆ , see red points in Figure 3b. After subtracting the quadrupole contribution from the experimental points, the remaining variation in CT line position (Figure 3b, purple) is solely caused by the chemical shift tensor, which can be determined from it. Due to the cylindrical symmetry of the tensor and the fact that it does not transform between its PAS and CRY frame (see Equation (5)), the exact orientation of the rotation axis in the crystallographic ab plane of AlN is indeterminate. For simplicity, the rotation axis can be assumed to be parallel to the b axis, and the orientation of the magnetic field vector in the CRY frame for each rotation angle ϕ can be expressed by: Inserting this (and Equation (5)) into Equation (8), we obtain the expression necessary for fitting the data in Figure 3b: For this fit, ϕ ∆ was kept fixed at the value derived from fitting Q Al , and the components of the chemical shift tensor of 27 Al determined thereby are P = (107.2 ± 0.3) ppm and R = (126.3 ± 0.3) ppm, with the full tensor listed in Table 1. The isotropic chemical shift of δ iso = (113.6 ± 0.3) ppm is in good agreement with a previously reported value [4], which was determined from a polycrystalline sample of AlN under magic-angle spinning (MAS), and after correcting for the second-order quadrupole shift (from the reported line position of 113.3 ppm at a 600 MHz spectrometer [4], the correction of ν (2) ai = −(3/500)(χ 2 /ν 0 ) ≈ −0.9 ppm needs to be subtracted), comes out to δ iso = 114.2 ppm. The chemical shift asymmetry parameter η CS = 0 and the orientation of the chemical shift eigenvectors follow the same symmetry restrictions as for the quadrupole coupling tensor described above.

14 N Quadrupole Coupling Tensor
For the determination of the quadrupole coupling tensor Q N and the chemical shift tensor δ N of 14 N in aluminum nitride, the same AlN crystal (Figure 1a) and goniometer axis as for 27 Al was used. Since a change of the solenoid coil was necessary to go from the resonance frequency of 27 Al to 14 N, the offset angle ϕ ∆ is slightly different and needs to be determined from the data fit again. Representative 14 N NMR spectra are depicted in Figure 4a, and at first glance, appear to show much broader lines than the 27 Al spectra. In fact, with fwhm ≈ 3 kHz, the resonance lines are only about one third as broad as those of 27 Al, since the gyromagnetic ratio of 14 N is 3.5 times smaller than that of aluminum, which scales down the homonuclear contribution of the dipolar coupling. The impression of broad lines for 14 N is chiefly because the shifts of its k = ±0.5 resonances caused by the quadrupolar interaction (≈ 300 ppm) are much smaller than those of 27 Al (≈ 8000 ppm), since these shifts scale with the quadrupolar moment of the nucleus, which is 20.44 mb for 14 N, but 146.6 mb for 27 Al [20]. The broad resonance lines of the 14 N spectra, combined with the relatively poor signal-to-noise ratio (due to the long relaxation time of T 1 = 1080 s [22]) make it difficult to precisely derive the line positions from the spectra. Therefore, all 14 N NMR spectra were deconvoluted, assuming combined Lorentz-Gauss functions (so-called Voigt profiles), to reliably obtain the line positions.
The splittings of the thus deconvoluted 14 N doublets are plotted over the rotation angle ϕ in Figure 5a. The quadrupole coupling tensor was determined by a fit of these splittings according to Equation (9) with ∆k = 1, giving the quadrupolar coupling constant χ = (8.19 ± 0.02) kHz and an offset angle of ϕ ∆ = −(0.74 ± 0.13) • . The full quadrupole coupling tensor, with the eigenvalues and corresponding eigenvectors in the PAS frame (Equation (2)), is summarized in Table 2. The quadrupolar asymmetry parameter η Q = 0, and the orientation of the eigenvectors are identical to the Q tensor of 27 Al. So far, only an upper limit of the quadrupolar coupling constant of 14 N in AlN was available in the literature, namely χ < 10 kHz determined from a polycrystalline powder sample [20].   To determine the chemical shift tensor for 14 N, the center of the two resonances, k = ±0.5, is 158 plotted over the rotation angle j In Figure 5b. The data exhibit quite some scatter, however, it has to 159 be kept in mind that for tracing the anisotropy of the 14 N chemical shift in aluminum nitride, we are 160 attempting to extract variations of the order of~90 Hz from resonance lines with fwhm ⇡ 3 kHz. As 161 stated above, the quadrupole interaction to second order for 14 N in AlN is neglectable small, other 162 than for 27 Al, since the nuclear quadrupole moment is almost ten times smaller for 14 N (20.44 mb) than 163 it is for 27 Al (146.6 mb). [28] 164 The full chemical shift tensor was thus directly determined from the data in Figure 5b

