The electronic band structure and projected density of states (PDOS) of AlN were calculated to elucidate its fundamental electronic behavior (Figure 5(a) and Figure 5(b)). As shown in the band structure (Figure 5(a)), the valence-band maximum (VBM)is located at the Γ point and the conduction-band minimum (CBM) also occurs at the same k-point, indicating that AlN exhibits a direct band gap (Γ-Γ transition). The identification of a direct band gap is based on the coincidence of the highest occupied and lowest unoccupied electronic states at the Brillouin-zone center. This result is consistent with recent experimental measurements and advanced first-principles calculations, which report wurtzite AlN as a direct ultra-wide-band-gap semiconductor with the band edges located at Γ [6][7].

Although the present calculations were performed using a semilocal exchange-correlation functional, which is known to underestimate the absolute magnitude of band gaps, the nature and k-space position of the band gap are reliably described. Recent quasiparticle GW studies and optical experiments place the direct band gap of AlN in the range of approximately 6.0 - 6.3 eV, confirming that the Γ-Γ transition remains unchanged across different theoretical formalisms [6][7]. Therefore, while the computed gap value is expectedly smaller, the predicted electronic topology remains physically meaningful and consistent with modern literature.

Further insight into the electronic structure is obtained from the PDOS shown in Figure 5(b). The upper valence band is dominated primarily by nitrogen 2p states, whereas the conduction-band edge is mainly composed of aluminum 3s and 3p orbitals. This orbital separation reflects the characteristic bonding nature of III-nitride compounds, where the valence band arises from anion p-state hybridization and the conduction band exhibits stronger cation character. Similar orbital distributions near the band edges have been reported in recent density-functional and k·p parameterization studies of AlN and related nitride alloys, confirming the reliability of the present electronic description [8][9].

The coexistence of a direct band gap at Γ (Figure 5(a)) and a clear separation of valence and conduction-band orbital characters (Figure 5(b)) underpins the optical transparency of AlN in the visible region and its suitability for deep-ultraviolet optoelectronic applications. At the same time, the large band gap and low intrinsic carrier concentration impose intrinsic limitations on electrical conductivity,

Figure 5. (a): Electronic band structure of wurtzite AlN; (b): Total and projected DOS for wurtzite AlN.

which is consistent with the generally weak thermoelectric performance observed in ultra-wide-band-gap nitrides. These electronic characteristics establish AlN as a reference material for high-power electronics, deep-UV photonics, and nitride-based heterostructure engineering [9].

The elastic constants were determined using the finite-strain energy method. Symmetry-adapted homogeneous strains appropriate for hexagonal crystals were applied to the relaxed equilibrium structure, and internal atomic coordinates were fully relaxed for each strained configuration. Strain amplitudes of ±0.5%, ±1.0% and ±1.5% were used, providing multiple strain points for each deformation mode. The total energies were fitted to a quadratic strain-energy relation, and the elastic stiffness coefficients were obtained from the curvature of the fits; energies rather than stresses were used for improved numerical stability. The calculated elastic constants were C11 = 374 GPa, C12 = 127.97 GPa, C13 = 98.00 GPa, C33 = 392 GPa and C44 = 125 GPa, which satisfy the Born stability criteria for hexagonal crystals.

The mechanical response of wurtzite AlN is governed by its elastic stiffness tensor Cij, which describes the linear relationship between applied stress and the resulting strain [10]. These elastic constants provide the fundamental link between atomic bonding and macroscopic mechanical behavior, including resistance to volume change, shear deformation and uniaxial loading. Accurate determination of Cij is therefore essential for evaluating the suitability of AlN for energy-harvesting and electromechanical applications, where mechanical robustness and elastic anisotropy directly influence device performance [11].

The elastic constants were calculated using the first-principles energy-strain method [12]. In this approach, small symmetry-preserving homogeneous deformations are applied to the equilibrium wurtzite structure and the total energies of the strained configurations are computed after relaxing all internal atomic coordinates [12]. Because the total energy can be expanded to second order in the strain components, the curvature of the energy-strain relationships yields the elastic stiffness coefficients [13]. This method provides a self-consistent and accurate description of elastic properties, particularly for anisotropic hexagonal crystals [14].

