First-Principles Study on the Mechanical Properties of Gd-Doped BCZT Ceramics

Abstract

Due to their remarkable piezoelectric characteristics, (BaCa)(ZrTi)O₃ (BCZT) ceramics exhibit vast potential for being employed in cutting-edge electromechanical apparatus. Extensive experimental studies have been conducted to better meet the practical needs of BCZT-based materials, focusing on their mechanical performance. However, there is a serious lack of research on the theoretical computational aspects. Here, first-principles calculations were utilized to evaluate the mechanical properties of BCZT-xGd ceramics. The structural models were established using the virtual crystal approximation (VCA) method. The investigated compounds demonstrate structural and mechanical strength, as evidenced by their negative formation energies and adherence to the Born stability criteria. Compared to pure BCZT, the substitution of Gd leads to a significant enhancement in the system’s elasticity and stiffness. The BCZT-0.05Gd with B-site doping demonstrates the highest level of Vicker’s hardness ($H_V$), with the noteworthy observation that the inclusion of Gd concomitantly augments its machinability performance. Upon the incorporation of the Gd element, the anisotropic elasticity in the systems gradually transitions into isotropic elasticity, which favors a more uniform stress distribution and consequently reduces sensitivity to the formation and propagation of microcracks. These results indicate that BCZT-xGd exhibits potential for application in electromechanical systems.

1. Introduction

Piezoelectric ceramics are pivotal functional materials for the conversion of mechanical vibrational energy, serving as vital constituents in the fabrication of various high-performance energy harvesters [1,2,3,4]. In light of the substantial mechanical cycling and intermittent impact loading encountered by energy harvesters, the piezoelectric ceramics layer necessitates a high level of strength and fracture toughness. In terms of lead-based piezoelectric ceramics, the discovery of lead zirconate titanate Pb(Zr, Ti)O₃ (PZT) and its family of materials in the 1950s was viewed as a major advancement in the realm of energy harvesting. Fracture behavior studies of PZT have revealed that fracture toughness (K_IC) strongly depends on the chemical composition and varies within the solid solution range of PbZrO₃ and lead titanate PbTiO₃ [5]. Furthermore, it is also significantly influenced by the microstructure, porosity, crystallography, and operating environment [6]. The fracture toughness (K_IC) attains its minimum magnitude at the morphotropic phase boundary, which could be associated with reduced elastic constants [7]. Phase transformations and domain switching can play an essential role in enhancing the toughness of the material, as illustrated by the results measured in a previous study for the composition at the ferroelectric–antiferroelectric phase boundary [6,8].

In the field of lead-free piezoelectric ceramics, the superior properties of perovskite-type (ABO₃) metal oxides, along with their manufacturing processes, which are largely analogous to those utilized for traditional lead-based piezoelectric ceramics, have made them a focal point of research [9,10,11]. Compared with other perovskite-type lead-free piezoelectric ceramics, BaTiO₃-based ceramics exhibit notable attributes, including an elevated dielectric constant, enhanced chemical stability, and commendable mechanical performance, making them potential substitutes for lead-based materials in certain applications [12,13,14,15,16]. Consequently, numerous experimental and theoretical calculation efforts have been devoted to investigating BaTiO₃-based materials, aiming to modulate their physical and chemical properties by introducing dopants. On the experimental side, hetero−structural BaTiO₃-based ceramics ((1−x)(0.8BT–0.2BNT)–xCZ) were successfully synthesized using the solid-state reaction method, and exhibit outstanding mechanical properties such as a Vickers hardness of approximately 9.7 GPa and a compressive strength of around 500 MPa [17]. Also, a structural correlation between energy storage and mechanical properties has been established, which is associated with the synergistic mechanisms of solid solution strengthening, densification strengthening, grain boundary strengthening, and twin boundary strengthening. On the theoretical calculation side, the structure and electromechanical properties of Ba_1–xSr_xTiO₃ (BST) (0 ≤ x ≤ 1) systems in paraelectric and ferroelectric phases were investigated utilizing first-principles calculations [18]. An analysis of the obtained elastic moduli indicated that Ba_1–xSr_xTiO₃ systems are elastically hard and mechanically stable. Moreover, the electromechanical properties of ferroelectric BST systems are considerably improved with Sr doping. These results show that Ba_1–xSr_xTiO₃ (BST) are good materials in the design and realization of electroacoustic devices such as microwave resonators and acoustic sensors.

