Investigation of Piezoelectric Properties of Wurtzite AlN Films under In-Plane Strain: A First-Principles Study

Abstract

This research article presents a comprehensive first-principles study on the piezoelectric properties of Wurtzite Aluminum Nitride (AlN) films under in-plane strain conditions. By calculating the piezoelectric tensor coefficients (e33, e31, and e15), we investigate the variation patterns of these constants with respect to in-plane strain. Our results indicate significant changes in the piezoelectric constants within the range of in-plane strain considered, exhibiting a linear trend despite opposite trends for e33 compared to e31 and e15. This study highlights the extreme sensitivity of AlN films’ piezoelectric performance to in-plane strain, suggesting its potential as an effective means for tuning and optimizing the piezoelectric properties of AlN-based devices.

1. Introduction

Aluminum nitride (AlN) with a wurtzite structure, as a crucial III–V nitride, exhibits exceptional physicochemical properties, including outstandingly high piezoelectric performance, excellent thermal stability, and chemical stability. Notably, as a typical piezoelectric material, it holds significant potential for applications in electronic devices and sensors [1]. With the rapid advancement of micro-electro-mechanical systems (MEMS) technology, micro-nano sensor devices and actuators based on AlN piezoelectric films have gradually come into the spotlight [2], occupying a pivotal position in high-performance electronic devices such as acoustic devices [3,4], biosensors [5], and energy harvesters [6]. The piezoelectric properties of AlN materials directly impact the sensitivity and efficiency of these devices, underscoring the importance of in-depth understanding and optimization of their properties [7]. The piezoelectric performance of AlN is quantitatively described by piezoelectric tensor coefficients (e33, e31, and e15), which accurately reflect the electromechanical coupling effect of the material under different strain conditions.

In practical applications, AlN films are often subjected to in-plane strains originating from the substrate or the surrounding environment. Such strains can significantly alter the lattice structure and electronic state density of the material, thereby influencing the manifestation of its piezoelectric properties. In-plane strain, as an effective means of regulating material properties, can modify the electronic structure and physical properties of materials by adjusting lattice constants and crystal symmetry. The variation in piezoelectric properties of AlN films under different in-plane strain conditions holds significant theoretical and practical implications for understanding the relationship between in-plane strain and piezoelectric properties, as well as providing new design ideas and methods for optimizing and adjusting AlN-based devices [8]. By investigating this relationship, researchers can potentially tailor the piezoelectric properties of AlN films to meet specific performance requirements in various electronic and sensor applications.

Despite some research having focused on the piezoelectric properties of AlN films, the majority of these studies have concentrated on theoretical calculations in the free state or at the macroscopic scale [9,10]. There is a lack of understanding regarding the specific effects under constrained conditions, particularly the mechanisms underlying the variation in piezoelectric properties of AlN films under in-plane strain conditions. The lack of a systematic theoretical framework and comprehensive first-principles calculations imposes limitations on the in-depth understanding of how the piezoelectric properties of AlN films change under in-plane strain.

Specifically, our preliminary research has uncovered the existence of two distinct deformation modes in thin film materials under in-plane strain: the elastic deformation mechanism without internal relaxation, and the free deformation mechanism that allows for sufficient internal relaxation [11,12]. These two modes can lead to significant differences in the crystal structure of the strained thin film, with, for instance, a potential 50% discrepancy in the Poisson’s ratio, as well as a range of structural-related properties such as stability, electronic structure, and optical properties. Given the well-known sensitivity of piezoelectric effects to crystal structure, a meticulous investigation into the variation patterns and underlying mechanisms of piezoelectric properties in typical piezoelectric materials like AlN under different deformation modes holds significant theoretical and practical importance.

In this study, we leverage first-principles calculation methods to systematically investigate the piezoelectric properties of AlN films under various in-plane strain conditions. By calculating the piezoelectric tensor coefficients (e33, e31, and e15), we explore how these coefficients vary with changes in in-plane strain. Through an in-depth exploration of the piezoelectric characteristics of AlN films under in-plane strain conditions, we aim to provide theoretical support for optimizing the design of AlN-based piezoelectric devices and offer new scientific insights for their applications in novel sensors, electronic devices, and beyond. This work fills a critical gap in the current literature and contributes to advancing the understanding and utilization of AlN films as high-performance piezoelectric materials.

2. Modeling and Computational Details

Aluminum nitride (AlN) adopts a wurtzite crystal structure, which is a type of hexagonal close-packed (hcp) arrangement with a point group of 6 mm and a space group of P63mc. This structure features four atoms per unit cell, with aluminum (Al) and nitrogen (N) atoms alternating along the [001] crystallographic direction, as illustrated in Figure 1.

