The elasticity and piezoelectricity of AlN containing charged vacancies

Abstract

Wurtzite aluminum nitride (w-AlN) piezoelectric film plays a key role in modern MEMS actuators and sensors. Density functional theory can be used to predict the elastic and piezoelectric properties of AlN containing various charged vacancies, thereby obtaining methods for improving the piezoelectric properties. The theoretical investigations suggested that the Young’s modulus of defectants are smaller than that of intrinsic AlN, indicating that charged vacancies can effectively improve the elastic properties of AlN. A high piezoelectric coefficient d₃₃ values were obtained in ${\mathrm{V}}_{\text{Al}}^{3-},$ ${\mathrm{V}}_{\mathrm{N}}^{1+}$ and ${\mathrm{V}}_{\mathrm{N}}^{3+}$ that may be a cause of the non-symmetrical charges assignation. This work suggests that the elasticity and piezoelectricity of AlN films can be effectively enhancing using the introduction of charged vacancies.

Keywords:

• AlN
• charged vacancy
• piezoelectricity
• elasticity
• DFT calculation

1. Introduction

Piezoelectric materials, capable of transforming mechanical energy into electrical signals or vice versa, play a crucial role as components in various piezoelectric sensors and actuator systems.^[1–12] Lead based piezoelectric materials (such as lead zirconium titanite (PZT)) have a high piezoelectric coefficient (d₃₃ ≈ 700 pC/N), but their toxicity and lower Curie temperature (T_c ≈ 180° C) limit their application in high-temperature sensor or actuator systems.^[13,14] Wurtzite aluminum nitride (AlN) thin film is one of the most promising candidates for piezoelectric applications.^[15,16] AlN exhibits many excellent physical and chemical properties, such as non-toxicity, high coupling constant (0.25) and high Curie temperature (T_c ≈ 450 °C).^[17–20] Importantly, AlN is good compatibility with CMOS manufacturing processes.^[21] Therefore, AlN thin films exhibits vast potential across various applications and researches, including sensor thin films such as piezoelectric and wearable sensors,^[22–33] transducers for energy harvesters,^[34–40] the actuators for optical MEMS actuators,^[41–46] as well as in Photonics materials, exemplified by AlN waveguides and their diverse applications.^[47–55] With the increasing demands, it becomes crucial to anticipate methods that can enhance the piezoelectric properties of AlN. This is essential for optimizing the properties and process development of AlN preparation.

Formation of various defects, such as vacancies, interstitials, substitutions, and dislocations are often generated during the growth of AlN crystals.^[56,57] Compared to other defects, the vacancy defect has a lower formation energy which is most likely obtain in crystals.^[57] Vacancy-type defects can introduce new characteristics to affect the material properties. Liu et al. studied the effect of nitrogen vacancy (V_N) on the magnetism of AlN, and they found that the magnetization of AlN can be tuned through defect engineering.^[58] Xu et al. found that aluminum vacancy (V_Al) play an important role in reducing the thermal conductivity of AlN due to the largest mass-difference scattering.^[59] The vacancy defects also affect the elastic properties of AlN. The vacancy defect-induced extra space could accommodate small displacements between particles under the external forces. Consequently, the crystal material with vacancy defect is easier to deformed. The improved elastic properties will play a positive role in the piezoelectric effect. Sharma et al. studied the influence of vacancies on the piezoelectric properties of AlN through experimental methods. They proposed that the polarization enhanced can be induced by charged defects in AlN, which can improve the d₃₃ value.^[60] Though several investigation of vacancy-type AlN (V-AlN) has been reported,^[61–63] the piezoelectric properties of V-AlN have rarely been discussed.

In the present study, theoretical studies based on density functional theory are performed to understand the effect of vacancies on the piezoelectric and elastic properties of AlN. By comparing the formation energy, electronic properties, Young’s modulus, and piezoelectric coefficients of V_N– and V_Al–AlN, the elasticity and piezoelectricity of these V-AlN were acquired. All the possible charge states of V-AlN, from 3⁻ to 3⁺, are included in this work. The scope of this paper is to fill some of the gaps in understanding the mechanism of piezoelectric properties in V-AlN.

