Ab-Initio Study of (111) to (001) Texture Transformation in Ag Thin Films

Abstract

First-principles density functional theory was used to study the (111) to (001) texture transformation in Ag thin films. The texture transformation was experimentally observed and studied during annealing of Ag thin films. Ag(111) showed the lowest surface energy of 0.031 eV/Ų, whereas Ag(001) showed a surface energy of 0.040 eV/Ų. Ag films grown along ⟨111⟩ and ⟨001⟩ exhibited epitaxial hardening and softening behavior, respectively. Further, we found that the strain-induced texture transformation of Ag thin films from (111) to (001) can also be realized under a hydrostatic compressive strain of 2.25% or a biaxial compressive strain of 3%. Our results agree greatly with previously reported experimental observations on texture transformation in Ag thin films.

Fig. 3 The total energies obtained under hydrostatic (solid line) and epitaxial (dashed line) strain for the (111)-oriented (circle) and (001)-oriented (square) bulk Ag as a function of in-plane lattice constants. Compressive strain regions from strain of −3% to 0 (lowest-energy structures) and tensile strain regions from 0 to strain of 3%.

1. Introduction

The preferred orientations for Ag thin films on substrates have been widely reported owing to the research interest in understanding the competition between strain and interface energy during epitaxial Ag grain growth. Floro et al. reported that Ag(111) and Ag(001) films grown on Ni(001) substrates have favorable twist interface energies of respectively 440 and 630 mJ/m² at twist angles of 26.6° and 0°.¹⁾ The (111) texture plane had the lowest twist interface energy (440 mJ/m²), leading to initial preferred orientations. The texture transformation from Ag(111) thin films to Ag(001) thin films was observed when the film thickness exceeded 1500 Å, regardless of liquid nitrogen temperature (−175°C) or annealing temperature (350°C, 400°C, or 600°C). Notably, thick Ag thin films favored the growth of the (001) grains despite the significantly high interface energy (630 mJ/m²) of Ag(001)/Ni(001). In another study, Thompson et al. showed that grain growth owing to the reduction of the total energy associated with grain boundaries can play a key role in a strain relief mechanism for strain-energy-minimizing textures.^2,3) They found that strain-energy-driven grain growth, coarsening or grain size increase, favors the growth of the (001) texture, where the texture transformation from (111) to (001) depends on the film thickness and the difference between the deposition temperature and grain growth temperature. Subsequently, the results from a computer simulation of two-dimensional grain growth confirmed that the (001) texture is favored in highly strained films and can develop only if the yield stress of the (111) texture is sufficiently high to prevent early yielding.⁴⁾ However, Greister et al. reported the occurrence of abnormal Ag grain growth and complete texture transformation from (111) to (001) both with and without substrates.^5,6) Moreover, they concluded that free-standing or stress-free films cannot explain strain-energy-driven grain growth affecting the growth of the (001) texture.

Baker et al. presented a new mechanism for the (111) to (001) texture transformation, according to which reduction in defect energy is the driving force for secondary grain growth.^7,8) Moreover, this driving force can cause the formation of (001) recrystallization nuclei, and it increases with film thickness. However, the density of the (001) recrystallization nuclei decreases with increasing film thickness, which indicates that more initial (001) recrystallization nuclei are present in intermediate-thickness Ag films. Interestingly, Ag films exhibit compression during heating from room temperature to 405°C, which suggests a more favorable strain state for texture transformation. Very recently, Ellis et al. reported that when no bulge pressure is applied (0 kPa), Ag thin films are compressive at the beginning of the texture transformation but become tensile after the transformation is complete.⁹⁾ Further, the volume fraction of the (001) texture component, as a function of time during annealing, at applied bulge pressures of 5 and 13 kPa is equal to that at 0 kPa. This finding indicates that the imposed strain energies have no effect on the texture transformation kinetics. However, the stress in Ag thin films at the beginning of the texture transformation increases (−83, −35, and 31 MPa) upon increasing the applied bulge pressure (0, 5, and 13 kPa, respectively). These experimental results suggest that the actual driving force is considerably larger than strain energy as the driving force, thus indicating that the texture transformation is insensitive to applied bulge pressure. Despite these findings, the fundamental-strain-induced texture transformation is still only vaguely understood. To address this, we calculated the surface energies of Ag(111) and Ag(001) thin films to determine the initial preferred orientation of the Ag texture. In addition, we performed epitaxial softening calculations to determine the preferred Ag texture orientations under tensile and compressive stresses. Finally, we elucidated a key theoretical explanation for strain-induced texture transformation.