Since the discovery of the exceptionally high piezoelectric performance (d₃₃~620 pC/N) of (1−x)Ba(Zr_0.2Ti_0.8)O₃–x(Ba_0.7Ca_0.3)TiO₃ (BCZT) ceramics [19], there has been an escalating and noticeable focus on employing doped modification methods for practical applications. (Ba_0.85Ca_0.15)(Zr_0.1Ce_xTi_0.9–x)O₃ (BCZCT) ceramics were successfully prepared using a solid-state sintering technique, and the observed soft ferroelectric nature of BCZCT was mainly attributed to the reduced crystalline anisotropy of the two phases [20]. Through a conventional solid-state sintering process, (Ba_0.85Ca_0.15)(Zr_0.10Ti_0.90)O₃–xBa (Cu_0.5W_0.5)O₃ (BCZT–BCW) ceramics were synthesized [21]. The hardness (78 MPa) and fracture toughness (4.7 MPa m^1/2) of the ceramics were achieved due to the improved density imparted by BCW additives. Furthermore, reduced graphene oxides (rGO) and PDMS have been added into BCZT–CuY ((Ba_0.85Ca_0.15)(Ti_0.9Zr_0.1)O₃–0.10wt%CuO_0.06wt%Y₂O₃) to increase their elasticity and flexibility [22]. The electrical outputs of BCZT–CuY/rGO–based piezoelectric nanogenerators (NGs) illustrate their potential for use in tiny energy harvesting. Although extensive experimental studies have been conducted on the mechanical properties of BCZT-based ceramics, there is still a lack of research on their theoretical calculations. Gd₂O₃, as a well-known rare earth dopant, has been utilized to enhance the electrical properties of BCZT ceramics, making it of significant value in electronic device manufacturing.

Hence, in this work, we aim to demonstrate a systematic analysis of the mechanical properties of Gd-doped (Ba_0.85Ca_0.15)(Zr_0.1Ti_0.9) O₃ (abbreviate as BCZT–xGd, x = 0–0.05) utilizing the first-principles calculation. For the purpose of comprehensively elucidating the underlying mechanism of BCZT and probing into the effects of diverse doping strategies on its performance, a series of theoretical calculations were meticulously carried out for both A-site and B-site doping scenarios in ABO₃ perovskites.

2. Computational Details

The calculations here were conducted using the density functional theory (DFT) formalism implemented in the Cambridge Serial Total Energy Package (CASTEP) [23]. The Perdew–Burke–Ernzerhof (PBE) function was employed within the framework of Generalized Gradient Approximation (GGA) to formalize the exchange-correlation energy [24,25,26]. A cut-off energy of 580 eV was selected to ensure convergence. The minimum energy tolerance was 10⁻⁶ eV per atom and the force tolerance was 0.001 eV/Å. Brillouin zone integration was conducted using the Monkhorst–Pack scheme convention, employing a 3 × 3 × 3 k-point mesh [27]. The optimized cell structures were obtained using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm as the minimization approach [28]. All structures were constructed utilizing the tetragonal BaTiO₃, and the virtual crystal approximation (VCA) method was adopted to add Gd for the replacement of Ba/Ca atoms or Ti/Zr atoms [29,30,31]. The pseudopotentials were generated based on the electron configurations of Ba 5s²5p⁶6s² states, Ca 3s²3p⁶4s² states, Ti 3s²3p⁶3d²4s² states, Zr 4s²4p⁶ 4d²5s² states, Gd 4f⁷5d¹6s² states, and O 2S²2p⁴ states, respectively. The pure and A/B-site-doped structures are presented in Figure 1, where different colors represent different elements, and the proportions of the colors represents their respective molar ratios.