2. Computational methods

V-AlN with various charge states are investigated using ab initio calculation based on density functional theory carried out in Vienna ab initio simulation package (VASP).^[64,65] The interaction between valence electrons and atomic nuclei was described by projector-augmented-wave (PAW) method.^[66,67] The exchange-correlation potential was the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) function.^[68] The electron wave function is expanded in plane waves up to a cutoff energy of 520 eV. The convergence criterion of total energy is set as 1 × 10⁻⁸eV. And the atomic coordinates are fully relaxed until the maximum force on a single atom is less than 0.02 eV/Å. The k-space integrations of 32- and 72-atoms wurtzite supercells are performed by using the Monkhorst-Pack sampling scheme with a 9 × 9 × 3 and 4 × 4 × 3 k-point mesh special, respectively.^[69] A similar k spacing is used in the defect calculation. The density-functional perturbation theory (DFPT) in VASP code was used to calculate elastic and piezoelectric stress tensors of AlN-based systems.^[70]

The pure wurtzite AlN primitive cell is firstly optimized. The lattice parameters were selected based on minimization of the energy of the perfect lattice, which was achieved at a = 3.13 Å and c/a = 1.60 for pure AlN (experiments: a = 3.11 Å and c/a = 1.60).^[71] Then 32- and 72-atoms supercells are created with the above lattice constants. Defects were created by the removal of a lattice atom of an appropriate type. The atoms were allowed to relax, but the lattice constants of supercells are fixed during geometry optimization calculations of defective configuration.^[72] The optimized structures of 32-atoms supercells are shown in Figure 1.

Figure 1. Atomic geometry of pure and native point defects in 32-atom AlN in the ideal (unrelaxed) structure. (a) Pure AlN, (b) V_N-AlN and (c) V_Al-AlN. Large blue and small gray spheres represent Al an N atoms, respectively. Vacancies are indicated by the red circles.

3. Results and discussion

The formation energy can explain the stability of defective crystals to a certain extent. For a given temperature and in the limit of low concentrations, the logarithm of the concentration is proportional to the formation energy. Therefore, a low formation energy value implies a high equilibrium concentration of defects, while a high value indicates a low equilibrium concentration.^[73] Following Zhang-Northrup formalism,^[74] the formation energy of a vacancy in charge state q is calculated from: (1) $$E_f^{\mathit{\text{def}}}=E_{\mathit{\text{tot}}}^{\mathit{\text{def}}}-E_{\mathit{\text{tot}}}^{\mathit{\text{bulk}}}+{\mu }_i+q\left[E_V^{\mathit{\text{def}}}+E_F\right]$$(1) where $E_{\mathit{\text{tot}}}^{\mathit{\text{def}}}$ is the total energy of the supercell containing a vacancy in charge state q and $E_{\mathit{\text{tot}}}^{\mathit{\text{bulk}}}$ is the total energy of the pure AlN bulk crystal in the same supercell. μ_i indicates the chemical potential of atoms that removed from pure supercell. To determine these quantities, we using the chemical potentials of the N₂ molecule and bulk Al depending on the assumed growth conditions (Al or N rich).^[74] Furthermore, the chemical potentials satisfy the boundary conditions: (2) $${\mu }_{\text{Al}}+{\mu }_{\mathrm{N}}=E\left(\text{AlN}\right)$$(2) (3) $${\mu }_{\text{Al}}\le E\left(\text{Al}\right)$$(3) (4) $${\mu }_{\mathrm{N}}\le \frac{1}{2}E\left({\mathrm{N}}_2\right)$$(4)

Furthermore, $E_V^{\mathit{\text{def}}}$ is the valence band maximum for the defect supercell and E_F is the Fermi energy, which is set to zero at the valence-band maximum. In this work, the formation energy is calculated in the infinite crystal, so $E_V^{\mathit{\text{def}}}$ can be instead by the valence band maximum in the bulk $E_V^{\mathit{\text{bulk}}}$ and without any additional corrections.^[75]

