Introduction

III-Nitrides are important direct bandgap semiconductors commonly used in power electronic and optoelectronic devices1,2,3,4,5,6,7,8,9,10,11. As the power density of modern III-Nitride based devices continues to increase, efficient thermal transfer has become critical to ensuring device reliability, prevention of overheating, and enhancement of device performance, especially as device sizes approach the nanoscale. In general, thermal transport in III-Nitride semiconductors is dominated by phonons, for which scattering resistance due to nano- and micro-scale structural defects can be significant. Since III-Nitride semiconductors are predominantly grown on substrates such as sapphire or silicon, a substantial lattice mismatch at the interface results in a high density of defects, which degrades the thermal conductivity and device performance12,13,14,15,16,17,18,19. Therefore, rigorous understanding of phonon behaviors and transport in the presence of defects is critical for designing enhanced high-power-density III-Nitride based devices.

However, correlating phonon transport with defects is challenging, particularly for obtaining mode-resolved phonon information, for two primary reasons. First, most extended defects, such as grain boundaries and dislocations, result in lattice strain and broken periodicity, which make correlation of specific defect atom vibrations with particular phonon modes difficult. With respect to simulation, poor periodicity also leads to oversized computational models and thus exaggerated numerical challenges. Second, localized defect vibrations can only be detected within a few nanometers of the defects, beyond the resolution of traditional measurement methods such as Raman scattering spectroscopy whose spatial resolution is constrained by laser wavelength.

To visualize defect-induced localized vibrations, here we focus on prismatic stacking fault (PSF) defects in GaN. PSF is a kind of stacking fault in (\(11\bar{2}0\)) crystallographic planes with a total displacement of 1/2\([10\bar{1}1]\), as identified by Drum et al.20 in AlN crystals, showing as 4 and 8 atom rings21,22. It has twofold rotational symmetry along the [0001] direction and translational symmetry along [\(10\bar{1}0\)] and [0001] directions, which exhibits better periodicity than other planar defects. The central symmetry of PSF reduces delocalized dipole signals which typically ranges from 20 to 200 Å23,24. It is known that dipole scattering is linked to the long-range Coulomb fields, which results from asymmetric stretching or deformation modes involving adjacent cations and anions25,26. Therefore, the reduction of delocalized dipole signals weakens the long-range Coulomb fields and thus make it possible to achieve atomic resolution. In addition, PSFs are actually often detected in polar and non-polar GaN films27,28 or GaN crystals grown on needle-shaped GaN seeds obtained under high nitrogen pressure29. They commonly occur at the junctions between two basal stacking faults (BSFs), forming closed domains or complex folded arrangements30,31,32,33, indicating that BSFs can be terminated by PSFs27,34. Therefore, it is important to study the vibrations of PSF since it is correlated with other planar defects as well.

In this study, we characterized vibrational modes of PSF in GaN using scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS). STEM and EELS, equipped with a monochromator and spherical aberration correction, provide high spatial and energy resolution, enabling nanoscale measurement of phonon properties of different materials35,36,37,38,39,40,41,42,43,44,45,46 and localized phonon modes of defects47,48,49,50,51,52,53 and interfaces51,52,53,54,55,56,57,58,59. Using these state-of-the-art tools, we are able to directly observe phonon energies and dispersions at defects and experimentally elucidate the influence of defects on phonon propagation. In both measurements and calculations, we identified three types of phonon modes around the PSFs: a localized defect mode (LDM), a confined bulk mode (CBM) and a fully extended mode (FEM). It is found that LDMs are attributed to vibration of atoms located at the center of the PSF. CBMs exist at atoms on both sides of the defect, but they are free of direct vibrational contributions from defect atoms, while FEMs are delocalized on all atoms over the entire system. In addition, phonon dispersion of bulk GaN was directly measured and compared with that of the PSF defect, demonstrating that the energy gap between acoustic (and folded acoustic-like modes) and high-frequency optic phonons in the PSF is smaller than that in defect-free GaN. This suggests that scattering of acoustic phonons by optic modes is more pronounced near the PSF than in bulk GaN60,61,62, which will lead to increased thermal resistance in the defect zone. Furthermore, the acoustic sound speeds extracted from the local PSF phonon dispersion is reduced compared with the bulk, which may lead to slower phonons and further enhanced thermal resistance in the defect zone63,64. Our study builds a bridge between defect-phonon interactions and thermal conductivity in materials with high defect densities through direct observation of defect phonon modes.