3. Result and Discussion

3.1. Structural Stability

Formation energy (${\Delta E}_{for}$) is an essential thermodynamic parameter that has significant implications in materials research, as it offers valuable insights into the stability, feasibility, and potential applications of chemical reactions [32]. Therefore, the stability and experimental feasibility of the BCZT-xGd systems were evaluated using the concepts of ${\Delta E}_{for}$. For disordered VCA structures, determining ${\Delta E}_{for}$ requires considering the influence of mixed atoms due to the lack of a reasonable energy standard for pure elements. Regarding the substitution of the Ba/Ca site, due to the consideration of Ba, Ca, and Gd atoms as virtual mixed atoms, the A-site atoms’ energy should be replaced with the total energy of ${{(\mathrm{B}\mathrm{a}}_{0.85}{\mathrm{C}\mathrm{a}}_{0.15})}_{1-\mathrm{x}}{\mathrm{G}\mathrm{d}}_{\mathrm{x}}$for the systems. For the substitution of the Zr/Ti site, Zr, Ti, and Gd are considered as mixed atoms and the B-site atoms’ energy can also be replaced with the total energy of ${{(\mathrm{T}\mathrm{i}}_{0.90}{\mathrm{Z}\mathrm{r}}_{0.10})}_{1-\mathrm{x}}{\mathrm{G}\mathrm{d}}_{\mathrm{x}}$. Therefore, the formation energy of the BCZT-xGd VCA structure with A-/B-site doping can be, respectively, obtained using Equations (1) and (2) as follows [33]:

$${∆E}_{for}=\frac{1}{5}{(E}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{{\left({\mathrm{B}\mathrm{a}}_{0.85}{\mathrm{C}\mathrm{a}}_{0.15}\right)}_{1-\mathrm{x}}{\mathrm{G}\mathrm{d}}_{\mathrm{x}}}-E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{{\mathrm{T}\mathrm{i}}_{0.90}{\mathrm{Z}\mathrm{r}}_{0.10}}-3E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{\mathrm{O}})$$

(1)

$${∆E}_{for}=\frac{1}{5}{(E}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{{\left({\mathrm{T}\mathrm{i}}_{0.90}{\mathrm{C}\mathrm{a}}_{0.10}\right)}_{1-\mathrm{x}}{\mathrm{G}\mathrm{d}}_{\mathrm{x}}}-E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{{\mathrm{B}\mathrm{a}}_{0.85}{\mathrm{C}\mathrm{a}}_{0.15}}-3E_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}^{\mathrm{O}})$$

(2)

where $E_{total}$ and $E_{solid}^{\mathrm{O}}$ refer to the total energies of the BCZT-xGd systems and the chemical potential of the corresponding oxygen atom in the bulk phase. The calculated formation energies of the BCZT-xGd compounds are shown in Figure 2a. It can be observed that the formation energies of all doped structures are negative, indicating the thermodynamic stable architecture. Notably, our findings demonstrate a decreasing trend in ${\Delta E}_{for}$ with decreasing doping concentrations, implying a more facile synthetic route for the preparation of doped systems in experimental contexts [34]. Additionally, it was observed that the formation energy of A-site doping demonstrates a lower magnitude of energy in comparison to B-site doping. This phenomenon is accompanied by a discernible trend, as illustrated in Figure 2b, showing an initial increase followed by a subsequent decrease in the differences in formation energies. Thus, it can be concluded that the experimental routes for A-site doping have greater potential for facile synthesis.