To simulate isolated defects, the formation energies and electronic structures of 32- and 72-atoms AlN supercell were calculated. The results obtained with the larger cell were very similar to the 32-atoms cell (As shown in Figures 2 and 3). Therefore, the majority of our calculations are performed using 32-atoms wurtzite supercells and it was found to be adequate in earlier work on similar systems.^[63]

Figure 2. Formation energies of vacancies with different charge states in AlN as calculated by DFT. Formation energies as a function of the Fermi level is shown for the extreme growth conditions: 32-atom supercell under (a) Nitrogen rich and (b) Aluminum rich; 72-atom supercell under (c) Nitrogen rich and (d) Aluminum rich.

Figure 2 presents the calculated defect formation energies for the various charge states of N- and Al-rich conditions as a function of the Fermi level (E_F) in w-AlN. The change in slope ration of the lines between Fermi level and defect formation energy represents a change in the charge state of the defect. Our calculations reveal that the formation energy of V_N-AlN has a very low formation energy under Al-rich conditions. The V_N in the triply positively charged state (${\mathrm{V}}_{\mathrm{N}}^{3+}$) has the lowest formation energy under p-type conditions (low E_F values). The formation energy of V_Al-AlN has a very low formation energy under N-rich conditions. The V_Al in the triply negatively charged state (${\mathrm{V}}_{\text{Al}}^{3-}$) has the lowest formation energy under n-type conditions (large E_F values). In thermodynamic equilibrium, the concentration of N vacancy under Al-rich conditions and the concentration of Al vacancies under N-rich conditions should therefore be relatively high. It is presenting a negative-U effect between triply and singly positive charge states owing the doubly positively charged nitrogen vacancy is unstable, which is consistent with previous reports.^[72,75–77]

Additionally, ${\mathrm{V}}_{\text{Al}}^{3-}$–AlN has the lowest formation energy in both N-rich and Al-rich. The formation energies of ${\mathrm{V}}_{\mathrm{N}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{3+}$– and ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN, respectively under Al-rich conditions are all low. Defect formation energy greater than zero indicates an endothermic process, in reverse implies an exothermic process. In other words, the formation energy of the material is less than 0 eV, meaning that the structure will be more stable. Stability is the basis for materials to have other properties. Therefore, the subsequent property studies mainly focus on the 3⁻ charge states V_Al-AlN, as well as 3⁻, 3⁺ and 1⁺ charge states V_N–AlN.

Electronic properties will directly determine various macroscopic properties of materials. Therefore, the density of states of pure AlN and four vacancy configurations are calculated as shown in Figure 3. It can be observed that AlN is a semiconductor, exhibiting a calculated band gap (E_g) of 5.97 eV (The experimental value is 6.28 eV.^[71] After introducing a vacancy in the bulk AlN, ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN maintain semiconductor properties and the band gap values have all decreased. The reduced E_g will favor the transition of electrons from valence band to conduction band, thereby improving electron mobility. This enhancement will be conducive to improve the energy conversion efficiency in AlN-based piezoelectric devices. The E_F of ${\mathrm{V}}_{\mathrm{N}}^{3-}$–AlN moves upward and passes through the conduction band. Consequently, it leads a metallic property, which will have a negative impact on the piezoelectric properties of the system. Therefore, the following piezoelectric and elastic studies will be mainly conducted on the ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN configurations.

Figure 3. The density of states of bulk AlN (a) and the 32-atoms supercell containing an Al or N vacancy (b–e). The density of states of bulk AlN (f) and the 72-atoms supercell containing an Al or N vacancy (g–j). Fermi level is set to zero.

