Generalized-stacking-fault energy and twin-boundary energy of hexagonal close-packed Au: A first-principles calculation.

Although solid Au is usually most stable as a face-centered cubic (fcc) structure, pure hexagonal close-packed (hcp) Au has been successfully fabricated recently. However, the phase stability and mechanical property of this new material are unclear, which may restrict its further applications. Here we present the evidence that hcp → fcc phase transformation can proceed easily in Au by first-principles calculations. The extremely low generalized-stacking-fault (GSF) energy in the basal slip system implies a great tendency to form basal stacking faults, which opens the door to phase transformation from hcp to fcc. Moreover, the Au lattice extends slightly within the superficial layers due to the self-assembly of alkanethiolate species on hcp Au (0001) surface, which may also contribute to the hcp → fcc phase transformation. Compared with hcp Mg, the GSF energies for non-basal slip systems and the twin-boundary (TB) energies for and twins are larger in hcp Au, which indicates the more difficulty in generating non-basal stacking faults and twins. The findings provide new insights for understanding the nature of the hcp → fcc phase transformation and guide the experiments of fabricating and developing materials with new structures.

In recent years, many new materials of nanoparticles have been synthesized and applied in potential domains. Note that there is often phase transformation among face-centered cubic (fcc), body-centered cubic (bcc) and hexagonal closed-packed (hcp) systems in mono-component nanocrystal superlattices, especially for the noble metal nanostructures123456789. This can be attributed to the diffusion of guest molecules (e.g. dodecanethiol)3, surface stress45, specific stacking processes depending on evaporation kinetics6 and high pressure78 etc. Solid Au is usually most stable as an fcc structure. Surprisingly, Huang et al.1 have reported the first in situ synthesis of pure hcp Au square sheets on graphene oxide, which possess an edge length of 200–500 nm and a thickness of ~2.4 nm (~16 Au atomic layers). The exclusive hcp phase, which is stable under ambient conditions, provides a significant basis for fabricating novel Au architectures with unique chemical and physical properties10. Recently, it is observed that a large value of initial residual stress drives the fcc/hcp phase transformation in [100]/{100} Au nanowires during the energy minimization process, as conducted by molecular dynamics simulation5. Moreover, Stoeva et al.11 reported that Au particles obtained by the solvated metal atom dispersion method predominantly organized into hcp nanocrystal superlattices with long-range translational ordering.

Note that structural defects such as point defects, stacking faults, dislocations, twins and grain boundaries can be observed in nanocrystal superlattices12, which are similar to the phenomena shown in classical crystals. In hcp Au nanocrystal, the stacking faults and twin defects were both indicated by transmission electron microscope (TEM)1. With the appearance of faulted stacking in hcp structures, the ideal hcp packing sequence (…ABABAB…) would exhibit partial fcc feature, e.g. the ABC sequence in I2 (…ABABCACA…) and T2 (…ABABCBABA…) stacking faults. Therefore, the stacking faults in one system can be actually regarded as the presence of a segment of the other2. Moreover, the mixed stacking of hcp and fcc planes was confirmed in Au plates, suggesting that the pure hcp phase will become less stable when the Au sheets grow thicker1. Since the close relationship between the stacking faults and phase transformation, it is necessary to investigate the defect properties of hcp Au, so as to evaluate the structural stability and growth mechanism of this new material.

Furthermore, the first synthesis of hcp Au stimulates us to explore its novel mechanical properties and to make comparisons with common hcp structures, e.g. Mg, in anticipation of developing new materials which are distinct from those natural crystals for wider application. On the atomic level, the plastic formability of hcp crystals is closely related to the ease of the formation of dislocations and twins. Note that the GSF energy, which quantifies the ability of the dislocation in a crystal to glide onto an intersecting slip plane, affects the nucleation and mobility of dislocations13 as well as the propensities to form twins14. Meanwhile, the TB energy is intimately connected with the mobility of twinning dislocations15, without referring to the atomic shuffling during the nucleation of twins in hcp crystals. Therefore, the TB energies mainly depict the stability of twin structures without involving the formation dynamics.

