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.

ideal hcp packing sequence (…ABABAB…) would exhibit partial fcc feature, e.g. the ABC sequence in I 2 (…ABABCACA…) and T 2 (…ABABCBABA…) stacking faults. Therefore, the stacking faults in one system can be actually regarded as the presence of a segment of the other 2 . 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 thicker 1 . 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 dislocations 13 as well as the propensities to form twins 14 . Meanwhile, the TB energy is intimately connected with the mobility of twinning dislocations 15 , 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 ( {1122} 1123 ) are considered respectively, as illustrated in Fig. 1. Meanwhile, the {1012} mirror reflection, the {1012} mirror glide as well as the {1011} mirror reflection and the {1011} 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 materials 17 , 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 processes 1-3 . 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.

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 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 monolayer [19][20][21][22] . This configuration is further confirmed by the Au top-site adsorption of alkanethiolate in experiments 23,24 .
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 {1120} and {1100} 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 CH 3 S 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 (Au a in Fig. 2a) and the underlying lattice atom (Au l in Fig. 2a) with r(S-Au a ) = 2.344 Å and r(S-Au l ) = 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 CH 3 S-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).
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 Au l and surrounding Au atoms (a) 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 monolayers 25 . Moreover, the adsorption of alkanethiolate surfactant results in increased tensile stress when the S-C bond is normal  49 . e Hector Jr., L. G. et al. 50 . to the surface, and therefore gives rise to a significant expansion of the Au lattice (a: 3.049 Å). This situation is remedied to some extent by the tilting of the S-C bond (a: 2.997 Å). The resulting directionality of the S-C bond leads to a preferred value of the C-S-Au l 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, c) 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 c 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 AE 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 ( = / = . a a 2 2 9677 fcc pri fcc Å). Moreover, the fcc AEhcp transformation of Au is proven to involve the compression of lattice along the ( ) 110 direction 22 , 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 {1122} 1123 ), which is responsible for accommodating the strains of c-and a-axis simultaneously, is also calculated. Meanwhile, the basal slip system (

< >
{111} 112 ). The GSF energy curves (γ GSF -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 γ GSF -curve is the unstable SFE γ us , which denotes the lowest energy barrier for dislocation nucleation 27 , while the local minimum indicates the intrinsic SFE γ is 28 . The computed γ us and γ is values are marked on the γ GSF curves and summarized in Table 2, together with those obtained by other first-principles predictions [29][30][31][32][33][34] . Note that the GSF energies for Mg 32 in different slip systems accord with the results calculated by others [29][30][31][32][33][34] , which proves the accuracy of our calculations.
In hcp Au and Mg, the γ GSF curves for different slip systems show similar trends in variation, e.g. the existence of γ us and γ is corresponding to approximately the same fault vector (abnormal: γ ( ) {0001} 1010 slip system, the γ us and γ is values are apparently lower for hcp Au than those for hcp Mg, and are even lower than those for fcc Au. Specially, the γ is values for the intrinsic (I 2 ) and twin-like (T 2 ) faults are even negative, implying that the I 2 (…ABABABCACACA…) and T 2 (…ABABABCBABAB…) can form extremely easily. In the basal < > {0001} 1120 slip system, the γ us 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 thicker 1 .
In the prismatic  Elastic property. For a hexagonal solid, there are five independent elastic constants, namely, C 11 , C 12 , C 13 , C 33 and C 55 . For a cubic crystal, there are only three independent elastic constants, namely, C 11 , C 12 and C 44 . The elastic constants for hcp Au, Mg and fcc Au are calculated and listed in Table 4. Note that    34 .
our calculated C ij values agree well with the results of Shang et al. 37,38 , in which the stability of pure elements was successfully discussed in terms of the elasticity. The acceptable tolerance between our results and other simulation 37,38 indicates the accuracy of our work. Based on the predicted C ij 's, the mechanical stability for a given structure can be judged according to Born's criteria 39,40 : − > C C 0 11 12 , 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 conditions 1 . 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 observation 1 .
Starting from C ij 's, the polycrystalline aggregate properties such as bulk modulus (B), shear modulus (G) and Young's modulus (E) are calculated according to the Voigt 41 (B v and G v ), Reuss 42 (B r and G r ) and Hill 43 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 experiment 1 . 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 < >

{0001} 1010 and < >
{0001} 1120 slip systems, the GSF energies (γ us ) 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

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's 45 (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-Pack 46 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 where E tot solid is the total energy of the elements in their ground-state crystal structures, E iso atom 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 {1120} and {1100} 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 area 47 , as shown in Formula (3).
where ( ) E u is the total energy of the supercell with the fault vector u, E 0 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 < > {0001} 1010 slip system, the intrinsic I 2 (…ABABABCACACA…) stacking fault is generated from the perfect hcp structure (…ABABABABABAB…), and the twin fault T 2 (…ABABABCBABAB…) can be obtained by the further shear of I 2 .
Moreover, the fcc Au slabs consisting of 13 (111) planes were also constructed to calculate the GSF energy in 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 material 34 , as expressed in Formula (4).
where E twin and E hcp correspond to the total energy of the supercells with and without twin boundaries, and A 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 E TB 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's 45 (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 scheme 46 . 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 strain 48 . 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 approximations 43 for averaging the elastic constants of the single crystal.
For hexagonal system, the Voigt values are expressed as follows 41 : And the Reuss values are calculated according to Formula (7) and (8) 42 : The Hill mean values are obtained by 43 :