Experimental setup

Figure 1a shows a schematic of the STEM-EELS used in our experiments. An electron beam is focused on the sample and an EELS aperture on the diffraction plane selectively collects scattered electrons with desired momentum transfers. The collected electrons are deflected by a magnetic prism and the vibrational signals of the sample are collected based on the loss of the electron beam kinetic energy. EELS is performed using a monochromated, aberration-corrected Nion U-HERMES200 STEM operating at an accelerating voltage of 60 kV. To maximize spatial resolution, a large beam convergence angle of 35 mrad and a collection angle of 25 mrad covering multiple Brillouin zones (BZs) are used (see illustration in Fig. 1b). The collected spectrum represents the local phonon density of states (DOS). To measure the phonon dispersion curves of the PSF in GaN, four-dimensional EELS (4D-EELS) was performed with a spatial resolution of ~2.4 nm and a momentum resolution of ~0.37 Å−1 using a convergence angle of 3 mrad54. We placed a slit aperture along the Γ-M-Γ direction on the diffraction plane (see depiction in Fig. 1c).

Fig. 1: Experimental setup and PSF structure.
figure 1

a Schematic of the experimental setup for phonon measurements. The electron beam is focused on the GaN sample with a PSF. The energy loss of the electron beam is analysed using an EELS spectrometer to determine the vibrational energy of the sample. b Schematic of the EELS aperture placement in reciprocal space. The green circle marks the on-axis geometry of the aperture with a 35 mrad convergence semiangle, and the yellow circle marks the position of the aperture with a 25 mrad collection semiangle. c Schematic of the experimental geometry, which shows the diffraction plane. The yellow rectangle illustrates the position of the slot aperture used in 4D-EELS. d HAADF-STEM image of the GaN PSF viewed along the [0001] zone axis. e GPA map of the PSF corresponding to the same area with (d), showing lattice strain component εxx.

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Results

A high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image of a PSF in GaN along the [0001] zone axis is shown in Fig. 1d. This defect often occurs when there is a deviation from the regular stacking sequence of atomic layers along specific crystallographic directions during the epitaxy of GaN. The surrounding area around the PSF is shown in Supplementary Figs. 1a–c. The PSF is a planar defect that extends into the [0001] zone axis with alternating 4-atom and 8-atom rings of Ga and N in the plane. Geometric phase analysis (GPA) was performed on the HAADF-STEM image in Fig. 1d to investigate the defect-induced strain. The εxx, εyy, and εxy distortion components obtained from the GPA are given in Supplementary Figs. 2. Strains in all directions are less than 1% and the predominant strain is concentrated on the εxx component. Figure 1e shows the εxx distortion component of the PSF, which demonstrates that the strain is located at the center of the PSF with a range of approximately 1 nm, diminishing to become negligibly small in the bulk GaN beyond this.