Though these defect parameters are significant in reflecting the mechanical properties of hcp Au, they are difficult to be investigated experimentally due to the structural instability. Moreover, a universal fcc to hcp phase transformation is possible only when the dimension of Au nanostructure decreases to a critical value, which also increases the difficulty in detecting the hcp Au phase by experiments. Alternatively, the first-principles method based on density functional theory (DFT)16 has emerged as a key technology in determining the properties of material, which enables the calculation of defect energies at a reasonable computational cost. Therefore, the main purpose of this paper is to study the energies of stacking faults and twin boundaries in hcp Au using first-principles method. For the calculation of GSF energy, the basal slip system ( and ), prismatic slip system () and pyramidal slip system ( and ) are considered respectively, as illustrated in Fig. 1. Meanwhile, the mirror reflection, the mirror glide as well as the mirror reflection and the mirror glide twin boundaries are taken into account when calculating the TB energy. In addition, the elastic properties of hcp Au, which are associated with the brittleness and ductility of materials17, are also calculated for the systematic study of hcp Au. For comparison, the corresponding physical parameters for fcc Au and hcp Mg are listed as well, in order to estimate the structural stability and application of hcp Au. Note that a lot of surfactant molecules adsorb on the surface of Au nanostructure during the synthesis processes123. Therefore, the effect of the thiol surfactant to the crystal lattice of hcp Au is also investigated. The intrinsic parameters in this calculation contribute to completing the theoretical database of hcp Au, and act as a guide to experiments of fabricating and developing materials with new structure.

Figure 1

Schematic representation of different slip systems for hcp crystals: (a) , and slip systems; (b) and slip systems.

Results and discussion

Structural parameters

The calculated equilibrium lattice constants and cohesive energy for fcc Au, hcp Au and Mg are listed in Table 1. For hcp Au, the structural parameters (a = 2.952 Å, c = 4.885 Å) in our work are in acceptable tolerance with the values of the experiment (a = 2.96 Å, c = 4.84 Å)1 and other simulation (a = 2.927 Å, c = 4.903 Å)18. The c/a ratio of hcp Au (1.655) is larger than that of Mg (1.608), implying larger anisotropy in lattice parameters for hcp Au. Note that the lattice constant for primitive cell of fcc Au ( Å) is very close to that of hcp Au (2.952 Å), which reveals the intrinsic link between the fcc and hcp phase. Moreover, the cohesive energy () of hcp Au (−3.2045 eV) is negative, indicating that the existence of this new phase is energetically favorable. According to the energetic results, the stability of crystal is sequenced as: fcc Au >hcp Au.

Table 1

Lattice constants and cohesive energy for fcc Au, hcp Au and Mg.

Structure	Lattice constant (Å)		Cohesive energy(eV/atom)
	a	c
fcc Au	4.197	-	−3.2064
fcc Auexp, a	4.0786	-
fcc Aucal, b	4.173	-	−3.2026
hcp Au	2.952	4.885	−3.2045
hcp Auexp, c	2.96	4.84
hcp Aucal, b	2.927	4.903	−3.2018
hcp Mg	3.221	5.178	−1.4411
hcp Mgexp, d	3.209	5.210
hcp Mgcal, e	3.19	5.17

aJette, E. R. et al.29.

bWang, Y. et al.18.

cHuang, X. et al.1.

dWalker, G. B. et al.49.

eHector Jr., L. G. et al.50.

Surfactant effect

In hcp Au nanostructure, a lot of surfactant molecules adsorbed on its surface12. In view of the molecule-surface interactions, the effect of the surfactant, e.g. thiols, to the lattice constant of hcp Au is investigated. Spontaneous formation of an ordered molecular over-layer on the gold surfaces has been found for the thiols, namely molecular self-assembly. A series of experiments has revealed the Au (111) surface releases gold adatoms that become incorporated into the monolayer19202122. This configuration is further confirmed by the Au top-site adsorption of alkanethiolate in experiments2324.

For hcp Au (0001) surface, the Au-adtom-induced self assembly of alkanethiolate species is presented analogy to the fcc Au (111) surface. Accordingly, a 4 × 2 slab (along the and direction) with 9 layers was constructed, separated by a 12 Å vacuum. Upon relaxation, the Au adatom locates at the twofold bridge site and the headgroup-S atom of CH3S occupies the Au top-site as shown in Fig. 2a,b. The distribution of Au adatom and alkanethiolate species on hcp Au (0001) surface is similar to that on fcc Au (111). This originates from the same in-plane atomic coordination on the two close packed planes. Meanwhile, the S atom is attached to both the Au adatom (Aua in Fig. 2a) and the underlying lattice atom (Aul in Fig. 2a) with r(S-Aua) = 2.344 Å and r(S-Aul) = 2.523 Å respectively. These values are very close to the 2.33 Å and 2.49 Å in the case of adoption of alkanethiolate on fcc Au (111)19. The binding energy of the CH3S- species and the Au substrate is 2.08 eV, which is smaller than the value of cohesive energy for hcp Au, i.e. 3.2045 eV (Table 1).

