Impact of isovalent and aliovalent substitution on the mechanical and thermal properties of Gd₂Zr₂O₇

Abstract

In this study, a density functional theory method is employed to investigate the effects of isovalent and aliovalent substitution of Sm³⁺ on the phase stability, thermo-physical properties and electronic structure of Gd₂Zr₂O₇. It is shown that the isovalent substitution of Sm³⁺ for Gd³⁺ results in the formation of Gd₂Zr₂O₇-Sm₂Zr₂O₇ solid solution, which retains the pyrochlore structure and has slight effects on the elastic moduli, ductility, Debye temperature and band gap of Gd₂Zr₂O₇. As for the aliovalent substitution of Sm³⁺ for Zr⁴⁺ site, a pyrochlore-to-defect fluorite structural transition is induced, and the mechanical, thermal properties and electronic structures are influenced significantly. As compared with the Gd₂Zr₂O₇, the resulted Gd₂Sm_yZr_2-yO₇ compositions have much smaller elastic moduli, better ductility and smaller Debye temperature. Especially, an amount of electrons distribute on the fermi level and they are expected to have larger thermal conductivity than Gd₂Zr₂O₇. This study suggests an alternative way to engineer the thermo-physical properties of Gd₂Zr₂O₇ and will be beneficial for its applications under stress and high temperature.

Introduction

The rare-earth zirconates, with chemical formula A₂Zr₂O₇ (A = Y or another rare earth elements)^{1, 2}, exhibit ordered pyrochlore-type structure or defect fluorite-type structure, which is mainly governed by the ionic radii of A³⁺ and Zr^{4+ 3}. They have attracted the attention of many researchers, due to their good chemical and mechanical stability, excellent catalytic activity, high ionic conductivity, ferromagnetism, luminescence as well as strong resistance to amorphization under irradiation^4,5,6,7,8. Owing to these outstanding properties, the rare-earth zirconates have a wide range of technical applications, e.g., ceramic thermal barrier coating^{9, 10}, oxidation catalyst^{5, 11}, solid electrolyte¹², hosts of actinides in nuclear waste¹³ and oxygen gas sensor¹⁴.

Of the zirconate pyrochlores, Gd₂Zr₂O₇ is of special interest due to its good thermo-physical properties^{15,16,17,18,19}. Shimamura et al. have measured the thermal expansion of a series of zirconate pyrochlores employing high-temperature X-ray diffraction and found that the thermal expansion coefficient of Gd₂Zr₂O₇ is larger than other zirconates during the temperature range of 400–1600 °C²⁰. The thermal conductivity of rare-earth zirconates has been investigated by Wang et al.²¹, who reported that the thermal conductivity of 1.15–1.43 W/mK for Gd₂Zr₂O₇ is lower than that of Yb₂Zr₂O₇, Dy₂Zr₂O₇ and 7YSZ between 25 °C and 800 °C. In recent years, a great number of investigations have been carried out on substitution of lanthanides for Gd-site to engineer the thermo-physical properties of Gd₂Zr₂O₇ ^22,23,24,25. Guo et al. synthesized (Gd_1-xYb_x)₂Zr₂O₇ (x = 0, 0.1, 0.3, 0.5, 0.7) using solid state reaction and suggested that the thermal conductivity of 0.88–1.00 W/mk for (Gd_1-xYb_x)₂Zr₂O₇ is about 20% lower than that of Gd₂Zr₂O₇ (1.18 W/mk) at 1400 °C²⁶. Wan et al. predicted that, among the (La_xGd_1-x)₂Zr₂O₇ (0 ≤ x ≤ 1) systems, the LaGdZr₂O₇ has the minimum thermal conductivity, which is about 20–25% lower than that of Gd₂Zr₂O₇ ¹⁸. In recent years, both Liu et al.¹⁹. and Pan et al.²³ reported that the thermal diffusivity of (Sm_xGd_1-x)₂Zr₂O₇ (0 ≤ x ≤ 1) are lower than those of pure Gd₂Zr₂O₇ and Sm₂Zr₂O₇. Especially, Sm₂Zr₂O₇-Gd₂Zr₂O₇ solid solutions have lower Young’s modulus than (La_xGd_1-x)₂Zr₂O₇ (0 < x < 1) at room temperature and larger thermal expansion coefficients than (Gd_1-xYb_x)₂Zr₂O₇ (x = 0, 0.1, 0.3, 0.5, 0.7) from 300 °C to 900 °C^{18, 19, 23, 26}. These investigations are mainly experimental studies, and the related theoretical investigations are relatively much fewer^{27, 28}.