Figure 4 depicts the three-dimensional (3D) charge density difference of pure AlN, ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN systems, which can aid in understanding the charge transfer between atoms and exploring the mechanisms of piezoelectricity. In the plot, blue/yellow regions indicate the depletion/accumulation of charges. It can be observed from the figure that in the several vacancy systems, N and Al atoms maintain ionic bonds because charges consistently accumulate on the N atoms while depleting on the Al atoms. Comparing the 3D charge density of intrinsic AlN, the atomic charge density near the Al vacancy increases, and the atomic charge density near the N vacancy decreases. Consequently, the charge distribution in V–AlN is no longer symmetrical, which should facilitate the generation of piezoelectric properties.^[78] When strain is applied, charge redistribution occurs, leading to charge polarization and promoting the generation of the piezoelectric effect.

Figure 4. The three-dimensional (3D) charge density difference of AlN with vacancy defects. (a) AlN, (b) ${\mathrm{V}}_{\text{Al}}^{3-}$–AlN, (c) ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN and (d) ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN.

The elastic constants of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN were calculated to analyze their elastic properties. The elastic stiffness coefficients of the three defective configurations are shown in Figure 5. It can be seen that the calculated elastic constant of the three systems satisfied the mechanical stability criterion of hexagonal structure (C₄₄>0, C₆₆>0, C₁₁>|C₁₂|, and (C₁₁+2C₁₂)C₃₃>2C₁₃²).^[79,80]

Figure 5. The elastic coefficients C₁₁, C₁₂, C_13, C₃₃, C₄₄ and C₆₆ of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN.

Young’s modulus (E) characterizes the difficulty degree of elastic deformation under certain stress. Therefore, their E were calculated based on the elastic constants.^[81] The Voigt bulk modulus (B_V), Reuss bulk modulus (B_R), Voigt shear modulus (G_V) and the Reuss shear modulus (G_R) of hexagonal lattices are: (5) $$B_V=\frac{1}{9}\left[2\left(C_{11}+C_{12}\right)+C_{33}+4C_{13}\right]$$(5) (6) $$B_R=\frac{\left[\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2\right]}{\left(C_{11}+C_{12}+2C_{33}-4C_{13}\right)}$$(6) (7) $$G_V=\frac{1}{30}\left(7C_{11}-5C_{12}+12C_{44}+2C_{33}-4C_{13}\right)$$(7) and (8) $$G_{\mathrm{R}}=\frac{5}{2}\left\{\frac{\left[\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2\right]C_{44}{C}_{66}}{3B_v{C}_{44}{C}_{66}+\left[\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2\right]\left(C_{44}+C_{66}\right)}\right\}$$(8)

Based on the approximated by Hill’s average, the Bulk moduli and Shear moduli can be obtained as (9) $$B=\frac{1}{2}\left(B_V+B_R\right)$$(9) (10) $$G=\frac{1}{2}\left(G_V+G_R\right)$$(10)

The E can be given by (11) $$E=\frac{9\mathit{\text{GB}}}{3B+G}$$(11)

The E of pure AlN and different charge states defectant are plotted in Figure 6(b). The results show that the values E of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN are inferior to those of intrinsic AlN, indicating that the defectants are more flexible. Among that, the singly positive charge states V_N–AlN has the smallest Young’s modulus. Theoretically, a small Young’s modulus is more beneficial to obtain piezoelectric voltages.^[78,82] Therefore, ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN monolayers shall possess the larger piezoelectric coefficients.

Figure 6. Young’s modulus and piezoelectric strain coefficients d₃₃ of pure AlN and different charged state defectants.