While AlN can also exist in a cubic structure, the wurtzite phase exhibits superior thermal stability at high temperatures. This enhanced stability stems from the unique atomic arrangement of the wurtzite structure, which maintains a more robust morphological integrity under elevated temperature conditions. Specifically, the alternating stacking of Al and N layers along the [001] direction creates strong covalent bonds that resist structural degradation, ensuring the stability of the crystal lattice even at high temperatures.

The wurtzite structure of AlN is thus particularly well-suited for applications requiring high-temperature performance, such as in electronic devices, sensors, and other high-temperature environments. The understanding of its crystal structure and the mechanisms underlying its thermal stability is crucial for the design and optimization of AlN-based materials and devices.

Based on the density functional theory (DFT), a first-principles plane-wave pseudopotential method was employed using the Materials Studio and Vienna Ab-initio Simulation Package (VASP) software for simulation calculations. The simulation process began with the structural optimization and electronic self-consistency calculations of the AlN material. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof for Solids (PBESOL) functional was adopted to treat the electronic exchange-correlation energy.

During the calculations, a cutoff energy of 1000 eV was used, and the Brillouin zone was sampled with a K-point grid density of 10 × 10 × 5. The structural optimization was performed with a force convergence criterion of −0.01 eV/nm, and the electronic relaxation was considered converged when the energy change fell below 1 × 10⁻⁶ eV. The valence electron configurations for Al and N atoms were specified as 3s² 3p¹ and 2s² 2p³, respectively.

After geometric optimization, the lattice parameters of the AlN crystal were found to be a = b = 3.125 Å and c = 5.008 Å, which is in good agreement with previous reports [9,10,13,14].

To calculate the piezoelectric constants using VASP, the INCAR file was configured with IBRION set to 8 to enable the finite difference method for strain-stress tensor calculation, and LEPSILON was set to TRUE to activate the calculation of the piezoelectric tensor. The POSCAR file, which contains the optimized lattice structure, was generated by importing the cell into VESTA software and then exporting it in the POSCAR format. It is crucial to ensure that the choice of pseudopotentials and their ordering in the POTCAR file match the atom ordering in the POSCAR file to avoid computational errors. The K-point mesh for the piezoelectric calculations was set to 10 × 10 × 10 to ensure adequate sampling of the Brillouin zone.

After completing the setup of the four files, proceed to calculate the piezoelectric coefficients of the geometrically optimized AlN. The piezoelectric matrix distribution for the wurtzite structure is as shown in the following matrix [10]:

$$\left(\begin{array}{c}0\\ 0\\ \mathrm{e}31\end{array}\begin{array}{c}0\\ 0\\ \mathrm{e}31\end{array}\begin{array}{c}0\\ 0\\ \mathrm{e}33\end{array}\begin{array}{c}0\\ \mathrm{e}15\\ 0\end{array}\begin{array}{c}\mathrm{e}15\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\end{array}\right)$$

(1)

Upon successful completion of the calculation, the total piezoelectric coefficient matrix of AlN is obtained by adding the ionic and electronic parts of the piezoelectric matrix in the OUTCAR file. Based on the above steps, the calculated piezoelectric coefficients of AlN are as follows: e31: −0.55 C·m⁻², e33: 1.41 C·m⁻², e15: −0.28 C·m⁻². A comparison with the literature values is presented in the following

Table 1. Piezoelectric coefficients of wurtzite AlN.

	e31 (C·m−2)	e33 (C·m−2)	e15 (C·m−2)
This work	−0.55	1.41	−0.28
Calculation [10]	−0.58	1.46	−0.29
Calculation [9]	−0.53	1.50	/
Calculation [15]	−0.60	1.46	/
Calculation [16]	−0.45	1.54	/
Experiment [17]	−0.58	1.55	/
Experiment [18]	−0.60	1.50	/

:

By comparing with the literature, it can be seen that the calculated piezoelectric coefficients of AlN in this study are consistent with other research, demonstrating the reliability of this study.

2.1. Calculation of Strain-Dependent Piezoelectric Coefficients of AlN

2.1.1. Free Relaxation Model

A series of geometric optimizations are performed on the previously optimized AlN-unit cell with applied in-plane strains of −5%, −3%, −1%, 1%, 3%, and 5% in the ab-plane. The methodology employed for these optimizations follows previous reports [11], with a key aspect being that during the optimization process, the lattice parameters a and b are fixed while c is allowed to relax freely, permitting internal relaxations to occur.

By applying these different levels of in-plane strain and optimizing the structure accordingly, we can calculate and analyze the mechanisms and trends governing the changes in crystal structure, cohesive energy, bandgap width, electron distribution, and piezoelectric coefficients of AlN under various strain conditions.

