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Matryoshka Phonon Twinning in alpha-GaN

Understanding lattice dynamics is crucial for effective thermal management in high-power electronic devices because phonons dominate thermal transport in most semiconductors. This study utilizes complementary inelastic X-ray and neutron scattering techniques and reports the temperature-dependent phonon dynamics of alpha-GaN, one of the most important third-generation power semiconductors. A prominent Matryoshka phonon dispersion is discovered with the scattering tools and confirmed by the first-principles calculations. Such Matryoshka twinning throughout the three-dimension reciprocal space is demonstrated to amplify the anharmonicity of the related phonon modes through creating abundant three-phonon scattering channels and cutting the phonon lifetime of affected modes by more than 50%. Such phonon topology effectively contributes to the reduction of the in-plane thermal transport, thus the anisotropic thermal conductivity of alpha-GaN. The results not only have significant implications for engineering the thermal performance and other phonon-related properties of alpha-GaN, but also offer valuable insights on the role of anomalous phonon topology in thermal transport of other technically important semiconductors.

Gallium nitride (GaN), one of the most important third-generation power semiconductors, excels in power density and high-temperature stability due to its wide bandgap and high thermal conductivity [1][2][3][4] (230 W m -1 K -1 at room temperature has been reported [5][6][7][8]), among many other favorable properties. To further miniaturize high-power electronics [4], great efforts have been devoted to studying the thermodynamics of -GaN (wurtzite structure, see Fig. S1a in Supplementary Information). However, the knowledge on its phonon dynamics remains limited, with key questions yet to be answered. For example, existing experimental measurements of the phonon dispersion relation of -GaN are limited to ambient conditions, and the phonon scattering processes and the temperature effects are mostly unexplored [9,10]. Moreover, the anisotropic thermal transport of -GaN along a (inplane) and c (out-of-plane) axis directions remains controversial due to the challenges in transport measurements [7,[11][12][13][14][15][16][17][18] and calculations [19][20][21]. Therefore, a more comprehensive understanding of the phonon dynamics is pivotal to investigating the thermal transport and other thermodynamics properties of GaN.
Here we report a novel in-plane Matryoshka-like phonon dispersion twinning, in which the optical and the acoustic phonon dispersions are like nesting dolls (see Fig. S2). Such behavior is observed in -GaN single crystals by both inelastic X-ray scattering (IXS) and inelastic neutron scattering (INS), and is confirmed by first-principles calculations. The Matryoshka phonon twinning provides a great magnitude of acoustic-optical scattering channels and contributes to the reduction of the in-plane thermal conductivity, leading to the anisotropic thermal conductivity of -GaN. This result is supported by the phonon lifetime measured through scattering linewidths and provides valuable insights into phonon topology engineering for thermal management.

GaN Single Crystal Samples:
High-quality GaN single crystals (un-doped, n-type, MTI Corporation [36]) used in this work were grown by the hydride vapor phase epitaxy (HVPE)-based method with low dislocation density (< 1  10 7 cm -2 ) (Fig. S1c). The quality of the crystals was checked with both X-ray and neutron diffraction. The full width at half maximum (FWHM) of the X-ray diffraction peak at (002) plane is around 0.10  0.02º (Fig. S1d).

Inelastic X-ray Scattering Measurements:
High-resolution IXS experiment was performed to measure the phonon dispersions of an α-GaN single crystal (250 μm thickness, the lower panel of Fig. S1c) at 50, 175, and 300 K. The measurements were conducted at 30-ID-C (the High-resolution Inelastic X-ray Scattering beamline, HERIX) at the Advanced Photon Source (APS) [37,38]. The incident photon energy was ~23.7 keV with an energy resolution ΔE of 1.2 meV and a momentum resolution of 0.65 nm -1 . The single crystal was attached to a copper post by varnish and the copper post was mounted in a closed-cycle cryostat. The IXS measurements were accomplished at 3 constant wave-vector mode in reflection geometry. The orientation matrix was defined by using Bragg peaks at (4 0 0), (0 0 4), (0 0 5), and (2 2 0) respectively.

Inelastic Neutron Scattering Measurements:
The INS measurements were carried out on a bigger single crystal (φ=5 cm, upper panel of Laboratory [39]. An incident energy of E i = 50 meV and a Fermi chopper frequency of 420 Hz were used to optimize instrument energy resolution. At the elastic scattering, the energy resolution is 2 meV. For INS, the [110] axis was set vertical, and scattering plane was selected to obtain both the in-plane and out-plane phonons at 14, 50, 300, and 630 K. When collecting the data, 1° step was used with a rotation range from -90° to 90° for 300 K, and 2° step was used with a rotation range from -70° to 50° for other temperatures. Data reduction was performed using the Mantid program [40]. The data were normalized by the accumulated incident neutron flux, and the detector efficiency correction was applied based on the incoherent scattering of the vanadium standard.