Mechanical stability of hexagonal AlN was evaluated using the Born stability criteria, which require that specific combinations of elastic constants remain positive in order to ensure resistance against all symmetry-allowed lattice distortions [15]. These conditions guarantee that the crystal is mechanically stable under small deformations.

To determine the elastic constants in practice, five linearly independent homogeneous strain modes were applied: 1) an in-plane biaxial strain that changes the basal lattice parameter a while keeping c fixed; 2) a uniaxial strain along the c-axis at fixed a; 3) a volume-conserving orthorhombic in-plane strain used to isolate C11 − C12; 4) a shear strain ε4 to obtain C44; and 5) a basal-plane shear strain ε6 to obtain C66 [16]. Fitting the resulting energy-strain relations yields the independent elastic constants consistent with hexagonal symmetry.

The elastic stiffness tensor was expressed in standard 6 × 6 Voigt notation [12]. For hexagonal AlN, only five elastic constants are independent: C11, C12, C13, C33, and C44, with C66 = (C11 − C12)/2 and C55 = C44 by symmetry [12]. The hydrostatic elastic response is described by the combination Czz = C11 + C12 +2C33 − 4C13 [17].

The resulting elastic constants fully characterize the stiffness, directional rigidity and shear resistance of wurtzite AlN and provide the basis for computing the bulk modulus, shear modulus, Young’s modulus, and elastic anisotropy, which collectively govern its mechanical reliability and energy-harvesting performance. The calculated elastic constants satisfy the Born stability conditions for hexag-onal crystals, as summarized inTable 1.

Table 1. Born stability criteria.

We obtained the bulk modulus (B), shear modulus (G), and Young’s modulus (E) of polycrystalline AlN using Voigt-Reuss-Hill averaging. As expected, a larger E indicates a stiffer solid; literature trends show E for rock-salt > E for wurtzite > E for zinc-blende (noting that rock-salt is a high-pressure, non-piezoelectric phase).

Table 2 summarizes the computed bulk modulus, shear modulus, and Pugh’s ratio from this study, and compares them to values from the existing literature.

Table 2. Computed bulk modulus, shear modulus, and Pugh’s ratio for w-AlN, with percentage deviations from literature values.

Table 3. Pugh’s ratio.

Poisson’s ratio, derived from B and G in Table 2, reflects bonding character: smaller Poisson’s ratio (0.1 - 0.25) is more covalent-like, whereas larger (0.25 - 0.35) is more ionic-like [18] The ductile or brittle nature of a material can be effectively evaluated using Pugh’s ratio (B/G), where values greater than 1.75 indicate ductility, while lower values correspond to brittle behavior (Pugh, 1954). In the present study, the calculated B/G value of approximately 1.60, as indicated in Table 3, lies well below the critical threshold, confirming the intrinsically brittle nature of AlN. This observation is consistent with previous experimental and first-principles investigations, which have reported AlN as a brittle ceramic material with high elastic stiffness and limited plastic deformation capability. Such inherent brittleness has important implications for mechanically active applications, where appropriate structural design and stress management strategies are necessary to ensure long-term reliability. Nevertheless, the high elastic stiffness of AlN enables efficient stress transfer and reversible elastic deformation under small strains, which is advantageous for piezoelectric and thermally driven energy-harvesting applications where cyclic elastic stability is required.

Finally, we note that hardness correlates more strongly with shear modulus (and with elastic/plastic descriptors than with bulk modulus alone. The brittleness of AlN means it can sustain only limited plastic deformation, but its high stiffness and mechanical stability (all Born criteria satisfied) ensure it can reliably endure the small elastic strains involved in energy-harvesting operation.

The elastic anisotropy of hexagonal AlN was evaluated using the Zener anisotropy factor [9]. For a completely isotropic material, the Zener factor equals unity, whereas deviations from unity indicate the degree of elastic anisotropy [19]. In the present study, the calculated anisotropy factor for AlN differs significantly from one, revealing pronounced elastic anisotropy between the basal plane and the crystallographic c-axis [14]. This behavior reflects the intrinsic directional dependence of bonding in the wurtzite crystal structure [14].