14 N Chemical Shift Tensor
The chemical shift tensor of 14 N can be calculated from the evolution of the center of the doublet with k = ±0.5 over the rotation angle, as plotted in Figure 5b. Fitting the data in Figure 5b according to Equation (11), with the offset angle kept fixed at the value derived from the quadrupole coupling tensor fit (ϕ ∆ = −0.74 ppm), gives P = −(291.6 ± 0.7) ppm and R = −(294.5 ± 0.6) ppm, with the full tensor listed in Table 2. The data in Figure 5b exhibit quite some scatter; however, it has to be kept in mind that for tracing the anisotropy of the 14 N chemical shift in aluminum nitride, we are attempting to extract variations of the order of ≈ 90 Hz from resonance lines with fwhm ≈ 3 kHz. Despite the scatter, about two thirds of all data points belong to the CS tensor fit function within the error margins of ±1.2 ppm. The resulting isotropic chemical shift δ iso = −(292.6 ± 0.6) ppm is in good agreement with the previously reported value of δ iso = 64.7 ppm [4], determined from a polycrystalline powder sample under MAS and referenced to an aqueous (NH 4 ) 2 SO 4 solution, with a 'NH + 4 ' solution resonance shifted −355 ppm relative to the 'NO − 3 ' solution used here [23]. Similar to the quadrupole coupling tensor, the asymmetry of the CS tensor with η CS = 0, as well as the eigenvector orientation follow the symmetry restrictions of the crystal lattice. Table 2. Quadrupole coupling tensor Q N (left), and chemical shift tensor δ N (right) of 14 N in the wurtzite structure of AlN, as determined from single-crystal NMR experiments. The orientation of the corresponding eigenvectors are listed in spherical coordinates (θ, ϕ) in the hexagonal abc crystal frame CRY. The errors of the experimental values reflect those delivered by the fitting routine.

14 N and 27 Al DFT Calculations
It has become customary within the solid-state NMR community to augment experimental results by comparing them to predictions derived from calculations using density functional theory (DFT) methods employing periodic plane waves [24]. To check how the quadrupolar coupling constants for 27 Al and 14 N derived from our precise single-crystal results compare to DFT predictions, we have performed such calculations for aluminum nitride, using the CASTEP code, see Section 4.3 for computational details. Table 3 shows the quadrupolar coupling constants χ calc determined by DFT calculations using the coordinates from X-ray diffraction data reported in the inorganic crystal structure database (ICSD) for a selection of different database entries. The variation of these entries concerns mostly the unit cell dimensions (see also below about geometry optimization), which is reflected in the varying unit cell volumes V cell listed in the table. On the left of Table 3, the calculation results are given from directly using the ICSD coordinates, the so-called single-point energy (SPE). We note that for this calculation mode, the DFT algorithm returns χ calc values within a wide scatter, mirrored by standard deviations of 37% for 27 Al and 73% for 14 N. Whereas a single structure might accidentally give numbers for χ calc that are practically identical to the experiment, as structure ICSD 34475 does here for AlN, a more systematic exploration would demand to take the arithmetic mean of the eight different structures. These mean values, χ calc ( 27 Al  It is however well documented in the literature that in order to obtain good agreement between DFT and experimental results, a geometry optimization (GO) of the crystal structure is usually necessary [31][32][33]. This was also done for AlN, taking the coordinates of the previously used ICSD database entries as a starting point. It should be noted that for AlN, only the unit cell parameters a, b, c may be geometry optimized, since both aluminum and nitrogen atoms are situated on a crystallographic special position, Wyckoff position 2b. As may be seen from the entries on the right in Table 3, the χ calc values are practically independent from the starting point after energy optimization, with a mean of χ calc ( 27 Al) = 1.7913 MHz and χ calc ( 14 N) = −19.5 kHz. This leads to small standard deviations (0.1% for 27 Al and 17% for 14 N), which seem to imply a high accuracy of the DFT results. However, the small standard deviations of the GO calculations reflect only on a high precision of the computational algorithm. The accuracy of calculation results is defined by comparison to the experiment [34], and is therefore quite low, since both experimental values (especially that of 14 N) are outside the standard deviation of the high-precision χ calc values.

Aluminum Nitride
The single crystal of aluminum nitride shown in Figure 1a was grown at IKZ, using physical vapor transport of bulk AlN in a TaC crucible with radio frequency induction heating. Further details may be found in Reference [8].