Figure 2

Schematic illustrations of the Au-adtom-induced self-assembly of alkanethiolate species on hcp Au (0001) surface: (a) side view (5 layers are shown) and (b) top view (2 layers are shown).

Note that the formation of bare hcp Au (0001) surface is accompanied by charge transfer from the dangling bonds into the in-plane bonds, which increases the attraction between the surface atoms. The adoption of S with strong electronegativity leads to a charge redistribution into the surface-molecule bonds, which weakens the attraction between surface Au atoms and increases the superficial lattice constant. For the adoption of alkanethiolate on hcp Au (0001) surface, the average in-plane distance between Aul and surrounding Au atoms () is increased from 2.952 Å to 2.997 Å. Such adsorbate-induced surface stress has also been confirmed in alkanethiolate-Au (111) self-assembled monolayers25. Moreover, the adsorption of alkanethiolate surfactant results in increased tensile stress when the S-C bond is normal to the surface, and therefore gives rise to a significant expansion of the Au lattice (: 3.049 Å). This situation is remedied to some extent by the tilting of the S-C bond (: 2.997 Å). The resulting directionality of the S-C bond leads to a preferred value of the C-S-Aul bond angle of 106.5°.

The covalent character of the S-Au bond also expands the inter-planar spacing between the superficial hcp Au (0001) planes slightly. The average distance between the A1 and A2 layers (see Fig. 2a, ) is 4.895 Å for the adsorption of alkanethiolate species, which is 4.890 Å on the bare hcp Au (0001) surface. Note that the titling of the S-C bond also alleviates the expansion of inter-planar spacing, i.e. the value of is 4.912 Å if the S-C bond is normal to the surface. The lattice parameters in the underlying layers (a = 2.952 Å, c = 4.887 Å) are similar to those of hcp Au bulk (a = 2.952 Å, c = 4.885 Å).

The adsorption of alkanethiolate molecules on hcp Au (0001) surface enlarges the gold lattice in the superficial layers (e.g. the three outer layers), while it has slight effect in the underlying layers. The extension of surface lattice constants may facilitate the hcp  Æ  fcc phase transformation, as the lattice parameter a of hcp Au (2.952 Å) is smaller than that for the primitive cell of fcc Au ( Å). Moreover, the fcc Æhcp transformation of Au is proven to involve the compression of lattice along the direction22, and it in turn is predicted to be an expansion process for the hcp → fcc transformation. However, more experimental evidences are in expectation.

GSF energy

To calculate the GSF energies of hcp Au, which reflects its bulk property, a 16-layer slab (32-atom) is constructed based on the optimized unit cell (a = 2.952 Å, c = 4.885 Å). Five slip systems for both hcp Au and Mg are taken into account, as shown in Fig. 1. Specifically, the three < a > -type slip systems (, and ) are considered. Moreover, the second pyramidal slip system (), which is responsible for accommodating the strains of c- and a-axis simultaneously, is also calculated. Meanwhile, the basal slip system () is considered, as the dissociation of perfect dislocations into partial dislocations is energetically favorable26:

For fcc Au, the GSF energies are calculated in the primary slip system ().

The GSF energy curves (-curve), which express the initiation of one whole stacking fault, were plotted against the applied shift vector for fcc Au, hcp Au and Mg in Fig. 3a–f. Hereinto, the maximum energy on the -curve is the unstable SFE , which denotes the lowest energy barrier for dislocation nucleation27, while the local minimum indicates the intrinsic SFE 28. The computed and values are marked on the curves and summarized in Table 2, together with those obtained by other first-principles predictions293031323334. Note that the GSF energies for Mg32 in different slip systems accord with the results calculated by others293031323334, which proves the accuracy of our calculations.

Figure 3

GSF energy curves for fcc Au in (a) slip system; for hcp Au and Mg in (b) , (c) , (d) , (e) and (f) slip systems.