2.1.2. Biaxial Strain Model

In the biaxial strain model, rather than allowing the lattice parameter c to relax freely for each in-plane strain as in the previous approach, the strain value of c under biaxial strain is determined based on the elastic constants of the equilibrium wurtzite AlN crystal using the relationship ν = −2C₁₃/C₃₃ [19]. Here, the elastic constants matrix of AlN is calculated, yielding C₁₃ = 96 GPa and C₃₃ = 355 GPa, from which the Poisson’s ratio ν is found to be −0.541.

Using this Poisson’s ratio, the strain value of c under various in-plane strains can be determined, thereby establishing the values of c for the biaxial strain model. This approach ensures that the change in c is consistent with the mechanical response of the AlN crystal, as defined by its elastic properties.

With these strained lattice parameters, the methods described earlier are then applied to calculate the changes in cohesive energy, bandgap, electron distribution, and piezoelectric coefficients of the wurtzite AlN crystal under different levels of biaxial strain.

3. Results and Discussion

The cohesive energy of a material, defined as the energy required to separate the atoms from their bonded state into neutral, free atoms, serves as a metric to evaluate the thermal stability of crystalline materials [20]. Figure 2 presents the variations in the cohesive energy of strained AlN under different computational models. As can be observed, regardless of the model used, the cohesive energy after strain exhibits a similar parabolic shape. However, the absolute values of the cohesive energy under the free relaxation model are larger compared to those under the biaxial strain model. This indicates that the wurtzite-structured AlN exhibits superior thermal stability under the free relaxation model compared to the biaxial strain model, a phenomenon that is consistent with the behavior observed in other wurtzite-structured binary compounds [11].

The higher cohesive energy in the free relaxation model suggests that when the lattice parameter c is allowed to freely relax in response to in-plane strain, the system finds a more stable configuration with stronger interatomic bonds. In contrast, in the biaxial strain model, where the value of c is constrained by the Poisson’s ratio derived from the elastic constants, the system may not achieve the same level of stabilization, resulting in a lower cohesive energy.

The parabolic shape of the cohesive energy curve under strain is typically attributed to the competition between elastic and anelastic effects. At low strain levels, the elastic response dominates, and the cohesive energy increases due to the strengthening of interatomic bonds. However, as the strain increases further, anelastic effects, such as bond-breaking and lattice distortions, become more pronounced, leading to a decrease in the cohesive energy.

The findings regarding the thermal stability of AlN under different strain models are important for understanding its behavior in devices where mechanical stress is present, such as piezoelectric sensors and actuators. By selecting the appropriate strain model, engineers can optimize the design of these devices to ensure their stability and reliability under operating conditions.

Figure 3 illustrates the fundamental behavior characteristics of the wurtzite AlN crystal under strain conditions, specifically how the out-of-plane strain (ε_zz) varies with changes in the in-plane strain (ε_xx). The clear linear relationship depicted in the figure demonstrates a direct link between the two: as the in-plane strain increases, the out-of-plane strain decreases linearly, consistent with previous research findings [21].

The presence of a negative slope is a significant indicator, suggesting that the material exhibits a stronger resistance to volume changes than to shape changes when subjected to strain. In other words, upon compression or tension, the material tends to maintain its volume stability more effectively than its shape.

Further analysis reveals that the slope under the free relaxation model (−0.31) is only 57.3% of the corresponding Poisson’s ratio (typically close to −0.5, but specific values depend on the material) under the biaxial strain model. This notable difference underscores the role of internal relaxation mechanisms in the wurtzite AlN crystal. Under in-plane strain conditions, internal relaxation slows down the rate of change in the c-axis (the axis perpendicular to the plane), enabling the crystal to more effectively resist the effects of external strain, thus maintaining higher thermal stability and experiencing smaller volume changes.

In contrast, Figure 3 also shows that under identical in-plane strain conditions, the c-axis response of the AlN crystal in the biaxial strain mode is more sensitive and rapid. This means that when the crystal is subjected to uniform strain in two directions simultaneously, the change in its c-axis is more pronounced, consistent with experimentally observed phenomena [22]. This finding highlights the complexity and diversity of material behavior under different strain conditions, which are of great significance for research and applications in materials science and engineering.

In summary, Figure 3 not only validates the basic behavior characteristics of the wurtzite AlN crystal under strain conditions but also reveals the crucial role of internal relaxation mechanisms in modulating these characteristics. These discoveries provide valuable insights for material design, optimization, and performance prediction.

The wurtzite-structured compounds are characterized by three primary piezoelectric coefficients: the longitudinal piezoelectric coefficient e33, the transverse piezoelectric coefficient e31, and the transverse shear piezoelectric coefficient e15. Among these, e33 describes the degree of mechanical strain response along the direction parallel to the applied electric field when the field is applied along the primary piezoelectric axis of the material. The piezoelectric coefficient e31 quantifies the mechanical strain response perpendicular to the electric field direction when the field is applied along the primary piezoelectric axis. Lastly, e15 represents the ratio of the transverse shear strain generated in the material to the applied electric field, and it is also a crucial parameter for piezoelectric performance.