Very recently, the Th⁴⁺ ion incorporation into Gd³⁺ and Zr⁴⁺ sites in Gd₂Zr₂O₇ was investigated by first-principles calculations²⁸. Unexpectedly, the aliovalent substitution of Th⁴⁺ for Gd³⁺ turns out to be thermodynamically stable and such substitution even results in better thermo-physical properties than the pure Gd₂Zr₂O₇ ²⁸. This thus arouses our interest that whether the aliovalent substitution of Ln³⁺ (Ln = lanthanide elements) for Zr⁴⁺ sites are energetically and mechanically stable or not? If yes, will the substitution of Ln³⁺ for Zr⁴⁺ sites cause different thermo-mechanical properties from the isovalent substitution of Ln³⁺ for Gd³⁺ sites? In this study, we choose Sm³⁺ as a model and investigate the phase stability and thermo-physical properties of Gd₂Zr₂O₇ with isovalent and aliovalent substitution of Ln³⁺ for Gd³⁺ and Zr⁴⁺ sites by employing the density functional theory (DFT) method. It reveals that the Sm_yGd_2-yZr₂O₇ retains the pyrochlore structure and the isovalent substitution of Sm³⁺ for Gd³⁺ sites influences slightly the mechanical and thermal properties of Gd₂Zr₂O₇. On the other hand, the aliovalent substitution of Sm³⁺ for Zr⁴⁺ sites induces pyrochlore-to-fluorite structural transition and affects significantly the elastic moduli, Debye temperature and thermal conductivity. The presented results provide a new way to tune the thermo-physical properties of Gd₂Zr₂O₇ and will have important implications in advancing the further related experimental and theoretical studies for its applications under high temperature.

Results and Discussion

Structural stability of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2)