Solid-State NMR Spectroscopy
Single-crystal NMR spectra were acquired on a BRUKER Avance-III 400 spectrometer at MPI-FKF Stuttgart, at a Larmor frequency of ν 0 ( 27 Al) = 104.263 MHz, and ν 0 ( 14 N) = 28.905 MHz, using a goniometer probe with a 6 mm solenoid coil, built by NMR Service GmbH (Erfurt, Germany). The 27 Al spectra were recorded with single-pulse acquisition, four scans and a relaxation delay of 20 s. For the 14 N spectra a spin-echo sequence [35] was employed to minimize baseline roll and the spectra were recorded with 16 scans and a relaxation delay of 300 s. All spectra were referenced to a dilute Al(NO 3 ) 3 solution at 0 ppm. The fit of the rotation pattern and deconvolution of the 14 N spectra were performed with the program IGOR PRO 7 from WaveMetrics Inc., which delivers excellent non-linear fitting performance.

DFT Calculations
All calculations were run with the CASTEP density functional theory (DFT) code [36] integrated within the BIOVIA Materials Studio 2017 suite, using the GIPAW algorithm [37]. The computations use the generalized gradient approximation (GCA) and Perdew-Burke-Ernzerhof (PBE) functional [38], with the core-valence interactions described by ultra-soft pseudopotentials [37]. Integrations over the Brillouin zone were done using a Monkhorst-Pack grid [39] of 16 × 16 × 8, with a reciprocal spacing of at least 0.025 Å −1 . The convergence of the calculated NMR parameters was tested for both the size of a Monkhorst-Pack k-grid and a basis set cut-off energy, with the cut-off energy being 1500 eV. Also, the possible contribution of pairwise dispersion interactions was checked by using the Tkatchenko-Scheffler method [40] as implemented in CASTEP, but no improvements were observed. The calculation results reported here therefore do not include dispersion interaction.
Geometry optimization (GO) calculations were performed using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [41], with the same functional, k-grid spacings and cut-off energies as in the single-point energy (SPE) calculations. Convergence tolerance parameters for geometry optimization were as follows: maximum energy 2.0 × 10 −5 eV/atom, maximum force 0.001 eV/Å, maximum stress 0.01 GPa/atom, and maximum displacement in a step 0.002 Å. Crystallographic data used in the calculations were taken from literature listed in Table 3.

Conclusions
In this work, both the chemical shift and quadrupole coupling tensors for 27 Al and 14 N in aluminum nitride have been determined to high precision by single-crystal NMR spectroscopy. To this end, a homoepitaxially grown AlN single crystal with known morphology was used, which allowed the rotation axis to be determined by optical alignment. Because of the high symmetry of wurtzite-type AlN, one full rotation pattern was sufficient to determine the NMR-interaction tensors in the crystal frame. The three-fold rotation axis on which both atom types are located enforces colinearity of the tensor eigenvectors with the crystallographic coordinate system, which simplifies data analysis. A simultaneous fit for the ST(3/2) and ST(5/2) splittings of 27 Al gave the quadrupolar coupling constant χ( 27 Al) = (1.914 ± 0.001) MHz, and fitting the 14 N doublet splitting resulted in χ( 14 N) = (8.19 ± 0.02) kHz. To extract the chemical shift tensor for 27 Al, the evolution of the central transition over the crystal rotation was tracked, and the contribution of the second-order quadrupolar shift was subtracted according to the previously determined quadrupolar coupling tensor. A fit over the thus corrected central transition positions resulted in an isotropic chemical shift of δ iso = (113.6 ± 0.3) ppm and an reduced anisotropy of δ ∆ = (12.7 ± 0.6) ppm. Due to the small quadrupolar moment of 14 N, its second-order quadrupolar shift in AlN is negligible, and the chemical shift tensor was directly fitted from the evolution of the 14 N doublet centers over the rotation angle. The resulting isotropic chemical shift is δ iso = −(292.6 ± 0.6) ppm and the reduced anisotropy is δ ∆ = −(1.9 ± 1.1) ppm.
For comparison, the quadrupolar coupling parameters of 14 N and 27 Al were also calculated using the CASTEP DFT code for a variety of previously reported X-ray structures. For both calculation strategies, i.e., single-point energy (SPE, where the coordinates are directly taken from XRD), and structures which were geometry optimized (GO) by the DFT code, agreement with the experimental values was relatively poor, leaving room for further improvement of these computational methods.