To describe the macroscopic mechanical behavior of polycrystalline AlN, the bulk and shear moduli were determined using the Voigt-Reuss-Hill (VRH) averaging scheme, which provides a reliable approximation by taking the arithmetic mean of the upper (Voigt) and lower (Reuss) bounds. The resulting Hill-averaged bulk and shear moduli were subsequently employed to estimate the corresponding Young’s modulus and Poisson’s ratio of the polycrystalline material [20]. Based on the calculated elastic moduli, AlN exhibits a Young’s modulus on the order of several hundred gigapascals, confirming its exceptionally high stiffness [18]. In addition, the obtained Poisson’s ratio is approximately 0.25, a value characteristic of materials with predominantly covalent bonding [21]. These mechanical parameters indicate that wurtzite AlN is a very stiff and only moderately compressible ceramic, in excellent agreement with both experimental measurements and previous theoretical studies of III-nitride materials.

Figure 6. Shear modulus (G) in the in xy, xz, yz plane and 3D.

Figure 7. Young’s modulus (E) in xy, xz, yz plane and 3D.

Figure 8. Shear compressibility in the xy, xz, yz plane and 3D.

We visualized the directional dependence of the elastic moduli to further assess anisotropy. Figure 6, Figure 7 and Figure 8 show stereographic projections (in the yz-plane for clarity) of the shear modulus, Young’s modulus, and compressibility, respectively [15]. These plots highlight the anisotropic mechanical response of hexagonal AlN, with stiffness differing along various crystallographic directions. Notably, the c-axis direction exhibits different elastic behavior compared to the basal plane, reflecting the lower crystal symmetry in that direction. Despite the anisotropy, AlN’s overall elastic response is dominated by its high stiffness and brittleness, which are advantageous for maintaining structural integrity in piezoelectric devices [22].

The elastic constants we obtained for w-AlN are in good agreement with previously reported values (within a few percent). The slight discrepancies can be attributed to differences in computational method or pseudopotentials used in literature. Our results confirm that wurtzite AlN is mechanically stable and very stiff. The relatively low Poisson’s ratio of 0.25 and Pugh’s ratio < 1.75 categorize AlN as a brittle material, meaning it will fracture with little plastic deformation. In the context of energy harvesting, this brittleness is not necessarily detrimental; in fact, a rigid, brittle material can sustain repeated elastic cycling strain and release without dissipating much energy through plastic deformation, which is desirable for a piezoelectric generator. The key is that the material must be operated within its elastic regime to avoid fracture (Figure 9).

From an energy-harvesting perspective, the elastic properties play a critical role in governing electromechanical coupling, as piezoelectric coefficients are intrinsically linked to both polarization response and elastic compliance. The high elastic stiffness of AlN ensures efficient mechanical-to-electrical energy transfer

Figure 9. Poisson’s ratio in the in xy, xz, yz plane and 3D.

under small applied strains, while its mechanical stability guarantees repeatable operation under cyclic loading. Although the intrinsic brittleness of AlN limits its tolerance to large deformations, its ability to sustain reversible elastic strains without plastic dissipation is advantageous for piezoelectric generator applications, provided that device operation remains within the linear elastic regime. In addition, robust elastic stability is essential for thermoelectric devices operating under temperature gradients, where thermal expansion-induced stresses can degrade performance over time. Therefore, the present elastic and anisotropic mechanical analysis provides an essential foundation for understanding and optimizing both the electromechanical and thermoelectric performance of AlN-based energy-harvesting systems.

The thermoelectric properties of AlN were investigated computationally based on the electronic structure obtained from density functional theory. The transport coefficients were evaluated within the framework of semi-classical Boltzmann transport theory, as implemented through post-processing of the DFT band structure under the constant relaxation time approximation Within this approach, the Seebeck coefficient, electrical conductivity and electronic contribution to thermal conductivity are derived from the energy-dependent band velocities and density of states, enabling temperature-dependent transport trends to be analyzed without requiring explicit scattering calculations [23].