Although the atomic structure of the PSF is clear in the HAADF-STEM image, three distinct atomic models (shown in Fig. 2a–c) can reproduce the same HAADF-STEM images seen by STEM along the [0001] zone axis. These PSF models include two kinds of double-position boundaries (DPB-I and DPB*-I) and an inversion domain boundary (IDB-I)21. DPB-I has a distinct stacking sequence with 1/2\([10\bar{1}0]\) horizontal shift across the boundary and contains seemingly unphysical Ga-Ga and N-N bonds at the boundary with the same bond length of 1.699 Å21. DPB*-I has an additional vertical displacement of 1/2[0001] and a total displacement of 1/2\([10\bar{1}1]\), forming Ga-N bonds at the boundary. IDB-I exhibits a total displacement of 1/2\([10\bar{1}0]\) with a lattice polarity inversion across the boundary and Ga-N bonds at the boundary. A previous theoretical study reported that the DPB-I has the highest domain wall energy of 246 meV/Å2, slightly lower than the energy of the two free \((11\bar{2}0)\) surfaces (256 meV/Å2), indicating a metastable state21. The IDB-I has a calculated domain wall energy of 122 meV/Å2, while the DPB*-I has the lowest energy with a value of 99 meV/Å21. This indicates that the DPB*-I should be the most stable and likely best representative of experiments. The existence of an N-vacancy can be ruled out, as the energy of DPB*-I with an N-vacancy at the 4-atom ring is 11.538 eV higher than that of the DPB*-I model. Then, we changed the zone axis to observe the atomic model of PSF in measurement. Since PSF contains a total displacement of 1/2\([10\bar{1}1]\), a 1/2 [0001] translation vector is not visible on a [\(10\bar{1}0\)] zone axis, and a 1/2 \([10\bar{1}0]\) translation vector is not visible on a [0001] zone axis. The entire displacement of PSF can only be detected along the \([11\bar{2}0]\) zone axis. The atomic models of DPB-I, DPB*-I, and IDB-I, as well as HAADF-STEM images from \([11\bar{2}0]\) zone axis, are shown in Supplementary Fig. 3. Since the PSF lies in the \((11\bar{2}0)\) plane, it cannot be seen edge on along the \([11\bar{2}0]\) zone axis. Therefore, the spatial resolution of the HAADF-STEM image is insufficient to make the N atoms visible. Nevertheless, the HAADF-STEM images showing semi-overlapping atomic structures with atoms projected in a hexagonal-like pattern22,27,30 match well with the DPB*-I atomic model, serving as supporting evidence.

Fig. 2: Three different PSF models and corresponding calculations.
figure 2

Atomistic schematics of (a) DPB-I, (b) DPB*-I, and (c) IDB-I viewed along the [0001] zone axis and the \([10\bar{1}0]\) zone axis. The calculated DOS were extracted from defect-free GaN (blue) and at the PSF (red) with atomic configurations of (d) DPB-I, (e) DPB*-I, and (f) IDB-I.

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From these atomic models, we determined the phonon dispersions and densities of states (DOSs) from first-principles calculations based on density functional theory (DFT) using the Quantum Espresso package65 within the local density approximation (LDA). These were compared with those of bulk wurtzite GaN calculations just as done in ref. 60, which demonstrated excellent agreement with phonon frequencies obtained from inelastic X-ray scattering66. PSF defect models were built into supercells (56 atoms) of the bulk GaN structure. Further details of the bulk calculations can be found in ref. 60 and for the defect structures in the Methods. Figure 2d–f show calculated DOSs of DPB-I, DPB*-I, and IDB-I compared with that of bulk GaN. The phonon DOS of bulk GaN exhibits four peaks at around 18–32 meV, 35–41 meV, 69–79 meV (attributed mostly to transverse optic (TO) modes, and 83–88 meV (attributed mostly to longitudinal optic (LO) modes). These peaks are also found in all three models, though with some broadening, demonstrating that the bulk like modes are captured in these supercells. Besides these bulk peaks, additional peaks occur in DPB-I around 45–50 meV, in DPB*-I around 60–65 and 95–100 meV, and in IDB-I around 60–65 and 98–111 meV, respectively. These additional vibrational peaks are attributed to defect modes originating from the PSF.