3.2. Elastic Constants

In order to investigate the mechanical properties of the BCZT-xGd system, a stress–strain method was employed to apply three normal strains (${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$) and three shear strains (${\gamma }_4$, ${\gamma }_5$, and ${\gamma }_6$) to the crystal, and we recorded the corresponding stresses (${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$, and ${\tau }_4$, ${\tau }_5$, ${\tau }_6$). Then, by solving Hooke’s law, independent elastic constants were obtained to further study the elastic properties of the system. The tetragonal crystal system of BCZT-xGd exhibits six distinct elastic constants ($C_{11}$, $C_{12}$, $C_{13}$, $C_{33}$, $C_{44}$, and $C_{66}$), which are presented in

Table 1. Calculated elastic constants $C_{ij}$ in GPa for tetragonal BCZT-xGd perovskites.

x	Sites	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{13}$	${\mathit{C}}_{33}$	${\mathit{C}}_{44}$	${\mathit{C}}_{66}$
0		279.21	95.45	73.99	117.77	109.14	116.13
0.01	A site	287.35	98.56	98.62	288.40	109.52	109.51
	B site	279.07	97.07	79.19	108.07	105.34	114.03
0.02	A site	289.31	98.67	98.67	286.85	104.81	104.85
	B site	287.25	98.55	97.14	262.06	110.73	110.63
0.03	A site	293.64	99.09	99.02	284.53	102.32	102.36
	B site	290.45	99.68	99.81	293.01	109.89	110.04
0.04	A site	297.80	98.48	98.49	298.75	99.70	99.76
	B site	290.13	98.81	98.85	292.20	110.25	110.23
0.05	A site	305.19	99.31	99.38	304.19	97.15	97.13
	B site	292.72	97.83	97.85	292.33	110.78	110.75

.

The Born stability criterion is commonly employed to assess the stability of crystalline materials subjected to stress. The elastic constants of the tetragonal symmetry must satisfy the Born stability condition to ensure the stability of the system, as expressed in the following equation [35,36]:

$$C_{11}>0,C_{33}>0,C_{44}>0,C_{66}>0$$

(3)

$$C_{11}-C_{12}>0,{C_{11}+C}_{33}-2C_{13}>0$$

(4)

$$2\left(C_{11}+C_{12}\right)+C_{33}+4C_{13}>0$$

(5)

The elastic constants of all samples fully comply with the aforementioned conditions, ensuring the mechanical stability of the system. The elastic constant of $C_{33}$ exhibits a higher sensitivity to the concentration of Gd³⁺ ions in comparison to the other elastic constants. For A-site doping, the incorporation of the Gd³⁺ ion into the BCZT component results in a significant increase in the $C_{33}$ value, which is approximately two and a half times greater than that of the pure BCZT material. In the case of B-site doping, except for a slight decrease in the sample with x = 0.01, the other samples are also close to the level of A-site doping. These results indicate that the incorporation of the Gd³⁺ ion plays a crucial role in enhancing the orientation-dependent elastic modulus of the material along the z-axis. For the pure BCZT structure, $C_{11}$ is significantly larger than $C_{33}$, indicating that the bonding energy in the [001] and [010] directions is much higher than in the [001] direction. For the doped systems, except for the B-site sample with x = 0.01, the differences between $C_{11}$ and $C_{33}$ decrease, reducing the differences in bonding energy in different directions. Moreover, the bonding within the (001) plane demonstrates enhanced rigidity and a higher elastic tensile modulus along the c-axis, due to the inequality $C_{11}+C_{12}>C_{33}$.

3.3. Elastic Modulus

In order to describe the response of the BCZT-xGd material under external forces, a series of key mechanical parameters, including the bulk modulus (B), shear modulus (G), Young’s modulus (Y), and Poisson’s ratio (ν), were calculated using elastic constants. The B modulus and G modulus can be estimated using Voigt–Reuss–Hill approximation, which is expressed by the following equations [37,38]:

$$B_{\mathrm{V}}=\frac{2}{9}\left(C_{11}+C_{12}+2C_{13}+\frac{C_{12}}{2}\right);B_{\mathrm{R}}=\frac{C^2}{M}$$

(6)

$$G_{\mathrm{V}}=\frac{\left(M+3C_{11}-3C_{12}+12C_{44}+6C_{66}\right)}{30}$$

(7)

$$G_R=\frac{15}{(\frac{18B_V}{C^2}+\frac{6}{(C_{11}-C_{12})}+\frac{6}{C_{44}}+\frac{3}{C_{66}})}$$