In this study, Sm substitution for both Gd-site and Zr-site in Gd₂Zr₂O₇ with different concentrations are considered, resulting in Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (y = 0, 0.5, 1, 1.5, 2). The geometrical structures of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ are optimized firstly. In order to test whether local density approximation (LDA)²⁹ or generalized gradient approximation (GGA)³⁰ is more appropriate to describe the exchange-correlation interaction between electrons, both LDA and GGA methods are employed to relax the structures of Gd₂Zr₂O₇ and Sm₂Zr₂O₇. For Gd₂Zr₂O₇, the lattice constant a ₀ and x _O48f obtained by LDA are 10.451 Å and 0.342, respectively, agreeing well with the experimental values of a ₀ = 10.472 Å and x _O48f  = 0.345^{31, 32}. As compared with the LDA results, the values of a ₀  = 10.641 Å and x _O48f = 0.339 calculated by GGA deviate much more from the experimental results. As for Sm₂Zr₂O₇, the LDA calculations yield a lattice constant of 10.531 Å and an x _O48f parameter of 0.34, which are also comparable with the experimental data of a ₀  = 10.514 Å and x _O48f  = 0.342^{31, 32}. On the other hand, the GGA results of a ₀  = 10.715 Å and x _O48f = 0.337 are in bad agreement with the experimental measurement. Obviously, the structural parameters obtained by LDA are in better agreement with experiments than the GGA method. The LDA method, thus, is employed in all the subsequent calculations. The calculated lattice constant a ₀ and x _O48f for Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ as a function of Sm content are plotted in Fig. 1a and b, respectively. As Sm substitutes for Gd-site in Gd₂Zr₂O₇, it is found that the resulted Sm_yGd_2-yZr₂O₇ compositions remain the ideal pyrochlore structure. Especially, the lattice constant a ₀ and the O_48f positional parameter  x _O48f increase and decrease linearly with the increasing Sm concentration, suggesting that he lattice parameters of Sm_yGd_2-yZr₂O₇ follow well the Vegard’s law, i.e., $a({{\rm{Sm}}}_{{\rm{y}}}{{\rm{Gd}}}_{{\rm{2}}-{\rm{y}}}{{\rm{Zr}}}_{{\rm{2}}}{{\rm{O}}}_{{\rm{7}}})=\frac{2-y}{2}\times a({{\rm{Gd}}}_{{\rm{2}}}{{\rm{Zr}}}_{{\rm{2}}}{{\rm{O}}}_{{\rm{7}}})+\frac{y}{2}\times a({{\rm{Sm}}}_{{\rm{2}}}{{\rm{Zr}}}_{{\rm{2}}}{{\rm{O}}}_{{\rm{7}}})$. These results indicate that the Gd₂Zr₂O₇-Sm₂Zr₂O₇ solid solution is formed by Sm substitution into Gd-site in Gd₂Zr₂O₇, agreeing well with the experimental observation¹⁹. Experimentally, Liu et al. also found that the lattice constants increase linearly with different compositions for (Sm_xGd_1−x)₂Zr₂O₇ system from x = 0 (Gd₂Zr₂O₇) to x = 1.0 (Sm₂Zr₂O₇), and the Gd₂Zr₂O₇ and Sm₂Zr₂O₇ ceramics are infinitely solid solution¹⁹. As Sm substitutes for Zr-site, i.e. Gd₂Sm_yZr_2-yO₇, the lattice constants increase more significantly than the case of Sm_yGd_2-yZr₂O₇ and there is a small deviation from the Vegard’s law. It is noticeable that the x _O48f of Gd₂Sm_yZr_2-yO₇ increases remarkably with the increasing Sm content instead of decreasing as the case of Sm_yGd_2-yZr₂O₇. In A₂B₂O₇ pyrochlores, the x _O48f positional parameter can be used to describe the degree of distortion of < B-O > octahedron and it is located within the range of 0.3125 to 0.375. Generally, the material with x _O48f closer to 0.3125 has the ordered pyrochlore structure and the one with x _O48f closer to 0.375 is more likely to undergo a transition from pyrochlore to defect-fluorite structure^{31, 33,34,35}. Our calculations show that the x _O48f for Gd₂Sm_yZr_2-yO₇ with high content of Sm approaches to be 0.375, implying that order-disorder transition may occur in the systems. The schematic view of optimized configurations of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ is illustrated in Fig. 2. As can be seen in the figure, Sm_yGd_2-yZr₂O₇ still maintains the pyrochlore-type structure, while Gd₂Sm_yZr_2-yO₇ exhibits defective fluorite-type structure due to the significant disordering of the anions. These results suggest that Sm substitution for Gd-site has minor effects on the pyrochlore structure of Gd₂Zr₂O₇, as evidenced by the small changes in the < Gd-O > and < Zr-O > bonding distances shown in Table 1. On the other hand, structural transformation from ordered pyrochlore to disordered defect-fluorite structure is induced by the substitution of Sm for Zr-site, which is accompanied by the weakened interaction of < Sm-O_48f  > and < Gd-O_8b  > bonds (see Table 1).

Figure 1

Variation of (a) lattice constants and (b) positional parameter x _O48f for Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2) as a function of Sm content.

Figure 2

Schematic view of the optimized configurations for (a) Gd₂Zr₂O₇, (b) Sm_0.5Gd_1.5Zr₂O₇, (c) SmGdZr₂O₇, (d) Sm_1.5Gd_0.5Zr₂O₇, (e) Sm₂Zr₂O₇, (f) Gd₂Sm_0.5Zr_1.5O₇, (g) Gd₂SmZrO₇, (h) Gd₂Sm_1.5Zr_0.5O₇, (i) Gd₂Sm₂O₇. The blue, purple, green and red spheres represent Sm, Gd, Zr and O atoms, respectively.

