*3.2. Vibrational Properties*

The Raman spectra of the AlScN/Al2O<sup>3</sup> system, shown in Figure 3, are complex because of the multiple spectral bands of different origins. In order to properly assign them, the spectra have to be approached by inspecting the film and the substrate separately. The spectrum of the aluminium oxide substrate indicates the single-crystalline c-plane oriented corundum by the numerous sharp peaks. The most pronounced ones are at about 416.7, 576.3, and 748.8 cm−<sup>1</sup> , corresponding to the A1*<sup>g</sup>* and 2 E*<sup>g</sup>* modes, respectively [23]. These spectral bands are present in all Raman spectra shown, indicating the penetration depth of the laserline, which spanned through the entire depth of a nitride film. The spectrum of pure AlN exhibits the first-order Raman-active modes, such as E2(low), E2(high), and quasi A1(LO) (labelled as "QLO") [33]; and the second-order modes—optic overtone [K3, M] and

A1-symmetry optic combination and overtone [34]. The spectral band at about 1189.4 cm−<sup>1</sup> is assigned to the overtone of the A1(TO) mode. The peak parameters, such as the spectral position and linewidth, are given in Table 1, to which we will refer from here on. According to the symmetry selection rules for the wurtzite-type crystals [35], the present combination of the visible spectral bands confirms the c-axis texture of the nitride films.

**Figure 3.** Raman spectra of AlScN films with various Sc concentrations. The spectra are stacked for clarity. The spectral bands of pure AlN are labelled by the dash lines to emphasise their low-frequency shifts in Al1−*x*Sc*x*N as a guide to the eye. The Raman spectrum of the Al2O<sup>3</sup> substrate revealed the bands unrelated to the nitride films.

The spectral position of the E2(high) mode in the spectrum of pure AlN (cf. Table 1) suggests a significant level of the tensile residual stress, which can be estimated to reach up to ca. 2 GPa, according to the Raman biaxial stress coefficient of <sup>−</sup>3.8 cm−<sup>1</sup> ·GPa−<sup>1</sup> and stress-free phonon frequency of 656.7 cm−<sup>1</sup> [36]. Such a high value of the residual stress, beyond the yield strength of AlN (ca. 0.3 GPa), however, suggests that the biaxial stress is not the only contribution to the peak position. Moreover, wafer bow measurements revealed the residual stress of ca. 1.1 GPa [37]. As we show below, another important factor is the hydrostatic stress caused by diverse point defects. This is corroborated by the large linewidth values of all AlN bands, being inversely proportional to the phonon lifetime limited by phonon scattering. The phonons can be scattered by other quasiparticles, such as phonons and electrons, or the lattice irregularities. Considering the columnar microsctructures of the films, we expect significant contributions to the phonon scattering in AlN by the grain boundaries and overall point defects.

**Table 1.** Peak position and linewdith (FWHM) of the Raman-active bands collected from the AlScN films with the various Sc concentrations.