Table 2

The unstable stacking fault energy ( ) and the intrinsic stacking fault energy ( ) for hcp Au, Mg and fcc Au in different slip systems.

Slip system	Structure	k-mesh	GSF energy (mJ/m2)
			Unstable SF ()	Intrinsic SF ()
	hcp Au32	5 ×8×1	γus (I2) 65.9	γis (I2) −10.5
			γus (T2) 67.5	γis (T2) −12.0
	Mg32	4 ×8×1	γus (I2) 95.8, 94.9b, 92c, 91f	γis (I2) 20.1 21.8a, 26.1b, 33c
			γus (T2) 120.3 111.2b	γis (T2) 38.2 37.1b
	hcp Au32	5 ×8×1	222.4	–
	Mg32	4 ×8×1	253.6, 276f	–
	hcp Au32	8 ×10×1	248.5	–
	Mg32	8 ×10×1	216.1, 231f	–
	hcp Au32	8 ×10×1	390.7	–
	Mg32	8 ×8×1	320.2 310d, 343f	–
	hcp Au32	8 ×10×1	640.1	309.0
	Mg32	8 ×8×1	449.5 485d	228.0 236d
	fcc Au13	10 ×17 ×1	γus (I) 83.6 94e	γis (I) 24.3 27e
			γus (T) 96.0	γis (T) 22.8

aWang, Y. et al.34.

bJette, E. R. et al.29.

cHan, J. et al.30.

dNogaret, T. et al.31.

eWu, X. Z. et al.32.

fPei, Z. et al.33.

In slip systems (Fig. 3a), (Fig. 3b) and (Fig. 3f), the curves keep saddle-shaped, in which the local maximum and minimum values both exist. However, there is no local minimum on the GSF energy curves of the (Fig. 3c), (Fig. 3d) and (Fig. 3e) system.

In system of fcc Au, the (83.6 mJ/m2) and (24.3 mJ/m2) values for the intrinsic I stacking fault are close to the results (94 and 27 mJ/m2) of Wu et al32. For some common fcc metals, the energy barrier () for the initiation of the I structure is: 169 mJ/m2 (Al)35, 180 mJ/m2 (Cu)35, 305 mJ/m2 (Ni)36, 111 mJ/m2 (Ag)32, 311 mJ/m2 (Pt)32 and 215 mJ/m2 (Pd)32. By comparison, the relatively low GSF energy for fcc Au indicates the ease of slip and thus, good plasticity.

In hcp Au and Mg, the curves for different slip systems show similar trends in variation, e.g. the existence of and corresponding to approximately the same fault vector (abnormal: in system). The GSF energies of hcp Au for five slip systems are sequenced as:  <  <  <  < .

In the basal slip system, the and values are apparently lower for hcp Au than those for hcp Mg, and are even lower than those for fcc Au. Specially, the values for the intrinsic (I2) and twin-like (T2) faults are even negative, implying that the I2 (…ABABABCACACA…) and T2 (…ABABABCBABAB…) can form extremely easily. In the basal slip system, the value of hcp Au is 12.3% lower than that of hcp Mg. This also indicates that the basal stacking faults can form more easily in hcp Au than in hcp Mg. Once the basal stacking faults generate, the segment of fcc crystal will appear, which is regarded as the beginning of hcp → fcc phase transformation. Accordingly, the hcp Au is predicted to be unstable. Moreover, the experimental evidence has been validated that hcp → fcc phase transformation of Au can proceed easily, e.g. when the Au sheet is exposed to an electron beam or grows thicker1.

In the prismatic as well as pyramidal and slip systems, the values of hcp Au are larger than those of hcp Mg by about 15.0%, 22.0% and 42.4% respectively. Therefore, the difficulties are increased in generating non-basal stacking faults for hcp Au compared with Mg.

TB energy

The mirror reflection, mirror glide, mirror reflection and mirror glide twin boundaries were constructed following Wang et al.’s work34, as illustrated in Fig. 4a–d respectively. The grey and gold balls in supercells represent the two kinds of stacking planes along the in hcp Au. The rotation angle between the hcp matrix and the twin is 87.4° and 124.8° for and twins, respectively.

Figure 4

Schematic illustrations of the twin boundaries: (a) mirror reflection, (b) mirror glide, (c) mirror reflection and (d) mirror glide. The grey and gold balls represent the two atomic layers along the direction conventionally used for the hcp structure.