Scattering Data Processing:
INS data shown in the present work were corrected for the sample temperature to obtain the dynamical susceptibility, χ″(Q, E)= S(Q, E)/[n T (E)+1], where n T (E) stands for the Bose distribution, S(Q, E) is the four-dimensional scattering dynamical structure factor [41] (1) where is the neutron scattering length, Q = k − k′ is the wave vector transfer, and k′ and k are the final and incident wave vector of the scattered particle. q is the phonon wave vector, ω s is the eigenvalue of the phonon corresponding to the branch index s, τ is the reciprocal lattice vector, d is the atom index in the unit cell, r d is the atom position, W d is the corresponding Debye-Waller factor, and e ds is the phonon eigenvectors. For given Q, the measured scattering spectra were fitted with a damped-harmonic-oscillator (DHO) model [42] ( where M is the effective mass, ω 0 is the bare phonon energy in the absence of damping forces, and 2γ is a damping factor that describes the phonon scattering rates. n is the Bose distribution. The bare phonon energy and phonon linewidths were extracted by deconvoluting with both instrument energy and momentum resolution functions, the latter of which eliminates the slope effect of highly dispersive phonons on linewidths.

Simulation Methods:
The first-principles calculations were performed based on the density functional theory (DFT) as implemented in the Vienna Ab Initio Simulation Package (VASP) [43]. The exchange−correlation energy was computed using the local-density approximation (LDA) functional, and the projector-augmented-wave (PAW) potentials were used (4s 2 4p 1 for Ga and phonon modes (q 1 ,  1 ) and (q 2 ,  2 ), for the mode (q 1 ,  1 ) on the Arc branch near  point, we can always find a TA 2 mode (q 2 =q Arc -q 1 ,  1 = Arc - 1 ) in the basal plane that enables such scattering (Fig. S9). This scattering behavior is consistent with recent report of weighted phonon scattering space in -GaN, which shows the emission process is significantly larger than that of the absorption process in the energy range from 20 to 40 meV (~5 THz to ~10 7 THz), (see Fig. S10) [6] in line with the energy range of the Arch branch in the Matryoshka twinning.

The impact of Matryoshka phonon twinning on the anharmonic scattering rate
Phonon linewidths of -GaN at 300 K were extracted from the IXS measurements by deconvoluting both the instrument energy and momentum resolution functions, the latter of which is equally important due to the steep phonon dispersion. Phonon linewidth is proportional to the total scattering rate from all processes [34,45] and dominated by anharmonic phonon-phonon interactions in -GaN. As shown in Fig. 4a, while the linewidths of most phonon modes are between 0.5 and 1.5 meV, the linewidths of the phonon modes on the TA 2 and the Arc branches are much broader and double of some other phonon modes.
These large linewidths indicate that the nested phonons are scattered more strongly than the others, leading to the phonon lifetime of nested modes is cut by more than 50%. This result can also be observed in the INS data (Figs. 4b and 4c), in which the linewidths of phonon modes on the TA 2 and Arc branches are much larger than those on the high energy optical (HEO) branch along  M direction and those in the HEO and LEO branches along The INS measurement is consistent with the Raman spectroscopy [46] result that the linewidth of the Arc branch at the  point is 0.8 meV (Fig. 4d). The anharmonic phonon scattering rate is expressed as [47] ( where V is the third-order interatomic potential,  qs is the phonon frequency of the phonon mode of wavevector q and branch index s, and n is Bose occupation. This anharmonic scattering rate is determined by both the third-order interatomic potentials and the phonon 8 scattering phase space: the former represents the scattering strength, and the latter represents the phase space of available scattering channels [48]. The large linewidths of the phonons involved in the Matryoshka twinning suggest that such behavior significantly promotes threephonon scatterings in -GaN. To elucidate the anharmonicity of the phonon modes on the Arc branch in -GaN, the evolution of the phonon energy at q = 0.1 along  M direction was fitted over a wide temperature range with the following expression [49] (4) where A and (0) are fitting constants, and  A (T) is the temperature-dependent phonon energy. The result in Fig. 5a shows a moderate anharmonicity below the Debye temperature (~636 K) [9], supported by the moderate difference between experimental isobaric [50] and calculated volumetric mode Grüneisen parameters (Fig. 5b), see the details in Supplementary   Information. Additionally, the frozen phonon potential for the phonon mode on the Arc branch at q = 0.1 along  M direction (Fig. 5c) only slightly departs from the harmonic behavior, indicating moderate anharmonicity (the phonon eigenvectors are shown in the insert in Fig. 5c). Considering such moderate anharmonicity, it is confirmed that the large linewidths of the phonon modes in the TA 2 and Arc branches are dominated by the enlarged scattering phase space from the vast scattering channels induced by the Matryoshka twinning.