The Raman spectra of the AlScN films evolve with the Sc concentration (Figure 3) in perfect agreement with the previous reports [12,15]. Namely, the peaks, which correspond to the E2(high) and QLO bands, shift towards lower frequencies, making the trend inversely proportional to the Sc concentration. We also observed a drastic enhancement of the bands above 900 cm−<sup>1</sup> , corresponding to the two-phonon modes, which may be related to the resonance enhancement in AlScN, the dielectric function of which drastically changes upon alloying [9]. Moreover, the expected high defect density promotes the midgap electronic states. The defect-state electron transitions in the visible light range can thus facilitate the resonance enhancement of the Raman scattering effect [38]. Apart from the three modes observed in the spectrum of the AlN film, two more bands at around 963.5 and 1071.8 cm−<sup>1</sup> are evident for AlScN. Their assignment to the combination of the acoustic and optic modes, for example, TA + A1(TO) or E2(high), requires further investigations. The assignment of the two-phonon modes to the "interference fringes" [15] can be ruled out by the fact that the peak positions (cf. Table 1) are also redshifted, though not as drastically as in the case of the first-order modes. Another observation concerns the overall amplification of the spectral background, which may be attributed to the relaxation of the Raman selection rules as a result of the lattice disorder, enabling the visibility of the phonon density of states (pDOS). This is the reason for the absence of the E2(low) mode in the spectra of AlScN. The relaxation of the momentum (*q*) conservation, observed as the enhancement of the background signal and typical broadening of the spectral features, is well known from the studies of other III-V solid solutions [33,39–42]. The origin of the relaxation is the substitutional disorder, which leads to the activation of the non-Γ-point (*q* 6= 0) phonon modes [41], which may either be seen as separate peaks or as peak tails contributing to the asymmetry of the pristine first-order modes [40]. Their actual spectral appearance is governed by the curvature of the phonon dispersion relations in the Brilloun zone. Given that AlScN pseudobinary alloys represent the amalgamation type of optical phonon behaviour [42], and since rock-salt ScN exhibits no first-order modes, no localisation of disorder structure is expected [41]. This implies the typical effects of the compositional fluctuation [43], and thus the asymmetric peak broadening and the non-linear linewidth variation with the Sc concentration, which is obvious from the retrieved peak characteristics (cf. Table 1). Due to the absent phonon dispersions for AlScN, we can approximate the behaviour of spectral modes using the ones of AlN [33,34], focusing on the two main first-order modes, E2(high) and QLO, visible in all spectra (Figure 3). One can clearly see that the dispersions of these two modes are dramatically different: while the E2(high) mode almost does not disperse from the zone-centre towards the edges, the QLO one spreads in the range of 200 cm−<sup>1</sup> in Γ-H direction. This difference explains the stark asymmetry of the QLO mode in the spectra of AlScN, while E2(high) remains symmetric even at the highest Sc concentration. The further theoretical studies would be necessary to verify this likely interpretation. We can hence conclude that the disorder-activated spectral features can be understood in the framework of typical compositional disorder, which originates from alloying, and the high-degree long-range ordering in the films is validated by the corresponding XRD patterns.

Thus, the evolution of the Raman bands due to Sc alloying can be traced only for two one-phonon bands and two-phonon bands (observed in AlN). Using the knowledge of the residual film stress in the AlScN films [37], we can estimate the shift of E2(high) mode as a function of Sc concentration, *x*, assuming that the Raman biaxial stress coefficient is invariable:

$$
\Delta \omega = -126.14 \cdot x
$$

The slope factor of −126.14 is obtained via a linear fit of the peak position values (cf. Table 1) with a subtracted contribution from the residual film stress. Comparing the slope to the one reported by Deng et al. [12], our slope factor is almost two times lower, which stems from the fact the Raman peak is visible at much higher Sc concentrations, confirming the high quality of these epitaxial films. Additionally, we surmise that the biaxial stress coefficient of the E2(high) mode in AlScN films may drastically differ from the one of AlN, which further

increases the uncertainty for the application of Raman spectra for the Sc concentration determination.

Apart from the one-phonon Raman bands, the Raman spectra of AlScN films also feature bands in the low-frequency region (Figure 4a). The suppression of the intense Rayleigh peak via the notch filter also enables us to observe the anti-Stokes side of the spectra. In the spectrum of AlN, we can clearly observe the E2(low) mode at 249.5 cm−<sup>1</sup> (on the Stokes side) and its anti-Stokes counterpart at <sup>−</sup>249.5 cm−<sup>1</sup> . The intensity of the Stokes-side Raman band is substantially higher than the one of the anti-Stokes counterpart, in accordance with the Bose–Einstein distribution of phonons [44]. The alloying of AlN with Sc leads to the emergence of the intense and broad band between 100 and 250 cm−<sup>1</sup> in the spectra of AlScN films. Moreover, its spectral position is seemingly proportional to the Sc content (Figure 4b). The redshifted Stokes and anti-Stokes maxima and the peak intensities differ negligibly. Although their relation to Sc alloying is clear, the underlying mechanism of their Raman activity can only be controversially discussed. As discussed earlier, the alloying practically increases the overall defect density, which might lead to amorphisation in its extreme case. The latter is known to relax the momentum conservation law of the photon–phonon scattering process, allowing the detection of the phonon density of state in Raman spectra [45,46]. According to the phonon structure calculated for pure AlN [34] and ScN [47] lattices, no considerable density is expected to be observed in the spectral range below 200 cm−<sup>1</sup> . This fact rules out the assignment of the bands to the pDOS and a possible resonant enhancement of the acoustic phonons by the bandgap narrowing or alloy-induced absorption in AlScN. Surmising no relation to a particular crystalline phase, their origin may have a purely geometry nature. This is corroborated by the similar peak intensity, which contradicts the ratio of the intensity values of the Stokes and anti-Stokes bands dictated by the Bose–Einstein statistics. The similar low-frequency bands were observed in the Raman spectra of various semiconductor nanoparticles [48,49] emerging due to the confinement of the spheroidal acoustic phonons [50,51]. The obvious linear dependence of the peak position on the Sc content (Figure 4b) resembles the *ωmax* ∝ *d* −1 relation, where *d* is a particle diameter [48] suggesting that alloying AlN with more Sc leads to an increase in the particles' average size. The existence of AlN nanoparticles can also be ruled out by the comparably large crystallites detected via the XRD study. Thus, the actual origin of the nanoparticles can only be assumed to stem from the rock-salt ScN phase. In addition, it is more energetically favourable for rock-salt phases to form the spherical nanoparticles due to the isotropy of their crystal structure.