To support this, we show the measured atomically-resolved STEM-EEL spectra of the PSF (red) and defect-free GaN (blue) in Fig. 3a. The range, from which the EEL spectra of PSF and defect-free GaN are extracted, is shown in Supplementary Fig. 1d. The finite temperature effect on phonon frequencies comparing 0 K DFT calculations and room temperature measurement is expected to be less than 1 meV67, and does not affect our conclusions. We note that the EELS signals of the defect-free GaN may be influenced by the PSF as the signals are extracted from an area ~1.5 nm from the PSF in the same measurement setup. There are three dominant EEL spectral peaks for defect-free GaN at 24, 39, and 77 meV. In comparison with the calculated DOS of defect-free GaN (see Fig. 2) and corresponding mode analysis, the two peaks at 24 and 39 meV correspond to acoustic phonon modes E2 and B168, respectively. The peak at 77 meV does not distinguish separate peaks for TO and LO phonons as seen in the calculations, likely due to the limited energy resolution of EELS measurements. Since the measurement is performed on-axis, the lack of resolution between the TO and LO peaks may also be a consequence of sampling surface phonon-polaritons that have energies between those of the TO and LO zone center phonons (Reststrahlen band)69. In comparison with defect-free GaN, the EEL spectrum of the PSF reveals two additional peaks in the energy ranges of 53–64 meV and 97–106 meV. Comparing the measured EELS spectrum with the calculated DOS of the three atomic models, the vibrational spectra of DPB*-I (Fig. 2e) best matches the experimental results. Additionally, to demonstrate that the calculated phonon spectrum of DPB*-I is not influenced by size effects due to PSF proximity in periodic supercell simulations, we calculated the DOS of a smaller model with 40 atoms, which reduces the normal GaN lattice on both sides of the PSF compared to that of the 56-atom DPB*-I model used here. As shown in Supplementary Fig. 4, the DOS for the two models of different sizes are remarkably similar, confirming that size effects are negligible for these structures. Therefore, from the phonon spectrum and the domain wall energy calculation discussed above, we conclude that the atomic structure of the GaN PSF is given by the DPB*-I model and restrict our further analysis to it.

Fig. 3: PSF on-axis EELS measurements and calculations.
figure 3

a EELS data extracted from the PSF (red) and bulk GaN (blue), the extracted range is shown in Supplementary Fig. 1d. b Calculated phonon spectra extracted from the PSF (red) and bulk GaN (blue). c Visualization of the atomic vibrations of two primary LDMs, a typical CBM and a typical FEM around the PSF. The upper figure shows the calculated vibrational eigenvectors. The arrows are the real parts of the eigenvectors showing a snapshot of the vibration amplitudes of the atoms. For FEM, the figure focuses on the main vibrations in the [0001] zone axis, overlooking the minor vibrations within the plane. Details about vibrations in [0001] and \([10\bar{1}0]\) zone axis are available in Supplementary Movies 1–10. The phonon energies around these calculated defect modes are shown in the red (LDM1 and LDM2), blue (CBM) and yellow (FEM) shaded regions of (b). The figures below are the corresponding experimental phonon intensity maps with the energy ranges shown in the red (LDM1 and LDM2), blue (CBM) and yellow (FEM) shaded regions of (a) and the HAADF-STEM images within the same areas.