Table 1 The bonding distances (Å) between atoms in Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2).

In the literature, it has been reported that the Madelung binding energy reduces with the increasing x _O48f and the thermal expansion increases with the decreasing Madelung binding energy²⁰. Considering that Gd₂Sm_yZr_2-yO₇ has larger x _O48f than Sm_yGd_2-yZr₂O₇, we thus suggest that smaller Madelung binding energy exists in the Gd₂Sm_yZr_2-yO₇ and they probably have larger thermal expansion coefficient. Meanwhile, the disordering of oxygen ions in Gd₂Sm_yZr_2-yO₇ may increase the phonon scattering, which will reduce the mean free path of the phonon and result in small phonon thermal conductivity.

Elastic constants and elastic moduli of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2)

In order to investigate the mechanical properties of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ compounds, we further calculate their elastic constants based on the optimized structures. For a cubic crystal, there are three independent elastic constants, i.e. C₁₁, C₁₂ and C₄₄, where C₁₁ refers to the uniaxial deformation along the [001] direction, C₁₂ is the pure shear stress at (110) crystal plane along the [110] direction, and C₄₄ is a pure shear deformation on the (100) crystal plane⁴. The values of these three elastic constants for all compounds are summarized in Table 2. For Gd₂Zr₂O₇, our calculated values of C₁₁, C₁₂ and C₄₄ are 325.6, 126.3 and 94.0 GPa, respectively, which are found to be close to the results of C₁₁ = 314.2 GPa, C₁₂ = 126.2 GPa and C₄₄ = 96.2 GPa for Sm₂Zr₂O₇. It should be pointed out that in this study the Gd 4 f and Sm 4 f electrons are treated as core electrons. In order to investigate if these f electrons will influence the mechanical properties of Gd₂Zr₂O₇ and Sm₂Zr₂O₇, we further consider the Gd 4 f and Sm 4 f electrons as valence electrons and carry out LDA + U calculations, in which the effective U values are taken to be 6 eV for Gd 4 f and 8 eV for Sm 4 f electrons⁴. For Gd₂Zr₂O₇, the LDA + U results of C₁₁ = 316.9 GPa, C₁₂ = 123.0 GPa and C₄₄ = 94.7 GPa are very similar to the LDA calculations. In the case of Sm₂Zr₂O₇, the calculated C₁₁ = 276.9 GPa, C₁₂ = 114.2 GPa, and C₄₄ = 112.6 GPa are deviated from the LDA results. Further investigation shows that the Sm₂Zr₂O₇ is elastically anisotropic, which is not consistent with the elastic isotropy of Sm₂Zr₂O₇. Hence, the Gd 4 f and Sm 4 f electrons are frozen in our calculations and all the calculations are carried out by the LDA method. These results indicate that Gd₂Zr₂O₇ and Sm₂Zr₂O₇ may exhibit very similar mechanical properties, agreeing well with the experiment carried out by Shimamura et al.²⁰. The elastic constants reported by Lan et al. employing GGA method are generally smaller than our LDA results, while they also found that the results of Gd₂Zr₂O₇ and Sm₂Zr₂O₇ are very similar to each other³⁶. We find that the mechanical stability criteria, i.e., (C₁₁-C₁₂) > 0, C₄₄ > 0, and (C₁₁ + 2C₁₂) > 0¹, are satisfied for all compounds, indicating that Sm-substituted Gd₂Zr₂O₇ compounds are mechanically stable.

Table 2 Elastic constants (C₁₁, C₁₂, C₄₄, in GPa), bulk modulus (B, in GPa), shear modulus (G, in GPa) and Young’s modulus (E, in GPa) of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2).