(8)

where $C^2=(C_{11}+C_{12})C_{33}-2C_{13}^2$, $M=\left(C_{11}+C_{12}+{2C}_{33}-{4C}_{13}\right)$, and through utilizing the Hill approximation method, the G modulus can be expressed as the average value between $G_V$ and $G_R$, whereas the B modulus can be calculated as the mean value of $B_V$ and $B_R$.

$$B=\frac{1}{2}\left(B_V+B_R\right);G=\frac{1}{2}\left(G_V+G_R\right)$$

(9)

The isotropic Y modulus and Poisson’s ratio (ν) can be obtained using the equations that relate them to the bulk modulus and shear modulus. The equations are as follows [38,39]:

$$Y=\frac{9BG}{3B+G};v=\frac{3B-2G}{2\left(3B+G\right)}$$

(10)

The B modulus refers to a material’s resistance to compression, and quantifies the material’s ability to withstand volumetric deformation. The Y modulus represents the elastic and stiffness properties of a material, whereas the isotropic G modulus reflects the material’s resistance to shear stress. The calculated B modulus, G modulus, Y modulus, Poisson’s ratio (ν), and Frantesvich ratio (G/B) for tetragonal BCZT-xGd systems are listed in

Table 2. Calculated bulk modulus (B) in GPa, shear modulus (G) in GPa, Young ‘s modulus (Y) in GPa, Frantesvich ratio (G/B), Poisson’s ratio (ν), machinability index (${\mu }_M$), and Vicker’s hardness ($H_{\mathrm{v}}$) in GPa for tetragonal BCZT-xGd.

x	Sites	B	G	G/B	Y	ν	${\mathsf{\mu }}_{\mathbf{M}}$	${\mathbf{H}}_{\mathbf{v}}$
0		116.16	89.02	0.77	212.73	0.19	1.06	18.11
0.01	A site	151.09	104.77	0.69	255.29	0.22	1.38	19.67
	B site	115.79	83.81	0.72	202.56	0.21	1.10	16.29
0.02	A site	151.49	102.16	0.67	250.23	0.22	1.45	18.75
	B site	148.79	103.52	0.70	252.10	0.22	1.34	19.49
0.03	A site	152.58	101.02	0.66	248.27	0.23	1.49	18.26
	B site	152.87	105.55	0.69	257.40	0.22	1.39	19.75
0.04	A site	153.90	101.14	0.66	248.89	0.23	1.54	18.17
	B site	152.09	105.83	0.70	257.71	0.22	1.38	19.93
0.05	A site	156.47	100.69	0.64	248.72	0.24	1.61	17.78
	B site	151.95	106.75	0.70	259.49	0.22	1.37	20.25

. It can be observed that the Y modulus values of all the materials are greater than the modulus values of the B modulus, indicating that BCZT-xGd materials are more prone to undergoing elastic deformation when subjected to force, rather than experiencing volumetric changes. Therefore, the BCZT-xGd system is less prone to breaking. When dopant is incorporated at the A site, the B modulus exhibits a positive correlation with the dopant concentration. For the BCZT-xGd with B-site doping, there is also an increase, apart from the component x = 0.1. This observation indicates that doped systems demonstrate greater resilience towards volumetric deformation, showcasing their enhanced capacity to withstand such mechanical strains. In comparison to the pure BCZT, with the exception of a minor reduction in the G and Y values observed in BCZT-0.01Gd due to B-site doping, the incorporation of Gd³⁺ ions in the doped systems leads to an overall increase in the G and Y moduli. Consequently, this phenomenon suggests a notable enhancement in the stiffness of the doped structures. The amplification of stiffness serves to mitigate energy dissipation, thereby rendering considerable advantages in applications demanding highly efficient energy transfer.