2. Computational Methods

A series of ab initio calculations based on the density functional theory (DFT) were performed to elucidate the epitaxial deformation behavior of Ag thin films. The calculations were performed using the Vienna ab initio simulation package (VASP),^10–12) in which the exchange–correlation functional was predicted using the generalized gradient approximation (GGA) through the Perdew–Wang (PW91) method to efficiently treat ion–electron interactions.^13,14) The electronic configuration for the valence electrons of Ag (space group: 225 Fm-3m) is 4d¹⁰5s¹. Two six-atom bulk models denoted by (111)- and (001)-oriented bulk Ag were constructed using the bulk Ag unit cell, as shown in Fig. 1(a) and 1(b), respectively, and then two surface models were constructed, as denoted by Ag(111) and Ag(001) in Fig. 1(c) and 1(d), respectively. The (111)- and (001)-oriented bulk Ag models constructed using a plane-wave cutoff energy of 400 eV and 50 and 63 irreducible k points generated from a 7 × 7 × 2 gamma-centered grid and a 11 × 11 × 5 Monkhorst–Pack grid, respectively, were fully relaxed to achieve a force accuracy of 0.0001 eV/Å.

Fig. 1

Atomistic representations of six-atom lattices of (a) (111)- and (b) (001)-oriented bulk Ag. Representative structures for (c) Ag(111) and (d) Ag(001) surface models with a (2 × 2) and ($\sqrt{2} \times \sqrt{2} $) basal dimensions, respectively. Schematic illustrations showing the (e) bi-axial and (f) tri-axial deformation of bulk models. The arrows point to principal directions of strain.

For calculating the surface energy, the Ag(111) and Ag(001) surface models were fixed at Ag₆ and (2 × 2) and ($\sqrt{2} \times \sqrt{2} $) basal settings, in which the basal dimensions were derived from the (111)- and (001)-oriented bulk Ag models, respectively, at the equilibrium lattice constant. The surface free energy σ of the Ag slab with repeated geometry is given by the following equation:   

\begin{equation} \sigma=\left(E_{\textit{slab}}-\sum n_{\textit{Ag}}\mu_{\textit{Ag}}\right)\Bigm/(2A), \end{equation} (1)

where E_slab is the total energy of the Ag(111) or the Ag(001) slab depicted in Fig. 1(c) and 1(d), respectively. Here, n_Ag and μ_Ag are the number of Ag atoms and the chemical potential of Ag, respectively. A is the surface area and the factor of 2 denotes two equivalent surfaces in the Ag slab. Optimized Ag(111) and Ag(001) surface atomic structures separated by a ∼40-Å-thick vacuum region were obtained using a plane-wave cutoff energy of 400 eV and 13 and 20 irreducible k points generated from a 5 × 5 × 1 gamma-centered grid and an 8 × 8 × 1 Monkhorst–Pack grid, respectively, to achieve a force accuracy of 0.0001 eV/Å.

The Ag films can be analyzed using the harmonic elastic theory. For this, the harmonic function q_harm is calculated as¹⁵⁾   

\begin{equation} q_{\textit{harm}}=1-\frac{B}{C_{11}+2\Delta\gamma_{\textit{harm}}}, \end{equation} (2)

where B = (C₁₁ + 2C₁₂)/3 is the bulk modulus and Δ = C₄₄ − 0.5(C₁₁ − C₁₂). C₁₁, C₁₂, and C₄₄ are elastic constants of the Ag unit cell. For a cubic lattice, the geometry factor γ_harm is given by   

\begin{align} \gamma_{\textit{harm}}(\varphi,\theta)&=\sin^{2}(2\theta)+\sin^{4}(\theta)\sin^{2}(2\varphi)\\ &=\frac{4}{5}\sqrt{4\pi}\left[K_{0}(\varphi,\theta)-\frac{2}{\sqrt{21}}K_{4}(\varphi,\theta)\right], \end{align} (3)

where φ and θ are the polar and azimuth angle, respectively, in the spherical polar coordinate system. K_l is the kubic harmonic of angular momentum l. For determining the epitaxial strain relaxation of Ag(111) and Ag(001) films, we apply anharmonic epitaxial strain on the (111)- and (001)-oriented bulk Ag, which can be quantified by the dimensionless parameter¹⁵⁾   

\begin{equation} q(a,\hat{\mathrm{G}})=\Delta E^{\textit{epi}}(a,\hat{\mathrm{G}})/\Delta E^{\textit{bulk}}(a), \end{equation} (4)

where ΔE^epi is the increase in epitaxial energy upon applying biaxial deformation on the bulk models, as shown in Fig. 1(e). ΔE^bulk is the increase in bulk energy upon applying hydrostatic or triaxial deformation on the bulk models, as shown in Fig. 1(f). $\hat{\text{G}}$ denotes the growth direction, and a is an in-plane lattice dimension.