While the fundamental softening mechanism of the one-phonon bands due to Sc alloying is understood [5,12], the means of their broadening are not sufficiently explained in the discussion of the phonon lifetime reduction [17]. From Table 1, it is clear that broadening of the one-phonon bands depends on the Sc concentration more then that of the two-phonon bands, which may be expected taking into account their probabilistic origin. We thus focus on the linewdiths, Γ, of the E2(high) and QLO modes, for which the correlation length (i.e., mean free path) can be estimated as follows

$$l = \frac{s}{\sqrt{\Gamma \omega\_0}}$$

where *s* is the dispersion parameter, having the same order as the acoustic velocity and *ω*<sup>0</sup> is the mode position [52]. The longitudinal acoustic velocity was estimated via the relation between the elastic constant, *c*33, and the film density, *ρ* [53], using the data obtained via the DFT simulation of AlScN pseudobinary alloys [54]:

$$V\_L = \sqrt{\frac{c\_{33}}{\rho}}$$

The theoretically obtained values of the film density agree well with the experimental ones, corroborating their use in our analysis [55]. The derived values of the acoustic velocity and

the phonon correlation length can eventually be used to assess the point defect density, *l* −3 , provided in Table 2.

**Figure 4.** (**a**) Raman spectra of AlScN films at the low frequency spectral region. The spectra are stacked for clarity. The spectral bands are labelled by the positions of their maxima. The asterisk marks the band related to the Al2O<sup>3</sup> substrate. (**b**) The positions of the low-frequency bands plotted as a function of the Sc concentration. The dashed lines are the linear fits.

**Table 2.** Material properties of the AlScN pseudobinary alloys, such as the elastic constant, *c*33, and the mass density, *ρ* [54] necessary to estimate the longitudinal sound velocity, *V<sup>l</sup>* , and the point defect density, (*l*) −3 . The mode position and linewidth are taken from Table 1.


The defect density value is two orders of magnitude larger in the case of AlScN with *x* = 0.42 of Sc when compared to pure AlN, which underlines the contribution of the point defects in the phonon scattering. This means that every 100th lattice position can present a certain defect, which, however, is still way too low to account for all substituting Sc atoms, comprising almost one half of Al atoms in the lattice of the sample with the highest Sc concentration. This example shows a very high sensitivity of the Raman scattering process, whereby even a minute presence of defects is reflected by the width of Raman peaks. Despite our assumption of the point defects due to alloying [56], another contribution to the defect density can be considered originating from the grain boundaries. The contributions from the phonon–phonon and phonon–electron scattering are discussed in the next section. To estimate the share of the scattering on grain boundaries, it is compelling to compare the phonon coherence length values to the average grain size in the AlScN films. The average size of the crystallites (i.e., coherently diffracting crystalline domains) can be estimated using XRD symmetric scans (Figure 1) via the well-known Scherrer Equation [57] assuming the shape factor of 0.9. We employed the AlScN (0002) reflection accessing the crystallite size in the out-of-plane direction, i.e., perpendicular to the film surface. Provided the anisotropy of the film growth due to the columnar structure of the nitride films, the inplane grain size can be retrieved from the AFM topography images via the grain analysis algorithms. The average grain size values are plotted together with the phonon coherence lengths as a function of the Sc concentration (Figure 5).

**Figure 5.** Average grain size and phonon correlation length as a function of the Sc content in the AlScN films. The continuous lines are given as guides to the eye.