The Poisson’s ratio (ν) typically represents the stability of a solid in terms of shear deformation, with values ranging from −1 to 0.5 for stable linear elastic solids. In general, a larger Poisson’s ratio indicates a solid’s good plasticity. As demonstrated in Table 2, the Poisson’s ratios of these compounds range from 0.19 to 0.24, falling within the range of −1 to 0.5, indicating that these BCZT-xGd compounds are stable linear elastic solids. Moreover, compared to pure BCZT, the addition of dopants to the system shows a slight increase in the Poisson’s ratio, reaching its maximum with a dopant content of x = 0.05. These findings indicate that the incorporation of Gd enhances the plasticity of the system compared to the undoped states. The Frantsevich G/B ratio serves as an indicator of a fractured material’s brittle/ductile behavior. When G/B < 0.571, the crystal exhibits ductile characteristics, whereas a G/B value greater than 0.571 indicates brittleness. Furthermore, Frantsevich emphasized that the critical value of ν is 0.26, and materials exceeding this critical value exhibit ductility. As shown in Table 2, the calculated G/B ratios are within the range of 0.70 to 0.80, and the values of ν are all less than 0.26 GPa, indicating that the studied compound is inherently brittle. By employing appropriate design and usage strategies, the fragility attribute can be effectively transformed into a reliable feature for practical applications. The Vicker’s hardness ($H_V$), a widely employed method for testing material hardness in experiments, involves embedding a diamond into the surface of the test specimen under a specific load and subsequently measuring the resulting indentation size to determine its hardness. To evaluate the hardness index of the systems, $H_V$ was determined using the following expression [40]:

$$H_V=\frac{(1-2\nu )Y}{6\left(1+\nu \right)}$$

(11)

It can be seen from the formula that the hardness of a material is contingent upon the values of Young’s modulus and Poisson’s ratio. Table 2 presents the computed values of $H_V$ for the BCZT-xGd systems. The obtained hardness value of pure BCZT is 18.11 GPa, slightly higher than that of BCZT bulk ceramics prepared via a solid-state reaction [41]. It is apparent from the data that BCZT-0.05Gd with B-site doping displays the highest level of hardness within this group of compounds. Moreover, the machinability index (${\mu }_M$) is increasingly crucial in industrial applications as it determines the optimal economic level of the material cutting force, plastic deformation, and machine utilization. This index can be determined using the following mathematical expression [42].

$${\mu }_M=\frac{B}{C_{44}}$$

(12)

After incorporating the Gd element, the doped system exhibits an increased value of ${\mu }_M$, indicating enhanced processing machinability.

3.4. Elastic Anisotropy

Elastic anisotropy refers to a phenomenon where materials exhibit distinct elastic properties in different directions, with their elastic behavior varying in response to changes in the direction of applied stress. Anisotropy induces an uneven stress distribution along distinct orientations, consequently augmenting the material’s susceptibility to microcrack formation and propagation. The elastic anisotropy of solids can be characterized through the utilization of several quantitative measures, namely the universal elastic anisotropic index ($A^U$), the percentage of compressible anisotropy ($A_{\mathrm{B}}$), and the percentage of shear anisotropy ($A_{\mathrm{G}}$). The percentage evaluation of $A^U$, $A_{\mathrm{B}}$, and $A_{\mathrm{G}}$ can be obtained using the following expressions [43,44]:

$${|A}^U|=|5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6|$$

(13)

$${|A}_B|=|\frac{B_V-B_R}{B_V+B_R}|\times 100\%$$

(14)

$$|A_{\mathrm{G}}|=|\frac{G_V-G_R}{G_V+G_R}|\times 100\%$$

(15)

where V and R denote the Voigt and Reuss approximations, respectively. Moreover, considering the tetragonal symmetry of crystal structures, the shear anisotropic factors ($A_1$, $A_2,A_3$) should also be considered, and their expressions are as follows [45]:

$$A_1=\frac{C_{44}\left(C_{11}+{2C}_{13}+C_{33}\right)}{C_{11}{C}_{33}-C_{13}^2}$$