The calculated elastic constants for both Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2) as a function of Sm content are plotted in Fig. 3. It is noted that Sm substitution for Gd-site results in very small changes in the elastic constants and the mixed Gd₂Zr₂O₇-Sm₂Zr₂O₇ phases have similar values with the pure states. As the Sm substitutes for Zr-site, the values of C₁₁ are strikingly decreased with the increasing Sm content, implying that the compositions with high content of Sm are more likely to undergo the uniaxial deformation along the [001] direction. On the other hand, the C₁₂ and C₄₄ slightly increase and decrease with the Sm incorporation, respectively. Generally, the changes in the elastic constants of Gd₂Sm_yZr_2-yO₇ are more significant than those of Sm_yGd_2-yZr₂O₇, meaning that the effects of Sm incorporation into Gd-site on the mechanical properties of Gd₂Zr₂O₇ are nearly negligible whereas Sm incorporation into Zr-site has remarkable influence. This may be mainly due to the fact that Gd³⁺ and Sm³⁺ have similar mass (Gd³⁺: 157.25 amu; Sm³⁺: 150.3 amu) and cation size (Gd³⁺: 1.053 Å; Sm³⁺: 1.079 Å) and Sm substitution for Gd-site affects the structural properties of Gd₂Zr₂O₇ slightly, while the mass and radius for Gd³⁺ are largely different from the mass of 91.22 amu and the radius of 0.72 Å for Zr^{4+ 19, 27} and an order-disorder phase transition has occurred in Gd₂Sm_yZr_2-yO₇.

Figure 3

Variation of elastic constants (C₁₁, C₁₂ and C₄₄) for (a) Sm_yGd_2-yZr₂O₇ and (b) Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2) with Sm content.

Based on the three elastic constants, the bulk modulus (B), Young’s modulus (E) and shear modulus (G) can be deduced under Voigt-Reuss-Hill (VRH) approximation³⁷, i.e., B = (C₁₁ + 2C₁₂)/3, E = 9BG/(3B + G), G = ((C₁₁-C₁₂ + 3C₄₄)/5 + 5(C₁₁-C₁₂)C₄₄/(4C₄₄ + 3(C₁₁-C₁₂)))/2^{1, 38,39,40}. The values of bulk modulus, Young’s modulus and shear modulus are shown in Table 2, together with available experimental and theoretical results in the literature^{20, 36}. As compared with the experimental measurement, our calculated elastic moduli for Gd₂Zr₂O₇ and Sm₂Zr₂O₇ are overestimated slightly. This may be resulted from the employed LDA method, which generally underestimates the lattice constant whereas overestimates the mechanical modulus⁴¹. Besides, the defects and impurities in the sample experimentally may also lead to the underestimated values of the B and G⁴¹. Variation of the elastic moduli for Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ with the Sm content is illustrated in Fig. 4. As expected, the elastic moduli for Sm_yGd_2-yZr₂O₇ vary slightly with the Sm content since the elastic constants for all compositions are very similar to each other. Different from the case of Sm_yGd_2-yZr₂O₇, the bulk modulus, Young’s modulus and shear modulus for Gd₂Sm_yZr_2-yO₇ all decrease sharply with the increasing Sm content, especially the Young’s modulus. Consequently, the Gd₂Sm₂O₇ has the minimum Young’s modulus of 87 GPa, minimum shear modulus of 31 GPa and minimum bulk modulus of 151 GPa. These results indicate that the Gd₂Sm₂O₇ has good compliance² due to the lowest bulk modulus and the lowest Young’s modulus, which will produce relatively smaller residual stresses in the coating system under the severe conditions and result in better thermo-mechanical stability²³.

Figure 4

Variation of elastic moduli for (a) Sm_yGd_2-yZr₂O₇ and (b) Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2) as a function of Sm content. B: bulk modulus; G: shear modulus; E: Young’s modulus.