The TB energies for hcp Au and Mg are summarized in Table 3, and the results of Mg are shown to agree with the data reported by Wang et al.34. Considering the and twin boundaries separately, the corresponding TB energies are close in values for mirror reflection and mirror glide-type twins. Therefore, it is predicted that the glide of the interfacial crystal planes have minor effects on the TB energies. Similar results have also been obtained in Wang et al.’s work34. Moreover, the twin structures with 15 and 17 layers are adopted here to investigate the effects of supercell sizes on the TB energies. For mirror reflection twin boundaries, increasing the supercell size by up to 34 atoms can only decrease the TB energies of hcp Au by about 7.8%. For mirror glide, mirror reflection and mirror glide twin boundaries, the TB energies of Au34 (17 layers) are slightly larger than those of Au30 (15 layers) by less than 3.7%. Accordingly, the calculated TB energies show weak dependence on the adopted supercell size.

Table 3

Calculated twin boundary (TB) energy for hcp Au and fcc Au (mJ/m2).

Twin system	Structure	k-mesh	TB energy (mJ/m2)
			This work	Previous calculation
mirror reflection	Au34 (17 layers)	8 ×4 ×1	175.2
	Au30 (15 layers)	8 ×4 ×1	190.0
	Mg34 (17 layers)	8 ×3 ×1	120.5	118.8 (40 atoms)a
	Mg30 (15 layers)	8 ×3 ×1	122.6	122.3 (20 atoms)a
mirror glide	Au34 (17 layers)	8 ×4 ×1	189.1
	Au30 (15 layers)	8 ×4 ×1	182.3
	Mg34 (17 layers)	8 ×3 ×1	112.5	120.8 (40 atoms)a
	Mg30 (15 layers)	8 ×3 ×1	116.8	125.3 (20 atoms)a
mirror reflection	Au34 (17 layers)	9 ×5 ×1	107.3
	Au30 (15 layers)	9 ×5 ×1	106.8
	Mg34 (17 layers)	8 ×4 ×1	80.8	84.2 (40 atoms)a
	Mg30 (15 layers)	8 ×4 ×1	83.1	85.5 (80 atoms)a
mirror reflection	Au34 (17 layers)	9 ×5 ×1	114.3
	Au30 (15 layers)	9 ×5 ×1	110.5
	Mg34 (17 layers)	8 ×4 ×1	80.6	84.2 (40 atoms)a
	Mg30 (15 layers)	8 ×4 ×1	80.1	81.0 (80 atoms)a

aWang, Y. et al.34.

The TB energies mainly reflect the stability of twin structures without referring to the nucleation course of twins. By comparison, the TB energies are larger in hcp Au than in hcp Mg for the four twin types, implying less stability of twin structures in the former case. Moreover, the TB energies in hcp Au are lower than TB energies, which is the same as the situation in Mg.

Elastic property

For a hexagonal solid, there are five independent elastic constants, namely, C11, C12, C13, C33 and C55. For a cubic crystal, there are only three independent elastic constants, namely, C11, C12 and C44. The elastic constants for hcp Au, Mg and fcc Au are calculated and listed in Table 4. Note that our calculated Cij values agree well with the results of Shang et al.3738, in which the stability of pure elements was successfully discussed in terms of the elasticity. The acceptable tolerance between our results and other simulation3738 indicates the accuracy of our work.

Table 4

Calculated elastic constants for hcp Au, Mg and fcc Au (GPa).

Material	C11	C12	C13	C33	C44	C66	Br	Bv	B	Gr	Gv	G	E
hcp Au	174.9	139.8	112.7	182.9	15.1	17.6	140.0	140.3	140.2	18.3	20.7	19.5	56.0
hcp Aucal, b	185.7	138.9	112.7	181.3	15.6	-	141.5	142.4	141.9	20.9	23.5	22.2	63.3
hcp Mg	63.1	22.2	22.7	66.3	22.6	20.5	36.4	36.4	36.4	21.4	21.5	21.5	53.8
hcp Mgcal, a	63.5	24.9	20.0	66.0	19.3	-	-	-	35.83	-	-	18.5	47.4
fcc Au	154.4	129.2	-	-	44.1	-	137.6	137.6	137.6	22.1	31.5	26.8	75.4
fcc Aucal, b	159.1	136.7	-	-	27.6	-	144.2	144.2	144.2	17.4	21.0	19.2	55.2

aGaneshan, S. et al.37.

bShang, S. L.et al.38.