(16)

$$A_2=\frac{C_{44}\left(C_0+{2C}_{13}+C_{33}\right)}{C_0{C}_{33}-C_{13}^2}$$

(17)

$$A_3=\frac{2C_{66}}{C_{11}-C_{12}}$$

(18)

$$C_0=\frac{C_{66}+\left(C_{11}+C_{12}\right)}{2}$$

(19)

where $A_1$, $A_2$, and $A_3$ correspond to the {100}/{010}, {1$\stackrel{\mathrm{-}}{1}$0}, and {001} shear planes, respectively. For elastic isotropic crystals, the anisotropic indexes are ${|A}^U|={|A}_B|=|A_G|=0$ and $A_1=A_2=A_3=1$ [46]. In contrast, for anisotropic crystals, their values deviate from 0 or 1. The greater the deviation of the factors from 0 or 1, the stronger the crystal’s anisotropy and the larger the differences in the elastic properties among different directions. Figure 3 presents the computed indexes of elastic anisotropy and the trend of the elastic anisotropy indexes of BCZT-xGd as a function of x variation. It is readily observable that, except for the composition with a doping level of x = 0.01 at the B site, the values of ${|A}_B|$ and $|A_{\mathrm{G}}|$ exhibit a decreasing trend with the rise in doping concentration. Notably, the attenuation in the $A_{\mathrm{G}}$ value surpasses that of $A_B$, yielding a considerably more pronounced descent, indicating the incorporation of the element Gd manifestly influences the variation in shear isotropic tendencies within the systems. In the context of the anisotropy index for universal elasticity, which takes into account both the bulk modulus and shear modulus, $A^U$ is shown to be more suitable for evaluating elastic anisotropy. Figure 3a illustrates the trend of $A^U$ variation in A-site-doped compounds, with pure BCZT exhibiting the highest $A^U$ value, thus affirming the presence of the highest elastic anisotropy. In terms of B-site-doped compounds, as depicted in Figure 3c, BCZT-0.01Gd demonstrates the utmost level of elastic anisotropy.

Moreover, it can be seen from the data that the percentages of variation in the isotropy of the shear modulus and the universal elastic anisotropy exhibit a remarkable level of congruence. Therefore, it can be concluded that shear anisotropy exerts a pivotal influence on the overall anisotropy. As for the shear anisotropy factors $A_i$(i = 1–3), the deviation from 1 is plotted in Figure 3b,d. It can be observed that BCZT-0.04Gd with A-site doping and BCZT-0.05Gd with B-site doping have the smallest deviation values in terms of $A_1$ and $A_3$. The low deviation values indicate the lowest levels of shear anisotropy along the {100}/{010} and {001} planes. Anisotropy of elasticity can be visualized through three-dimensional surface representations, illustrating the dependence of the mechanical modulus on different directions. When a three-dimensional surface exhibits spherical geometry, it signifies the presence of isotropic characteristics. Conversely, deviations from a spherical shape are indicative of the degree of elastic anisotropy. The elastic anisotropy of bulk modulus B, bulk modulus (B), Young’s modulus (Y), and shear modulus (G) via 3D surface construction can be obtained through the following expressions [47].

$$\frac{1}{B}=(S_{11}+S_{12}+S_{13})-(S_{11}+S_{12}-S_{13}-S_{33})l_3^2$$

(20)

$$\frac{1}{E}=S_{11}(l_1^4+l_2^4{)+(2S}_{13}+S_{44})(l_1^2{l}_3^2+l_2^2{l}_3^2)+S_{33}{l}_3^4+(2S_{12}+S_{66})l_1^2{l}_2^2$$

(21)

$$\frac{1}{Y}=\frac{1}{3G}+\frac{1}{9\mathrm{B}}$$

(22)

In the above equations, $S_{ij}$ represents the usual elastic compliance constants, which are listed in

Table 3. Calculated elastic compliance matrix $S_{ij}$ of BCZT-xGd systems.