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Using the DPB*-I model (Figs. 2e, 3b), we now explore the origin of the phonon modes in the measured EEL spectra (Fig. 3a). Figure 3c shows calculated atomic vibrational eigenvectors for the DPB*-I supercell at energies of 37, 62, 68 and 99 meV corresponding with measured peaks in EELS intensity. HAADF-STEM images and phonon intensity maps are also shown for these energy ranges. As demonstrated by the HAADF-STEM images and intensity maps, the peaks located at 58–62 and 99–105 meV are localized at the center 4-atom ring of the PSF and then rapidly decay within 0.75 nm, particularly for the higher frequency vibrations. These peaks are most likely correlated with localized defect phonon modes (LDM), which are labelled LDM1 and LDM2, respectively. From calculations, these LDMs are predominantly governed by the vibrations of N atoms in the 4-atom rings, while the other atoms remain largely stationary as shown in the corresponding calculated vibrational eigenvectors. LDM1 originates from vibrations of the 4-atom rings resembling a phonon vortex with rotational vibrational character around the center of the 4-atom rings. This vibrational eigenvector is similar to that previously found in asymmetric Si impurities in graphene70. The lower energy of LDM1 is most likely due to the wide space between N atoms caused by their tangential vibrations. LDM2 encompasses two degenerate vibrational modes, where the N atoms in the 4-atom ring undergo vibrational motions towards each other. Conversely, the higher energy of LDM2 probably result from the close proximity of N atoms as they vibrate towards each other. We extracted the intensity attenuation of LDM1 and LDM2 as a function of distance from the defect core and determined their influence range. Supplementary Fig. 5 displays the integrated phonon intensities of LDM1 and LDM2 as a function of position. The stronger phonon intensity correlates with atoms in the HAADF-STEM image, showing the atomic spatial resolution of phonon detection. LDM1 and LDM2 are highly localized at the center of 4-atom ring, with full-widths at half maximum (FWHMs) of 0.93 nm and 0.50 nm, respectively. The larger FWHM of LDM1 compared to LDM2 may be attributed to that the energy range of LDM1 is close to the optical phonon energy of bulk GaN, resulting in the inclusion of a small number of bulk phonons. Therefore, the range of LDM2 may more accurately reflect the range of localized phonon modes. Furthermore, the measured EEL spectra extracted from the 8-atom ring are shown in Supplementary Fig. 6. LDM1 and LDM2 do not exist at the 8-atom ring as the vibrations occur on the 4-atom ring. The distinction observed in the EEL spectra between the 4-atom ring and the 8-atom ring highlights the exceptional spatial resolution achieved in our study. On the contrary, the other modes are delocalized as demonstrated by the broad smearing in their intensity maps. For the intensity peak in the 32–36 meV range where the EELS intensity map becomes weaker in the center of the PSF, we show the calculated eigenmotion of a typical vibrational mode in this energy range. This mode is confined to the bulk and is predominantly governed by the vibration of Ga atoms away from the PSF while the atoms in the 4-atom ring remain static, which is called the confined bulk mode (CBM). In a sense, this mode is the inverse of the LDMs that were confined to the PSF. This mode is in the energy range of the bulk B1 mode, thus weakening of phonon intensity at the PSF in the EEL spectra is not significant; however, its confinement can be observed in the EELS intensity mapping. At the 8-atom ring, the weakening of phonon intensity in CBM is clear, as the vibration is away from this area. We note that the calculated behavior of this mode may be affected by the finite size of the supercells used. For the 68–72 meV energy range, a different mode type is typical, one that has eigenvibrations extended across the entire system, bulk and PSF, which is called the fully extended mode (FEM). This results in the experimental EELS intensity in the PSF region and the bulk GaN region being the same. The EELS intensity in the 8-atom ring is also the same as bulk GaN. Further visualizations and animations of the LDMs and above-described modes are given in the Supplementary Movies 1–10. As phonon-polaritons have energies between the TO and LO phonons (Reststrahlen band), none of the discussed modes exist within this range, except perhaps the FEM. Additionally, off-axis EELS measurements give similar results as the on-axis data (Supplementary Fig. 7), further supporting the conclusion that surface-phonon polaritons are not playing a significant role here.

Lastly, we explore the dependence of vibrational energy on momentum at the PSF with nanometer resolution using four-dimensional EELS (4D-EELS). We placed a slot from Γ to M (ΓMΓ line) along the \([11\bar{2}0]\) direction (see details in Methods). Measured phonon dispersions with the beam located at defect-free GaN and the PSF are shown in Fig. 4a, b, respectively. The calculated EEL spectra based on DFT and the scattering cross-section54 of bulk GaN and the DPB*-I supercell are shown in Fig. 4c, d for comparison. The calculated PSF supercell structure is orthorhombic rather than the hexagonal GaN structure. Although the limited EELS energy resolution prevents full mode quantification, the overall features show reasonable agreement with the calculated results. The energy range of experimental acoustic and optical phonons are guided for the eye by white dotted lines. It is noted that the energy range of the optical phonon dispersion increases at the PSF, contributing to the LDM1 and LDM2 observed in Fig. 3a and further confirming the existence of localized defect modes at the PSF, with an apparent rise in optical phonon energy. It is also shown that the energy gap between acoustic (and folded acoustic-like modes) and optical phonons at the PSF is smaller than that in defect-free GaN, which may enhance phonon-phonon scattering in this region60. The estimated sound velocities of the PSF and defect-free GaN (slopes of white lines in Fig. 4a, b; see details in Methods) demonstrate that the PSF sound speed is approximately 74% that of defect-free GaN (see details in Methods). This sound speed ratio is highly qualitative because it does not exclusively sample low frequency acoustic phonons, but also includes folded acoustic-like modes. The calculations also show an increase in acoustic phonon energy near the Γ point, while the slope near the Γ point decreases. More specifically, the sound speeds for the PSF are 3894 m/s (TA1), 4627 m/s (TA2), and 7034 m/s (LA), compared to those of GaN, which are 4188 m/s (TA1), 4424 m/s (TA2), and 7501 m/s (LA), giving a reduction for the PSF sound speed of 96.5%. The numerical discrepancy between the calculations (reduction to 96.5%) and experimental results (reduction to 74%) may be due to the calculations representing mode behaviors across the entire defect supercell structure, while the experiments focus specifically near the PSF. In addition, the intensity of acoustic phonons is stronger than that of optical phonons in both calculations and measurements. Both the smaller energy gap between acoustic and high frequency optical phonons and reduced phonon velocities should lead to decreased thermal conductivity in the region of the PSF. Although the impact of defects on the reduction of macroscopic thermal conductivity is dependent on defect density17,18,19, disordered interfaces are expected to facilitate more energy transfer through increased mode-mixing as they relax momentum conservation rules, acting as phonon bridges from one material to the other58, and leading to reduced interfacial resistance. PSFs have higher symmetry with clean interfaces that preserve phonon momentum conservation conditions that lead to less bridging and thus greater thermal resistivity than defects with atomic-level disorder.