Based on the predicted Cij’s, the mechanical stability for a given structure can be judged according to Born’s criteria3940: , and for hexagonal system; and , and for cubic system.

Note that hcp Mg and fcc Au, which have stable structures at room temperature, satisfy the above Born criteria for mechanical stability. Meanwhile, the new material hcp Au also accords with the Born’s criteria, implying the mechanically metastable. The results of elastic constants confirm the possibility of fabricating pure and stable hcp Au under ambient conditions1. Although the elastic property demonstrates that hcp Au is metastable, the phase transformation from hcp to fcc is still predicted to proceed easily according to the GSF energies and the surfactant effect, which is also in good agreement with the experimental observation1.

Starting from Cij’s, the polycrystalline aggregate properties such as bulk modulus (B), shear modulus (G) and Young’s modulus (E) are calculated according to the Voigt41 (Bv and Gv), Reuss42 (Br and Gr) and Hill43 approximations and indicated in Table 4. Surprisingly, the G of hcp Au (19.5 GPa) is comparable to that of hcp Mg (21.5 GPa) and fcc Au (26.8 GPa). Moreover, the values of B are similar between hcp Au (140.2 GPa) and fcc Au (137.6 GPa), and are far larger than that for hcp Mg (36.4 GPa). Therefore, hcp Au possesses potential application prospect, and it is of great significance to in situ synthesize the pure hcp Au in experiment1. Meanwhile, more methods are still in urgent need involving hindering the hcp → fcc phase transformation and stabilizing hcp Au structure.

In summary, we perform systematic first-principles calculations to predict the structural parameters, GSF energies, TB energies and surfactant effect to hcp Au, in anticipation of evaluating the structural stability and mechanical properties of this new material. Originating from the self-assembly of alkanethiolate species on hcp Au (0001), a slight extension of surface lattice is found, which may be related to the ease of the hcp → fcc phase transformation. Furthermore, the comparisons are made among hcp Au, fcc Au and hcp Mg. In the basal and slip systems, the GSF energies () are apparently lower for hcp Au than those for hcp Mg and even for fcc Au. Accordingly, the basal stacking faults with partial fcc feature may generate more frequently in hcp Au, which facilities the transformation to an fcc phase. In the prismatic as well as pyramidal and slip systems, the larger values for hcp Au than for hcp Mg indicate the increased difficulties in generating non-basal stacking faults in the former case. Moreover, the TB energies of and twins are larger in hcp Au than in hcp Mg, implying less stability of twin boundaries in hcp Au. The mechanically metastable of hcp Au is proved in terms of Born’s criteria, which verifies the existence of this new phase, however it still shows great tendency to transform to the fcc phase because of the easy operation of basal stacking faults and the surfactant effect. The calculated values can serve as the input for the future simulation of the growth process of these planar defects, and contribute to guiding the experiments of fabricating and developing materials with new structure.

Methods

Methods and parameters for calculation

The calculation of total energy in this work was performed by the Cambridge Sequential Total Energy Package code (CASTEP)44 based on density-functional theory (DFT), in which the Perdew Wang’s45 (PW91) version of the generalized gradient approximation (GGA) was employed as exchange correlation functional. The plane-wave cutoff was set to 400 eV. The optimization was performed through the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique with the convergence tolerances: the energy change less than 5 × 10−6 eV/atom, the Hellmann–Feynman force within 0.01 eV/Å and the maximum displacement less than 5 × 10−4 Å. The Brillouin zone integration was sampled using dense Monkhorst-Pack46 k-point meshes.

The structural properties (equilibrium lattice constants, cohesive energy) of hcp Au, Mg and fcc Au were evaluated by full optimization on both equilibrium volume and atomic positions. The k-point meshes were samples as: 18 × 18 × 12 for hcp Mg and hcp Au, and 12 × 12 × 12 for fcc Au. The cohesive energy () of the pure elements was computed according to Formula (2):

where is the total energy of the elements in their ground-state crystal structures, is the energy of isolate atom, and n is the number of atoms in the crystal.

To evaluate the surfactant effect to the lattice constants of hcp Au, a 4 × 2 slab (along the and direction) with 9 layers was constructed, separated by a 12 Å vacuum. The plane-wave basis cutoff energy was 400 eV. The 2 × 2 × 1 k-point was utilized according to the Monkhorst-Pack scheme.