The impact of Matryoshka phonon twinning on the anisotropic thermal conductivity
The anisotropic thermal conductivity of -GaN recently attracted intensive attention because it is critical for heat management in power electronics, where -GaN is a key component. However, the answer remains controversial: some reported that the out-of-plane thermal conductivity is greater than the in-plane one while others showed opposite results [5,6,19,20] Recent reports suggest that the in-plane thermal conductivity is smaller than the 9 out-of-plane one, with a relatively strong anisotropy of 12% [19,20]. Here, we will investigate the roles of phonon group velocity and scattering rates on the anisotropic lattice thermal conductivity.
The temperature-dependent acoustic phonon group velocities were extracted from the measured phonon spectra (Fig. 2), as shown in Table 1

Conclusion
Through a combination of inelastic X-ray/neutron scattering experiments and firstprinciples calculations, a novel Matryoshka phonon twinning is found throughout the basal plane of the Brillouin zone in -GaN. This phenomenon creates a vast number of threephonon scattering channels and leads to enhanced anharmonic phonon scattering, shown by anomalously large phonon linewidths. The strong in-plane phonon scatterings due to the Matryoshka phonon twinning suppress the in-plane thermal conductivity of -GaN and enhance the thermal transport anisotropy, while its acoustic phonon group velocities remain fairly isotropic. Amplification and suppression of such Matryoshka phonon twinning by phonon engineering could provide a valuable means to control lattice thermal transport in many related materials, such as electronic devices, thermoelectrics, and thermal barriers.

Conflict of interest
The authors declare no conflict of interest.

Acknowledgements
The work at Beijing Institute of Technology is supported by the National Natural Science

Direction
Temperature (K) LA TA 2 v g (m/s)

Structure and sample information of -GaN.
Fig . S1 shows the crystal structure of -GaN and the single crystal sample information.
GaN adopts the wurtzite () structure (hexagonal) under ambient conditions, composed of two interpenetrating hcp lattices of Ga and N atoms, with one Ga atom and four neighboring N atoms forming a tetrahedral structure (Fig. S1a). Fig. S1b shows the Brillouin Zone of -GaN. Fig. S1c shows the single crystal samples for the INS (upper) and IXS (lower) measurements, respectively. Fig. S1d shows the rocking curve of our sample, which indicates high crystalline quality.             -M (1/2, 0, 0) and K (1/3, 1/3, 0) are the high symmetry points of the Brillouin zone, and T (4/9, 1/9, 0) and S (7/18, 2/9, 0) are points on the path from M to K. (b) The projection contours of the TA 2 and Arc branches on the basal plane from the first-principles calculation (DFT) and INS results. The white dash line and the black solid line indicate the TA 2 and Arc phonon surfaces' energy contours, respectively. The difference between the contours of the TA 2 and Arc phonon surfaces is 2.5 meV. The color scale indicates the phonon surface energies.

Schematic of the three-phonon scattering channels enhanced by the Matryoshka dispersion twinning in -GaN.
The twinning of the Matryoshka-like dispersion potentially provides a vast number of three-phonon scattering channels in -GaN by reducing the momentum transfer constraint.
For example, in the three-phonon emission process where any phonon mode of the Arc branch (q Arc , ω Arc ) decays into two phonon modes, for one mode of the Arc branch around Γ point (q Arc0 , ω Arc0 ), we can always find a TA 2 mode (q Arc -q Arc0 , ω Arc -ω Arc0 ) in the basal plane that enables such the scattering (see Fig. S9). In contrast, for phonon modes at higher or lower energy, the number of emission channels will be less, and the heat carrier (mainly acoustic phonon) involved will be scattered less strongly. Fig. S9. Schematic of the three-phonon scattering channels enhanced by the Matryoshka dispersion twinning in -GaN. The two cones represent the Arc and the TA 2 phonon energy surfaces. The red, purple, and yellow dots represent the phonon modes on the Arc phonon surface, the phonon mode around  point of the Arc surface (conic top), and the phonon modes on the TA 2 phonon surfaces.

Calculated weighted phase space of phonon scattering in -GaN.
Fig. S10 shows the weighted phase space of three-phonon scattering in -GaN, which is reproduced from ref. [2] under the author's permission. The weighted phase space below 40 meV is much larger than that above 70 meV. Moreover, the values of the emission process are larger than those of the absorption process, especially in the energy range from 20 to 40 meV.
Such behavior is consistent with the reported Matryoshka phonon twinning. The black circles indicate the total weighted phase space, the red circles indicate the absorption process, and the blue circles indicate the emission process.

Grüneisen parameter calculations.
Grüneisen parameters describe how much the phonon frequency shifts with changes in volume. The mode Grüneisen parameter is defined as [3] (1) where V 0 is the equilibrium volume,  j (q) is the phonon energy of wavevector q and branch index j. Phonon dispersions calculated with the volumes 1% larger and smaller than the equilibrium volume are used to extract the quasi-harmonic Grüneisen parameters, as implement in Phonopy code [4].
The experimental mode Grüneisen parameter was obtained based on the isobaric Grüneisen parameter  P (T) concept, which is usually used to quantify the temperature dependence of the phonon energy shifts and is defined as [5] (2) where  is the thermal expansion coefficient of -GaN [6], and  i is the phonon mode energy. The difference between quasi-harmonic and isobaric Grüneisen parameters is a good indication of phonon anharmonicity [7]. In Fig. 5b, the mode Grüneisen parameters at 300 K show only a minor discrepancy between the two. This result implies that the measured isobaric Grüneisen parameters of -GaN contain contributions mostly from the quasiharmonic model instead of the anharmonicity [8].