The thermoelectric transport coefficients were obtained by post-processing the converged DFT band structure using the BoltzTraP2 code within the semiclassical Boltzmann transport framework under the constant relaxation time approximation (CRTA). The calculations were performed for intrinsic (undoped) wurtzite AlN by scanning the chemical potential around the Fermi level; therefore carrier concentrations arise only from thermally excited electrons across the wide band gap. Electrical conductivity is reported in the σ/τ form, since the carrier relaxation time τ is not explicitly calculated. The very small conductivity values therefore reflect the intrinsic ultra-wide-band-gap nature of AlN rather than scattering limitations. The lattice thermal conductivity was not computed from first-principles phonon transport but interpreted based on the known strong covalent bonding and light atomic masses of III-nitride materials, consistent with experimentally reported high thermal conductivity. This methodology explains the combination of moderate Seebeck coefficient, extremely low electrical conductivity, and consequently very small thermoelectric figure of merit (ZT) obtained for wurtzite AlN.

The thermoelectric response of a material is governed by three interrelated quantities: the Seebeck coefficient S (voltage generated per unit temperature difference), the electrical conductivity σ and the thermal conductivity κ (here primarily lattice thermal conductivity, since AlN is an insulator). The interplay of these quantities determines the dimensionless thermoelectric figure of merit, ZT = S²σT/κ [24].

Figure 10 shows the calculated Seebeck coefficient as a function of temperature. The negative sign of S over the investigated temperature range indicates dominant electron-type transport, consistent with the conduction-band characteristics obtained from the DFT electronic structure. The magnitude of S increases with temperature, reflecting enhanced thermal excitation of charge carriers in this wide-band-gap semiconductor. However, a large Seebeck coefficient alone does not guarantee good thermoelectric performance and must be considered together with σ and κ.

The temperature dependence of the thermal conductivity is presented in Figure 11. AlN exhibits a high lattice thermal conductivity due to its strong covalent bonding and light constituent atoms, which permit long phonon mean free paths. With increasing temperature, κ decreases as phonon-phonon Umklapp scattering becomes more significant. Despite this reduction, the thermal conductivity remains relatively large across the examined temperature range, which is unfavorable for thermoelectric energy conversion because it limits the maintenance of temperature gradients.

Figure 12 illustrates the calculated electrical conductivity as a function of temperature. Owing to the ultra-wide band gap of AlN, the intrinsic carrier concentration is extremely low at low temperatures, resulting in very small σ values. As temperature increases, σ rises gradually due to thermal activation of electrons across the band gap, although the increase remains limited by strong phonon scattering at elevated temperatures. Consequently, the electrical conductivity remains orders of magnitude lower than that required for efficient thermoelectric materials.

The combined temperature dependence of S, σ and κ yields the thermoelectric figure of merit shown in Figure 13. Although moderate Seebeck coefficients are

Figure 10. Graph of see-beck coefficient against temperature.

Figure 11. A graph of Thermal conductivity against temperature.

obtained at elevated temperatures, the simultaneously low electrical conductivity and high thermal conductivity result in a very small power factor (S²σ) and correspondingly low ZT values across the entire temperature range

The computational results show that pristine AlN exhibits intrinsically poor thermoelectric performance due to its ultra-wide band gap, which suppresses electrical conductivity and its high lattice thermal conductivity, which limits temperature-gradient retention. As a result, the thermoelectric figure of merit remains very low across the investigated temperature range. Nevertheless, these same characteristics provide excellent thermal stability and mechanical robustness, making AlN well suited for piezoelectric energy-harvesting applications. Enhancement of thermoelectric performance in AlN would require controlled doping to increase carrier concentration or alloying strategies, such as AlGaN or Sc-doped AlN, to reduce lattice thermal conductivity while preserving the non-centrosymmetric wurtzite structure and piezoelectric functionality. Accordingly, compositionally engineered AlN-based systems represent a promising route toward multifunctional energy-harvesting materials.

Figure 12. A graph of electrical conductivity against temperature.

Figure 13.A graph of ZT against temperature.