Figure 4 displays the variations of the piezoelectric coefficients e31, e33, and e15 of wurtzite AlN under two different strain models as a function of in-plane strain. The figure prominently illustrates the extreme sensitivity of the piezoelectric properties of AlN thin films to in-plane strain. Under the biaxial strain model, the curves for the piezoelectric coefficients e31, e33, and e15 are relatively steep, indicating a faster response speed of these coefficients to strain compared to the free relaxation model. In contrast, under the free relaxation model, the piezoelectric coefficient curves are relatively gentle, suggesting a slower response speed to strain compared to the biaxial strain model.

Within the range of in-plane strain considered, significant changes in the piezoelectric coefficients are observed for both models. As the applied strain varies from −5% to 5%, the values of e31 and e15 decrease, while e33 exhibits an opposing but still linear trend. The piezoelectric properties of wurtzite AlN are among the key indicators determining its application performance. The sensitivity of the piezoelectric coefficients to strain shown in the figure can explain the variability in piezoelectric performance observed in experimentally prepared AlN thin films. It also hints at an effective approach for widely tuning the piezoelectric properties of this material through strain engineering, where specific piezoelectric responses in wurtzite AlN thin films can be achieved based on the working curves presented in the figure.

The electronic properties of crystalline materials are primarily determined by the types of elements that compose the material and the topological arrangement of these elements, which define the overall distribution of various fields, such as the electric charge (electric field), and thus the electronic structure of the material [14]. Applying strain to a material involves adjusting the positions of atoms, inevitably affecting its electronic structure.

Figure 5 demonstrates the changes in the bandgap values of AlN under the free relaxation model and the biaxial strain model as a function of applied strain. It is evident from the figure that the bandgap values vary significantly with the applied strain. In the case of tensile strain, both the free relaxation model and the biaxial strain model exhibit a consistent trend of decreasing bandgap values. However, under compressive strain, the behavior differs. For the free relaxation model, the bandgap initially increases with increasing strain and then decreases. In contrast, under biaxial strain, the bandgap value continuously increases with increasing strain.

These observations indicate that strain engineering can be employed as an effective means to tune the properties of AlN materials. By applying carefully controlled strains, researchers can manipulate the bandgap and other electronic properties of AlN, potentially enabling the development of materials with tailored characteristics for specific applications. This capability is of significant interest for the advancement of microelectronics, optoelectronics, and other fields where precise control over material properties is crucial.

Figure 6 presents the differential electron density of wurtzite-structured AlN under various strain models. The differential electron density maps reveal that wurtzite-structured AlN contains two distinct types of Al-N covalent bonds: one that is parallel to the c-axis (designated as B1) and another that lies approximately within the ab-plane (designated as B2). At equilibrium, the differences between these two bonds are minimal, as evidenced by their very similar bond lengths and valence electron cloud density values.

Upon the application of in-plane strain, compressive strain leads to a decrease in electron density between the atoms of B1 bonds and an increase in electron density for B2 bonds. Conversely, tensile strain produces the opposite effect, with B1 bonds exhibiting an increase in electron density and B2 bonds showing a decrease.

When comparing the different strain models, it is notable that the amplitude of changes in electron density is smaller under the free relaxation model. This indicates that the electron density distribution exhibits a higher degree of robustness in this model, potentially explaining why the changes in piezoelectric properties are more modest. The stability of the electron density distribution under free relaxation conditions may suggest that the material is better able to maintain its intrinsic electronic structure against external perturbations, such as applied strain.

This insight into the behavior of differential electron density under strain is valuable for understanding and predicting the performance of AlN-based devices under varying conditions. By tailoring the strain conditions and strain models, researchers can optimize the electronic properties and piezoelectric response of AlN for specific applications, such as sensors, actuators, and energy-harvesting devices.

4. Conclusions

The article concludes that through a detailed first-principles investigation, significant variations in the piezoelectric coefficients of wurtzite aluminum nitride (AlN) have been observed under different strain conditions. Specifically, the study reveals that free relaxation results in enhanced thermal and structural stability of the material, while in-plane strain offers a powerful means to tailor its crystal structure and piezoelectric performance. These findings provide valuable insights into the strain-dependent properties of wurtzite AlN and underscore its potential for high-performance piezoelectric devices, such as sensors, actuators, and transducers. The study’s results contribute to the advancement of microelectronics, energy harvesting, and sensing technologies by offering a theoretical foundation for the optimization of AlN-based devices.