Fig. 4: Experimental and simulated phonon dispersions at the GaN PSF.
figure 4

Experimental phonon dispersion line profiles of (a) bulk GaN and (b) the GaN PSF defect, acquired with the slot aperture placed along the Γ-M- Γ line, which is shown in the yellow rectangle in Fig. 1c. The white dotted curves are eye guides for the approximate boundaries of the acoustic and optical phonon signals to show the energy gap. The white dots mark the points with the highest intensity at that momentum and are fitted to a polynomial curve, whose slope gives a semi-quantitative representation of the average sound speed as the acoustic branches have the only appreciable dispersion in this frequency range near the zone center. The corresponding simulation results of (c) bulk GaN and (d) the PSF defect based on DFT calculations and scattering cross section.

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Discussion

In summary, defect phonon modes around a PSF in GaN were visualized through STEM-EELS. By comparing the EELS data with DFT-derived phonon DOSs of three possible PSF models (DPB-I, DPB*-I and IDB-I) and calculated energetics, we conclude that the atomic structure of the PSF is DPB*-I. Through comparison of experimental and theoretical results, three types of PSF-associated modes are uncovered: those confined to the PSF (LDMs), those confined to the bulk (CBMs), and those extended over both (FEMs). Furthermore, we measured the phonon dispersions near and at the PSF using 4D-EELS. The energy gap between the acoustic (and folded acoustic-like modes) and optical branches is reduced in the PSF, which is caused by an LDM, indicating the possibility of enhanced phonon-phonon scattering over that in bulk GaN. Additionally, the average sound speed, determined semi-quantitatively from measurement fits, in the PSF is reduced to approximately 74% of that in defect-free GaN, and supported by DFT calculations. We conjecture that the phonon behaviors detected here may be applicable to similar extended defects such as edge dislocations. No previous experimental detection of the phonon modes only extended on one side of defect exist, which is similar to the partially extended modes at the interface between boron nitride and diamond54.This is due to the high symmetry present on both sides of PSF, in contrast to interfaces between two different materials. Further exploration is necessary to detect symmetry breaking defects, which often exhibit strong strain fields and are difficult to visualize. Overall, our study gives mode-resolved microscopic details regarding phonon behaviors near extended defects, providing valuable insights into defect engineering for targeted phonon-derived applications such as thermal management.