Elastic anisotropy, ductility and Debye temperature of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2)

The elastic anisotropy, which is correlated with the possibility of the appearance of microcracks, is an important mechanical property of materials^{42, 43}. For a crystal, the elastic anisotropy of materials can be evaluated by the A^U (universal elastic anisotropy index), which can be calculated by A^U = 5 G^V/G^R + B^V/B^R-6⁴⁴, with V and R representing the Voigt and Reuss approximation, respectively^{38, 40}. The A^U value of zero refers to isotropic mechanical properties, otherwise defines the anisotropy⁴⁴. The calculated results are shown in Table 3. Obviously, the values of A^U are very close to zero in the case of Sm_yGd_2-yZr₂O₇ (0 ≤ y ≤ 2), which indicates that all compounds are of elastic isotropy. However, the crystals show strong anisotropy in the Gd₂Sm_yZr_2-yO₇ (0.5 ≤ y ≤ 2) system, as the A^U values have large deviation from zero and they increase with the increasing Sm content. The sharp increase from 0.920 for Gd₂Sm_1.5Zr_0.5O₇ to 4.738 for Gd₂Sm₂O₇ may be caused by the pyrochlore-to-defect fluorite structural transition.

Table 3 Pugh’s indicator (G/B), elastic anisotropy index (A^U), sound wave velocity (v _m , in m/s), Debye temperature (Θ, in K) and Poisson’s ratio (σ) of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2).

Another important mechanical property of materials is the ductility, which is often evaluated by the Pugh’s indicator (G/B ratio)⁴⁵. The Pugh’s indicator of 0.5 is a boundary of brittleness or ductility, i.e., if G/B > 0.5, the material tends to be brittle; otherwise, the material is ductile⁴⁶. The calculated Pugh’s indicators are presented in Table 3. For the compounds of Sm_yGd_2-yZr₂O₇, the values are close to 0.5, and are located within the range of 0.499 to 0.505. As for Gd₂Sm_yZr_2-yO₇ (y = 0.5, 1, 1.5, 2), the G/B values of 0.205–0.428 are much smaller. Obviously, the Gd₂Sm_yZr_2-yO₇ compositions have better ductility than the Sm_yGd_2-yZr₂O₇ compounds. The Poisson’s ratio (σ) can also be used to evaluate the relative ductility of materials. Generally, the σ values are close to 0.1 and 0.33 for brittle covalent material and ductile metallic material, respectively^{46, 47}. The σ values are 0.286 and 0.284 for Gd₂Zr₂O₇ and Sm₂Zr₂O₇, respectively, which are comparable with the experimental data of 0.274–0.276 for Gd₂Zr₂O₇ and 0.277–0.278 for Sm₂Zr₂O₇ ^{20, 48}. As shown in Table 3, the Poisson’s ratio are ~0.285 for Sm_yGd_2-yZr₂O₇ and vary from 0.313 to 0.404 for Gd₂Sm_yZr_2-yO₇, meaning that the latter compositions are more ductile, which is consistent with the results obtained from the Pugh’s indicator.

In this study, the Debye temperature that is related to the hardness and thermal expansion coefficient of materials^{20, 49} is also estimated for Sm-contained Gd₂Zr₂O₇ by $\Theta =\frac{h}{{k}_{B}}{[\frac{3n}{4\pi }(\frac{{N}_{A}\rho }{M})]}^{\frac{1}{3}}{v}_{m}$. Here, h is the Planck’s constant, k _B is the Boltzmann’s constant, n is the number of atoms in molecular, N _A is the Avogadro’s constant, ρ is the density, M is the molecular mass and v _m is the sound wave velocity. The v _m can be deduced by ${v}_{m}={(\frac{3{({v}_{t}{v}_{l})}^{3}}{2{v}_{t}^{3}+{v}_{l}^{3}})}^{\frac{1}{3}}$, where ${v}_{l}={(\frac{B+4G/3}{\rho })}^{\frac{1}{2}}$ is the longitudinal sound velocity and ${v}_{t}={(\frac{G}{\rho })}^{\frac{1}{2}}$ is the transverse sound velocity⁴¹. As one can see from Table 3, Sm substitution for Gd-site has slight effects on the Debye temperature of Gd₂Zr₂O₇ and all the compositions have very similar results. However, as Sm substitutes for the Zr-site, the Debye temperature decreases considerably with the increasing Sm content. The Gd₂Sm₂O₇ has the lowest Debye temperature of 341.7 K, which is about 44.3% lower than that of Gd₂Zr₂O₇. These results suggest that Sm incorporation into Zr-site causes weaker interaction of chemical bonds and the Gd₂Sm_yZr_2-yO₇ with high content of Sm will have much larger thermal expansion coefficient than Gd₂Zr₂O₇.