3. Results and Discussion

The relaxed lattice constants of the optimized Ag unit cell are a = 4.12 Å. It was used to calculate elastic constants, C₁₁ = 1.70, C₁₂ = 1.16, and C₄₄ = 0.64 GPa, which closely matched the values, C₁₁ = 1.31, C₁₂ = 0.97, and C₄₄ = 0.51 GPa, obtained experimentally in Ref. 16. For the optimized (111)-oriented bulk Ag, the following values were found: lattice constants a₍₁₁₁₎ = b₍₁₁₁₎ = 2.92 Å, c₍₁₁₁₎ = 14.38 Å and bond angles α₍₁₁₁₎ = β₍₁₁₁₎ = 90°, γ₍₁₁₁₎ = 120°. For the optimized (001)-oriented bulk Ag, the corresponding lattice constants and bond angles were as follows: a₍₀₀₁₎ = b₍₀₀₁₎ = 2.90 Å, c₍₀₀₁₎ = 12.71 Å and α₍₀₀₁₎ = β₍₀₀₁₎ = γ₍₀₀₁₎ = 90°. Subsequently, the Ag(111) and Ag(001) surface models were fixed at the in-plane lattice constants derived using the optimized (111)- and (001)-oriented bulk Ag (i.e., a₍₁₁₁₎ = b₍₁₁₁₎ = 2.92 Å and a₍₀₀₁₎ = b₍₀₀₁₎ = 2.90 Å, respectively). The surface energies of Ag(111) and Ag(001) were 0.031 and 0.040 eV/Ų, respectively, which agree with other first-principles calculated values of 0.039 eV/Ų for Ag(111) and 0.044 eV/Ų for Ag(001).¹⁷⁾ Our results reveal that the Ag(111) surface is energetically more favorable than Ag(001); this finding is in agreement with many other experimental findings on initial preferred (111) orientations.^1–9)

In Fig. 2, the epitaxial softening function, calculated by substituting C₁₁ = 1.70, C₁₂ = 1.16, and C₄₄ = 0.64 GPa in eq. (2), represents that q_harm(⟨111⟩) and q_harm(⟨001⟩) are elastically hard and soft directions, respectively. These results indicate that the growth strain decays softly in the (001) texture plane, suggesting that the strain relaxation mechanism within the (001) texture region is more favorable than in the (111) texture region and thus leads to (001) recrystallization. This result is in agreement with other experimental findings.^7,8)

Fig. 2

A three-dimensional plot of a harmonic functional of the unit cell of Ag with hollows along ⟨001⟩ indicating the epitaxial softening behavior and bulges along ⟨111⟩ indicating the epitaxial hardening behavior in a fully relaxed Ag bulk for any strain.

Figure 3 shows the detailed hydrostatic and epitaxial curves for the (111)- and (001)-oriented bulk Ag models obtained under strain in the range from −3% to 3%. The results in Fig. 3 indicate that the lowest (111)- and (001)-oriented bulk energies of −16.48 and −16.36 eV correspond to the equilibrium in-plane lattice constants 2.95 and 2.92 Å, respectively. Further, beyond the hydrostatic compressive strain of 2.25%, the (111)-oriented bulk is no longer stable, whereas the (001)-oriented bulk is stable. In the case of biaxial deformation of bulk models, the (111)-oriented bulk becomes clearly unstable beyond the biaxial compressive strain of 3%. These results suggest that (111) Ag thin films become stiffer under hydrostatic compressive strains greater than 2.25% or biaxial compressive strains greater than 3%. This leads to an increase in the driving force for the strain-induced texture transformation in Ag thin films from (111) to (001). These results confirm the experimental observation that the growth of the (001) texture is often kinetically driven by increasing film thickness. Ag(001) is the epitaxial-strain-stabilized texture plane, but the epitaxial growth on the (001) face is limited by high surface energy (0.040 eV/Ų). Therefore, it is not surprising that the (001) texture is present in intermediate-thickness Ag films.^7,8)

Fig. 3

The total energies obtained under hydrostatic (solid line) and epitaxial (dashed line) strain for the (111)-oriented (circle) and (001)-oriented (square) bulk Ag as a function of in-plane lattice constants. Compressive strain regions from strain of −3% to 0 (lowest-energy structures) and tensile strain regions from 0 to strain of 3%.

A plot of the anharmonic behavior of constituent $q(a,\hat{\text{G}})$ versus in-plane lattice parameters a is shown in Fig. 4. The plot indicates that q(a, ⟨111⟩) softens as compressive strain and q(a, ⟨001⟩) softens as tensile strain; these trends are similar to those reported previously.¹⁵⁾ Moreover, the (111) texture is considerably stiffer than the (001) texture, indicating that the (111) texture is more elastically anharmonic than the (001) texture. This result is consistent with the previously reported experimental observation that the strain in Ag thin films varies from compressive to tensile as the texture transforms from (111) to (001).^8,9)

Fig. 4

The anharmonic epitaxial softening for bulk Ag as a function of in-plane lattice constants indicating that the (111)-oriented bulk Ag are more elastically anharmonic than the (001)-oriented.

4. Conclusion

The surface energy and epitaxial softening behavior of Ag thin films were studied using first-principles DFT–GGA calculations. Our results showed that Ag(111) is energetically more favorable than Ag(001) from the viewpoint of surface energy. By studying the harmonic and anharmonic elasticity of (111)- and (001)-oriented Ag bulk, we found that the strain-induced texture transformation of Ag thin films from (111) to (001) can be obtained by applying a hydrostatic compressive strain of 2.25% or a biaxial compressive strain of 3%. Our findings are in excellent agreement with previously reported experimental observations on texture transformation in Ag thin films.

Acknowledgments

Computational studies were performed using the resources of the National Center for High Performance Computing.

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