Methods

Sample preparation and characterization

Wurtzite quasi-van der Waals epitaxial (vdWe) GaN films were grown on a graphene/sapphire substrate using plasma-assisted molecular beam epitaxy (MBE). The in situ RHEED image of GaN films is shown in Supplementary Fig. 8. The streaky RHEED patterns indicate a flat surface, where no nitrogen reconstruction can be detected. As the growth progresses, nucleation islands coalesce and eventually form a GaN film consisting of a prismatic stacking fault (PSF) defect. A transmission electron microscopy (TEM) lamella was cut by a Thermo Fisher Scientific Helios G4 UX focused ion beam (FIB) system. To determine the atomic arrangement of the GaN planar defect, scanning transmission electron microscopy (STEM) measurements were performed using a spherical aberration-corrected FEI Titan Cubed Themis G2 300.

Electron energy loss spectroscopy (EELS) data acquisition

STEM-EELS measurements were performed using a Nion HERMES 200 microscope operating at 60 kV at room temperature equipped with both a monochromator and an aberration corrector. 3D-EELS datasets were obtained with a convergence semiangle of 35 mrad and a collection semiangle of 25 mrad. The STEM-EELS on-axis dataset was acquired in a 100 × 30 pixels mapping area covering a region of 3 × 0.9 nm that included the PSF. The dwell time for each pixel was set to 200 ms, resulting in a total acquisition time of approximately 10 minutes for each dataset. The sample drift during the acquisition was typically less than 1 nm. The energy resolution achieved under these conditions was approximately 10 meV, while the spatial resolution was approximately 0.2 nm.

For the acquisition of the 4D-EELS datasets, a convergence semiangle of 3 mrad was used, and a slot aperture with an aspect ratio of 16:1 was placed along the Γ-M-Γ direction. To increase the signal-to-background ratio and avoid strong central diffraction spots, the slot aperture was displaced away from the central Γ point. The typical dwell time per pixel was set to 15 s, resulting in a total acquisition time of approximately 32 minutes for each dataset. The energy resolution achieved under these conditions was approximately 15 meV.

Ab initio calculations

Density functional theory (DFT) calculations for the bulk and defect systems were performed with the Quantum Espresso package65 within the local density approximation (LDA). The calculated bulk wurtzite structural parameters (a = 3.164 Å, c = 5.150 Å, and u = 0.377) and phonon frequencies were determined exactly as in ref. 60. These give good agreement with measured phonon frequencies from inelastic x-ray scattering66.

Model interfacial PSF structures (see Fig. 2) were built using the bulk lattice parameters in 56-atom orthorhombic unit cells that consisted of the PSF structure (16 atoms) and two bulk hexagonal ring structures on either side of the interface (20 atoms each). Variations between the different model arrangements were described above. The use of periodic boundary conditions (as in typical DFT calculations) with this unit cell gives periodic PSF interfaces with an 11.32 Å centre-to-centre distance between them. This provides enough space to nearly recover the bulk vibrational behaviour for atoms sandwiched between the two interfaces. From these 56-atom unit cells, 3 × 1 × 2 supercells (336 atoms) were built (in-plane parallel to the interface × in-plane perpendicular to the interface × cross-plane parallel to the interface) to avoid spurious periodic forces in the DFT simulations.

Harmonic interatomic force constants (hIFCs) were constructed by finite difference methods for numerical differentiation of the interatomic forces71. These forces were determined from DFT electronic structure calculations of various supercell images, each with a strategically displaced unit cell atom (0.05 Å from equilibrium) along different Cartesian directions. Given the hIFCs, the phonon frequencies (\({\omega }_{{{\bf{q}}}\,\, j}\)) and eigenvectors (\({{{\boldsymbol{\varepsilon }}}}_{{{\bf{q}}}\,\,{jk}}\)) were determined via diagonalization of the dynamical matrices for a given wavevector (\({{\bf{q}}}\)), where \(j\) represents the phonon polarization (168 in total) and \(k\) represents the atom72. Note that long-range Coulomb corrections were not explicitly included for the defect dynamical matrices. Nonetheless, the Ga and N vibrations away from the interface exhibited similar behaviour as that in the bulk calculations with polar corrections. The vibrational density of states (\({{\rm{DOS}}}(\omega )\)) and atom-projected \({{\rm{DOS}}}(\omega,k)\) were obtained via:

$${{\rm{DOS}}}(\omega )={\sum}_{k}{{\rm{DOS}}}\left(\omega,k\right)={\sum}_{{{\bf{q}}}\,\,{jk}}{\left|{{{\boldsymbol{\varepsilon }}}}_{{{\bf{q}}}\,\,{jk}}\right|}^{2}\delta (\omega -{\omega }_{{{\bf{q}}}\,\,j})$$
(1)

where the wavevector sum is over the first Brillouin zone and the energy delta function is approximated by a Lorentzian of smearing width 0.15 meV.