Calculation of the GSF energy

The GSF energy can be obtained by incrementally shifting the upper half crystal along the slip direction and calculating the energy differences per unit area47, as shown in Formula (3).

where is the total energy of the supercell with the fault vector u, stands for the energy of the perfect lattice, and A represents the area of the fault planes. The stacking-fault vector u varies from 0.0b to 1.0b with a step of 0.1b for each slip system; hereinto b is the corresponding Burgers vector. During geometry optimization, all atoms in supercells were allowed to be relaxed along z-axis, i.e. the direction normal to the slip planes. For the calculation of the GSF energy, the k-point meshes in different systems are listed in Table 2.

A 32-atom hcp Au supercell containing 16 layers was constructed to calculate the GSF profiles, as illustrated by Huang et al.1. Meanwhile, a large vacuum width of 15 Å was added to accommodate the out-of-plane relaxations and to improve the calculation efficiency. The supercells with the same size were also employed for Mg. In the slip system, the intrinsic I2 (…ABABABCACACA…) stacking fault is generated from the perfect hcp structure (…ABABABABABAB…), and the twin fault T2 (…ABABABCBABAB…) can be obtained by the further shear of I2.

Moreover, the fcc Au slabs consisting of 13 (111) planes were also constructed to calculate the GSF energy in slip system. The intrinsic I (…ABCABCBCABCA…) stacking fault can be generated from the perfect fcc structure (…ABCABCABCABC…), and the two-layer twin fault T (…ABCABCBABCAB…) is formed basing on the further shear of I.

Calculation of the TB energy

The TB energy is depicted as the energy difference between the supercell containing twin boundary and the equivalent in bulk material34, as expressed in Formula (4).

where and correspond to the total energy of the supercells with and without twin boundaries, and represents the area of twin boundary. Note that full periodic boundary conditions were applied in our DFT calculations for both the twins, and therefore, two twin boundaries exist in each supercell: one is in the middle of supercell and the other is on its top/bottom edge. Accordingly, the is divided by 2 in Formula (4). The effect of the supercell sizes on the calculated TB energies was studied. The built supercells with different sizes are listed in Table 3, as well as the corresponding k-point meshes.

Calculation of the elastic constants

The calculation of the elastic constants was performed by the CASTEP code. The Perdew Wang’s45 (PW91) version of the generalized gradient approximation (GGA) was employed as exchange correlation functional. The plane-wave cutoff was set to 400 eV. The Brillouin zone was sampled on 18 × 18 × 12 k-point mesh for hcp Mg and hcp Au and 12 × 12 × 12 for fcc Au based on the Monkhorst-Pack scheme46. The criteria for the convergence of optimization on atomic internal freedoms were selected as: the energy difference within 1 × 10−6 eV/atom, the maximum force within 0.002 eV/Å and the maximum displacement within 1 × 10−4 Å.

The elastic stiffness coefficients were determined from a linear fit of the calculated stress as a function of strain48. The ground-state structure was strained according to symmetry-dependent strain patterns with varying amplitudes. Subsequently, the stress tensor was computed after a re-optimization of the internal structure parameters, i.e. a geometry optimization with fixed cell parameters. The elastic stiffness coefficients are then the proportionality coefficients relating the applied stress to the computed strain. Two positive and two negative amplitudes were used for each strain component with the value of 0.001 and 0.003 respectively.

The B and G are calculated using the Voigt-Reuss-Hill approximations43 for averaging the elastic constants of the single crystal.

For hexagonal system, the Voigt values are expressed as follows41:

And the Reuss values are calculated according to Formula (7) and (8)42:

The Hill mean values are obtained by43:

For cubic system, the elastic properties (B, G, E) are calculated as follows43:

Author Contributions

C.W. and H.Y.W. conceived and designed the experiments. C.W., F.Q. and T.L.H. performed the model construction and energetic calculations. C.W., H.Y.W. and X.N.X. performed the data analysis. C.W., H.Y.W. and Q.C.J. co-wrote the paper. All authors discussed the results and commented on the manuscript.

Additional Information

How to cite this article: Wang, C. et al. Generalized-stacking-fault energy and twin-boundary energy of hexagonal close-packed Au: A first-principles calculation. Sci. Rep. 5, 10213; doi: 10.1038/srep10213 (2015).