x	Sites	${\mathit{S}}_{11}$	${\mathbf{S}}_{12}$	${\mathit{S}}_{13}$	${\mathit{S}}_{33}$	${\mathit{S}}_{44}$	${\mathit{S}}_{66}$
0		279.21	95.45	73.99	117.77	109.14	116.13
0.01	A site	0.0042	−0.0011	−0.0011	0.0042	0.0091	0.0091
	B site	0.0053	−0.0006	−0.0057	0.0318	0.0110	0.0091
0.02	A site	0.0042	−0.0011	−0.0011	0.0042	0.0095	0.0095
	B site	0.0042	−0.0011	−0.0012	0.0047	0.0090	0.0090
0.03	A site	0.0041	−0.0010	−0.0011	0.0043	0.0098	0.0098
	B site	0.0042	−0.0011	−0.0011	0.0041	0.0091	0.0091
0.04	A site	0.0040	−0.0010	−0.0010	0.0040	0.0100	0.0100
	B site	0.0042	−0.0011	−0.0011	0.0091	0.0091	0.0091
0.05	A site	0.0039	−0.0010	−0.0010	0.0103	0.0103	0.0103
	B site	0.0041	−0.0010	−0.0010	0.0090	0.0090	0.0090

. $l_1$, $l_2$, and $l_3$ represent the directions of cosines.

Figure 4 and Figure 5 depict three-dimensional surface plots of the bulk modulus (B), Young’s modulus (Y), and shear modulus (G) for A-site- and B-site-doped compounds, respectively. It can be observed that the B, Y, and G moduli of BCZT exhibit strong anisotropy owing to the non-conformity of its three-dimensional graph to the ideal spherical configuration. These results are consistent with the original theoretical calculation results, and the high piezoelectric property presented in the BCZT system is mostly due to the enhanced anisotropy [48]. Notably, for the B modulus, a discernible compression along the z-axis is observed, which can be attributed to the relatively lower value of $C_{33}$ compared to $C_{11}$. Upon the incorporation of the Gd³⁺ ion for the A site, the three-dimensional surfaces of all doped systems approach an ideal spherical shape, indicating near-isotropic characteristics. However, in terms of B-site doping, there still exist non-ideal spherical surface features of the B, Y, and G moduli for BCZT-0.01Gd similar to the pure BCZT, indicating limited influence on enhancing material isotropy. As the doping concentration surpasses x = 0.01, the B, Y, and G graphs exhibit pronounced three-dimensional surface spherical characteristics akin to those observed in A-site doping. The transition from anisotropy to isotropy in the systems can be explained by the significant increase in $C_{33}$, and these findings are consistent with the results of the aforementioned anisotropy factor. It is worth noting that the enhanced isotropic elasticity in the systems favors a more uniform stress distribution and consequently reduces sensitivity to the formation and propagation of microcracks.

4. Conclusions

In conclusion, the mechanical properties of Gd-doped BCZT ceramics were thoroughly investigated through a first-principles calculation approach. The results of the formation energies reveal a more facile synthetic route for the preparation of doped systems in experimental contexts. The incorporation of the Gd³⁺ ion plays a crucial role in enhancing the orientation-dependent elastic modulus of the material along the z-axis, and the bonding within the (001) plane demonstrates enhanced rigidity and a higher elastic tensile modulus along the c-axis, due to the inequality $C_{11}+C_{12}>C_{33}$. The Young’s moduli values (with a range of 202.56~259.49 GPa) of all the materials are greater than the values of the bulk moduli (with a range of 116.16~156.47 GPa), indicating that BCZT-xGd materials are more prone to undergoing elastic deformation when subjected to force, rather than experiencing volumetric changes. After the incorporation of Gd, there is exists a significant enhancement in the system’s elasticity and stiffness. The calculations of the Vicker’s hardness (20.25 GPa) of BCZT-0.05Gd with B-site doping displays the highest level of hardness.

The doped system exhibits an increased value (with a range of 1.10~1.61) of ${\mu }_M$ compared to pure BCZT (1.06), indicating enhanced processing machinability. After the inclusion of Gd³⁺ ions, the system’s elastic anisotropy disappears, displaying characteristics close to those of isotropy. This study provides a foundation for understanding the mechanistic behaviors of electromechanical systems, and offers valuable insights into the potential improvements and limitations of new electromechanical devices based on lead-free BCZT-based materials.