The 4D-EELS DFPT calculation was conducted using Quantum ESPRESSO, employing the local density approximation and norm-conserving pseudopotentials. For wavefunctions, the kinetic energy cutoff was set to 44 Rydbergs (Ry), while charge density and potential cutoff was 395 Ry. The scattering cross section is represented as40,42,73:

$$\frac{{d}^{2}\sigma }{d\omega d\Omega }\propto {\sum}_{{{\bf{q}}}j }{\left|{F}_{{{\bf{q}}}j}\right|}^{2}\left[\frac{n_{{{\bf{q}}}j}+1}{{\omega }_{{{\bf{q}}}j}\left({{\bf{q}}}\right)}\delta \left(\omega -{\omega }_{{{\bf{q}}}j}\right)+\frac{n_{{{\bf{q}}}j}}{{\omega }_{{{\bf{q}}}j}}\delta \left(\omega+{\omega }_{{{\bf{q}}}j}\right)\right]$$
(2)

where \(n_{{{\bf{q}}}j}\) is the phonon occupancy number. The Dirac delta function is represented by δ(x). The coupling factor is given by:

$${F}_{{{\bf{q}}}j}\propto \frac{1}{{q}^{2}}{\sum}_{k}\frac{1}{\sqrt{{M}_{k}}}{e}^{-i{{\bf{q}}}{\cdot}{{{\bf{r}}}}_{k}}{e}^{-{W}_{{{\bf{q}}}k}}{Z}_{{{\bf{q}}}k}\left[{{{\bf{\epsilon}}}}_{{{\bf{q}}}jk}\cdot {{\bf{q}}}\right]$$
(3)

where \({M}_{k}\) is the mass of atom \({k}\), \({Z}_{{{\bf{q}}}k}\) is the effective charge, \({{{\bf{r}}}}_{k}\) is the real-space position, and \({e}^{-2{W}_{k}\left({{\bf{q}}}\right)}\) is the Debye-Waller factor74. \({Z}_{{{\bf{q}}}k}\) was calculated as per literature73, with atomic form factors constructed from parameters in the literature75.

EELS data processing

For each 3D-EELS dataset, the EEL spectra were initially aligned using normalized cross-correlation to correct for beam energy drift. Then, the spectra were normalized by the zero-loss peak (ZLP) total intensity. To eliminate Gaussian noise, a block-matching and 3D filtering algorithm was applied76,77. The EELS background was fitted using a Pearson function25 and subtracted. To enhance the resolution and reduce the broadening effect caused by finite energy resolution, Lucy-Richardson deconvolution was employed. The statistical factors of the background-subtracted spectra were further corrected, which resulted from the different scattering probability of phonons with different frequencies78, to make it more comparable to the real phonon DOS.

For each 4D-EELS dataset, EEL spectra were first registered by their normalized cross correlation to correct possible beam energy drift and momentum distortion. After the alignment, we applied block-matching and 3D filtering (BM3D) algorithm to remove Gaussian noise76,77. For each momentum and each energy channel, the data is denoised in two spatial dimensions, where the noise level is individually estimated based on high-frequency elements in the Fourier domain. To enhance the resolution and reduce the broadening effect caused by finite energy resolution, Lucy-Richardson deconvolution was employed. The white dots in Fig. 4a, b highlight the points with the highest intensity of acoustic phonons at their respective momentum values, and these dots are then fitted with a polynomial curve for visualization. We obtain the sound speed ratio between PSF and GaN by averaging the slope ratios for the two systems on 6 different momentum ranges from \(\Gamma \sim 0.3{{\rm{M}}}\) to \(\Gamma \sim 0.5{{\rm{M}}}\).