The piezoelectric strain coefficients d₃₃ value of materials directly affects the conversion efficiency between electrical energy and mechanical energy in devices such as piezoelectric transducers. Therefore, the piezoelectric stress matrix e_ij is calculated and d₃₃ is obtained using the formula d₃₃=e₃₃/C₃₃. The e₃₃ and d₃₃ value for pure AlN and three charged vacancy configurations are presented in Figure 6(a,c). Evidently, compared with intrinsic AlN, the d₃₃ of the three vacancy configurations has changed significantly. The ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN all have large d₃₃ values. In particular, the d₃₃ value of ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN is 243.692 pC/N, which is 48 times higher than intrinsic AlN. This result is consistent with the analysis in the Young’s modulus section. The above results demonstrate that the presence of these three charged vacancies (${\mathrm{V}}_{\text{Al}}^{3-},$ ${\mathrm{V}}_{\mathrm{N}}^{1+}$ and ${\mathrm{V}}_{\mathrm{N}}^{3+}$) play a significant role in enhancing the piezoelectric properties of AlN. Our results prove the conclusion that “Charged vacancies are the cause of improving the piezoelectric coefficient of AlN.” proposed by Sharma et al.^[60] Importantly, the calculations predict three types of charged vacancies that could enhance the piezoelectric properties of AlN. By comparing with other methods for enhancing the piezoelectricity of AlN (Table 1), it can be observed that the presence of charged vacancies can significantly enhance the piezoelectric and elastic properties of AlN. This insight can provide theoretical guidance for preparing materials with improved piezoelectric characteristics.

Table 1. d₃₃, d₃₃ Increase rate and E of AlN-based piezoelectric materials.

Materials	d33 (pC/N)	d33 increase rate	E (GPa)	Ref
Pure AlN	∼5.120	–	∼310.000	^[^20^,^83^,^84^]
${\mathrm{V}}_{\text{Al}}^{3-}$–AlN	34.821	6.916	237.801	–
${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN	243.692	48.400	120.535	–
${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN	100.993	20.058	222.859	–
ScxAl1-xN	24.600	∼4.731	–	^[^84^]
ScxAl1-xN	–	–	217.100	^[^85^]
(MgHf) xAl1-xN	16.417	3.205	∼208.000	^[^20^]
(MgZr) xAl1-xN	–	∼2.855	–	^[^86^]
(MgTi) xAl1-xN	–	∼2.498	–	^[^86^]
(MgHf) xAl1-xN	–	∼1.988	–	^[^86^]
Cr xAl1-xN	–	1.732	–	^[^87^]
Ta xAl1-xN	8.200	1.952	–	^[^83^]

4. Conclusions

In summary, the elastic properties and piezoelectric properties of different charges AlN with N and Al vacancies have been studied using first-principles method based on DFT. ${\mathrm{V}}_{\text{Al}}^{3-},$ ${\mathrm{V}}_{\mathrm{N}}^{3-},$ ${\mathrm{V}}_{\mathrm{N}}^{1+}$ and ${\mathrm{V}}_{\mathrm{N}}^{3+}$ were determined that the most likely existing vacancies in AlN through calculating the defect formation energy. The electronic structure and charge density of the stable configuration defects were studied. The symmetrical of AlN has been broken thought introducing charged defects. The band gaps of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN become smaller and maintain semiconductor properties. Consequently, compared with intrinsic AlN, charged vacancy AlN has lower Young’s modulus and good flexibility. The piezoelectric properties of ${\mathrm{V}}_{\text{Al}}^{3-}$–, ${\mathrm{V}}_{\mathrm{N}}^{1+}$– and ${\mathrm{V}}_{\mathrm{N}}^{3+}$–AlN are all significantly enhanced. The piezoelectric coefficients d₃₃ of ${\mathrm{V}}_{\mathrm{N}}^{1+}$–AlN are 48 times that of intrinsic AlN, which is a relatively good performance among piezoelectric film materials. Therefore, this investigation demonstrates that the piezoelectric materials with considerably high d₃₃ can be obtained by introducing charged vacancies. This conclusion can provide effective guidance for the preparation process of AlN piezoelectric films. Due to its excellent elastic properties, and beneficial piezoelectric performance, AlN with vacancies in different charge states possess great potential in piezo-electronics, sustainable energy, and sensors applications.

Nomenclature
• •	=	• • •

Acknowledgements

We extend our sincere appreciation to Hong Zhou, Dongxiao Li and Hengning Chen of NUS for invaluable guidance in this research. And the authors acknowledge support from Bianshui Riverside Supercomputing Center (BRSC).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Key Research and Development Program of China under [Grant number 2018YFF01011200] and Chongqing Science and Technology Commission; the Natural Science Foundation Project of Chongqing under [Grant cstc2020jcyj-msxmx0563].