Electronic structure of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (0 ≤ y ≤ 2)

In order to investigate how Sm incorporation influences the electronic structure of Gd₂Zr₂O₇, the atomic projected density of state (DOS) distribution of Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ are analyzed and plotted in Fig. 5 and Fig. 6, respectively. In Fig. 7a and b, the orbital projected DOS for Gd₂Zr₂O₇ and Sm₂Zr₂O₇ are also presented. For Gd₂Zr₂O₇, the valence band maximum (VBM) are mainly contributed by O 2p states hybridized with Zr 4d, Gd 5d and Gd 5p orbitals, and the conduction band minimum (CBM) are mainly contributed by Zr 4d states and O 2p states. The obtained band gap of 2.55 eV is comparable with the value of 2.71 eV reported by Wang et al.⁵⁰. For Sm₂Zr₂O₇, the DOS distribution is similar to that of Gd₂Zr₂O₇, i.e., the VBM are mainly contributed by O 2p states hybridized with Zr 4d, Sm 5d and Sm 5p states, and the CBM are mainly dominated by Zr 4d states and O 2p states. Considering that Sm and Gd are heavy atoms and the spin-orbit coupling (SOC) may affect the band gap and electronic structure of the investigated systems, we further calculate the density of state distribution of Gd₂Zr₂O₇ and Sm₂Zr₂O₇ employing the LDA + SOC method. A comparison of LDA and LDA + SOC results for both compounds is illustrated in Fig. 8. For Gd₂Zr₂O₇, the band gap value of 2.64 eV by LDA + SOC is close to the value of 2.55 eV by LDA method. In the case of Sm₂Zr₂O₇, the band gap values are 2.9 eV and 2.83 eV for LDA + SOC and LDA calculations, respectively. As shown in Fig. 8, the atomic projected and total DOS obtained by LDA with and without spin-orbit coupling exhibit very similar characters for both compounds. These results suggest that the spin-orbit coupling has slight effects on our results and such effects are thus not considered for the mixed states. When Sm substitutes for Gd-site, the Sm 5d orbitals also contribute to the VBM and interacts with the oxygen. Meanwhile, the insulating character of Gd₂Zr₂O₇ is kept and the band gap is broadened with the increasing Sm content, i.e., 2.62, 2.68, 2.76 and 2.83 eV for Sm_0.5Gd_1.5Zr₂O₇, SmGdZr₂O₇, Sm_1.5Gd_0.5Zr₂O₇ and Sm₂Zr₂O₇, respectively. In addition, the hybridization of O 2p and Zr 4d is slightly affected by the incorporation of Sm. As for Gd₂Sm_yZr_2-yO₇, the hybridization of O 2p states with Zr 4d, Gd 5d and Sm 5d orbitals also dominates the VBM. It is interesting to find that Sm incorporation causes electrons distributing on the fermi levels and the number of electrons increases with the increasing Sm content. This results in an insulating-to-metallic transition and the Gd₂Sm_yZr_2-yO₇ has much stronger electronic conductivity than the pure state. In the meantime, the Gd₂Zr₂O₇ and Sm_yGd_2-yZr₂O₇ do not have any magnetism, whereas the Gd₂Sm_yZr_2-yO₇ compositions exhibit strong ferromagnetic states due to the different charge states of Sm and Zr.

Figure 5

Projected density of state distribution for Sm_yGd_2-yZr₂O₇ (y = 0, 0.5, 1, 1.5, 2). The Fermi level is located at 0 eV.

Figure 6

Projected density of state distribution for Gd₂Sm_yZr_2-yO₇ (y = 0, 0.5, 1, 1.5, 2). The Fermi level is located at 0 eV.