The out-of-plane grain size of AlN grains was determined to be twice as large compared to the in-plane size, which agrees with the c-axis texture and the columnar growth of the nitride. The difference in the grain size values becomes smaller with the higher Sc content, reaching the equal magnitudes for the samples with *x* > 0.3 due to the larger grains seen via AFM (Figure 2) and the reduction of the grains in the orthogonal direction. The decrease in the out-of-plane grain dimension was observed by a similar trend in the phonon correlation length values, determined for the E2(high) and QLO modes separately. Despite the different polarisation of the Raman modes, the phonon correlation length of the double-degenerate E2(high) mode follows the one of the out-of-plane QLO modes, suggesting the phonons also travel less along the basal plane of the AlScN lattice. Even though this interpretation contradicts the positive trend of the in-plane size growth shown in Figure 5, the discrepancy may arise due to the worm-like surface morphology facilitated by clustering of individual grains. Thus, the similar trends in the correlation length values estimated using both Raman modes indicated the isotropic distribution of the scattering centres, suggesting point defects to be responsible for the low correlation length values. The phonon correlation length was estimated to be one order of magnitude lower than the average grain size in the out-of-plane direction overall, indicating that the lattice irregularities are ubiquitous and they become even more pronounced with the addition of Sc. We can hence speculate that the grain boundaries play a lesser role in the scattering of the optical phonons. Provided that the point defects also contribute to the hydrostatic stress in the lattice, high point defect density influences the spectral position of the mode itself, which explains the "impossibly high" residual biaxial stress in the AlN sample.

### *3.3. Temperature-Dependent Raman Measurements*

Figure 6 shows the position and the linewidth of the E2(high) mode obtained from the Raman spectra recorded at the elevated temperatures. The softening of the mode with the film heating was observed for all the AlScN films with different Sc contents (Figure 6a), which indicates the thermal expansion of the lattice due to the increase in the anharmonic phonon–phonon interactions, as expected. The phonon softening in this narrow temperature range is linear [58], conveniently enabling the temperature monitoring of the thin films. The linear approximations of the temperature coefficient [58] for the E2(high) mode, shown in Figure 6a, varied with the Sc content. Compared to the one in AlN (0.0222 K−<sup>1</sup> ·cm−<sup>1</sup> [59]), the increase in the coefficient values can be seen reaching up to 0.1157 K−<sup>1</sup> ·cm−<sup>1</sup> for the highest Sc content. The five-fold difference suggests the dramatic changes in the anharmonic potential in AlScN corresponding to the stronger interatomic interaction [60], which contradicts the widely observed softening of the onephonon bands as a result of bond length increase [12,15]. Additionally, the softening of the material should be accompanied by the lowering of the coefficient of the thermal expansion (CTE), but the experimental results showed the opposite behaviour [20]. The mechanism behind the ascending temperature coefficient should be related to the increase in the bound charge carrier density, realised through the alloying with Sc and the defect formation. It is instructive to see that the film with the least Sc added resulted in a temperature coefficient value (0.0143 K−<sup>1</sup> ·cm−<sup>1</sup> ) lower than the one in AlN. The similarly low temperature coefficient values were found for CVD-grown AlScN with 20% of Sc [19]. Thus, we can conclude that the temperature coefficient is not only a function of the Sc concentration, but it also depends on the defect density, which may significantly alter the temperature monitoring.

**Figure 6.** (**a**) E2(high) mode position and (**b**) FWHM of AlScN films with various Sc concentration plotted as a function of temperature. The values given in (**a**) are the Raman temperature coefficients determined from the linear fit of the data.

In contrast to the clear trends in the peak position, the peak linewidth (FWHM), also known as the damping constant, weakly increases with the temperature increase (Figure 6b). Moreover, it may seem inconsistent between the AlScN films with different Sc concentrations. Despite the large peak widths, the determination error increases proportionally to the alloying degree owing to the possible contributions from the enhanced signals of the pDOS. Assuming a considerable uncertainty in the FWHM values, we conclude that the damping constant values failed to show any temperature dependence, and the fluctuation of the values was caused by a fitting uncertainty exclusively. This interpretation leads to the conclusion that the low phonon lifetimes as a result of the considerable point defect density cloak the anharmonic phonon–phonon scattering contribution usually responsible for the temperature-related peak broadening [61].