Figure 7

Projected density of state distribution for Gd₂Zr₂O₇ and Sm₂Zr₂O₇. The Fermi level is located at 0 eV.

Figure 8

Projected and total density of state distribution for Gd₂Zr₂O₇ and Sm₂Zr₂O₇ obtained by the LDA and LDA + SOC methods. The Fermi level is located at 0 eV.

Obviously, Sm incorporation into Gd-site and Zr-site of Gd₂Zr₂O₇ causes very different electronic structures, i.e., the Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ are mainly of insulating and metallic characters, respectively. These results indicate that the thermal conductivity of Sm_yGd_2-yZr₂O₇ are mainly contributed by phonons, while both phonons and electrons contribute to the thermal conductivity of Gd₂Sm_yZr_2-yO₇.

Summary

In this work, a systematic study based on the DFT method is carried out to investigate the effects of Sm substitution for Gd-site and Zr-site in Gd₂Zr₂O₇ on its structural stability, mechanical properties, Debye temperature and electronic structures. It is shown that the Sm_yGd_2-yZr₂O₇ compositions keep the pyrochlore structure and their lattice parameters follow well the Vegard’s law, indicative of the formation of Gd₂Zr₂O₇-Sm₂Zr₂O₇ solid solution. On the other hand, Sm substitution for Zr-site influences the structure significantly and a pyrochlore-to-defect fluorite structural transition occurs. The Sm_yGd_2-yZr₂O₇ compositions are of elastic isotropy and their elastic moduli, ductility, Debye temperature and band gap vary slightly with the Sm content. However, the Gd₂Sm_yZr_2-yO₇ compounds show strong elastic anisotropy and their bulk, Young’s and shear moduli all decrease sharply with the increasing Sm content. Consequently, the Gd₂Sm₂O₇ has the minimum Young’s modulus of 87 GPa, minimum shear modulus of 31 GPa and minimum bulk modulus of 151 GPa. Meanwhile, both the Pugh’s indicator and Poisson’s ratio suggested that the Gd₂Sm_yZr_2-yO₇ have better ductility than the Sm_yGd_2-yZr₂O₇. As the Sm substitutes for Zr-site, the Debye temperature decreases considerably with the increasing Sm content and the Debye temperature of 341.7 K for Gd₂Sm₂O₇ is about 44.3% lower than that of Gd₂Zr₂O₇. In addition, the insulating character of Gd₂Zr₂O₇ is kept in the system of Sm_yGd_2-yZr₂O₇, while the Gd₂Sm_yZr_2-yO₇ compositions exhibit metallic characters. Our calculations demonstrate that substituting Sm for Zr-site is an effective approach to tailor the mechanical and thermal properties of Gd₂Zr₂O₇.

Methods

In this work, first-principles total energy calculations within the DFT framework are carried out. All calculations are performed with the Vienna Ab-initio Simulation Package (VASP)^{51, 52}. The interaction between electrons and ions is described by the projector augmented wave method^{52, 53}. All computations are based on a supercell containing 88 atoms. The convergence criteria for total energies and forces are 10⁻⁶ eV and 10⁻⁶ eV/Å, respectively. The structural relaxation is carried out at variable volume. In order to determine the values of cutoff energy and k-point sampling, a series of test calculation has been carried out. Figure 9 shows the variation of total energy of Gd₂Zr₂O₇ and Sm₂Zr₂O₇ with cutoff energy and k-point sampling, which leads to our calculation being performed with a 2 × 2 × 2 Monkhorst-Pack k-mesh for Brillouin-zone integrations and a cutoff energy of 600 eV for plane wave. For Sm_yGd_2-yZr₂O₇ and Gd₂Sm_yZr_2-yO₇ (y = 0.5, 1, 1.5), the structure models are constructed by the special quasirandom structure approach^54,55,56.

Figure 9

Variation of the total energy of Gd₂Zr₂O₇ and Sm₂Zr₂O₇ with (a) k-point sampling (the cutoff energy is fixed at 650 eV) and (b) cutoff energy (the k-point sampling is fixed at 2 × 2 × 2).