In this work, we use density functional theory to calculate the entire generalized stacking fault energy (GSFE) surface for eight transition metals with a face-centered cubic structure: Ag, Au, Cu, Ir, Ni, Pd, Pt, and Rh. Analysis of the 112 GSFE curves finds that the displacements corresponding to the unstable stacking fault energy are larger than the ideal value for all eight metals except Ag and Cu. Over the entire surface, Pt is found to not possess well-defined local maxima or minima, suggesting spreading in favor of dissociation of the dislocation core, unlike the other seven metals. Our calculations also reveal that at a large 112 displacement, where atoms on two {111} adjacent planes are aligned, an anomalous local minimum occurs for Ir and Rh. The oddity is explained by relatively large, localized atomic displacements that take place in the two metals to accommodate the alignment that do not occur in the other six metals. In addition to the fully calculated surfaces, we characterize a continuous 11-term Fourier-series function, which provides a particularly excellent representation of the GSFE surfaces for Ag, Au, Cu, Ni, and Pd.

Many industrially relevant metals have a face-centered cubic (FCC) crystal structure.1,2 The plastic response of FCC metals is governed by the motion of dislocations and deformation twins on specific crystallographic planes.3,4 A well-recognized characteristic of dislocations in FCC systems is that their atomic cores can be extended in their glide planes, spanning from a fraction to several tens of nanometers in width, depending on the metal.5 In this core region, the dislocation generates a heterogeneous distribution of atomic displacements across the glide plane. The extent of the dislocation core, whether wide or narrow, is fundamental to understanding its mobility and the type of dislocation processes (i.e., cross slip or climb) and plastic deformation mechanisms (wavy slip, deformation twinning) that can be expected to prevail in a given FCC metal.6,7 Many studies have been devoted to estimating or characterizing the size of the dislocation core in FCC systems.

Most models or experimental studies interpret the dislocation core as an energetically stable structure.8 The total Burgers vector of the dislocation prefers to dissociate into two Shockley partial dislocations with smaller length Burgers vectors,9,10 thereby lowering the total strain energy, but at the expense of creating a faulted region, named the intrinsic stacking fault (ISF), in between the partials, which increases the core energy. The excess energy corresponding to the fault is the intrinsic stacking fault energy (ISFE), γisf. The size of the individual Shockley partial dislocations and the ISF width separating them are considered the two primary structural length scales constituting the dislocation core.11 

Due to the fine characteristic length scales of dislocation cores, calculations for core structures usually involve atomic-level simulations. In this class of modeling, ab initio techniques can best account for the nonlinear, anisotropic interactions between the core atoms. However, since the cores of most FCC metals generally span a few nanometers, application of density functional theory (DFT) would be too computationally intensive. One exception is Al, which has a narrow core and for which a number of studies have used full DFT calculations,12,13 reporting ISF widths in the range of 0.66–1.28 nm. For other pure FCC metals, in which ISFs are wider than those in Al, molecular dynamics or molecular statics (MS) is the method of choice. However, the accuracy of the calculated core structure from these methods depends on the reliability of the interatomic potential used and whether it was developed for defect calculations.14–16 For the same FCC metal, the same dislocation character, and the same measurement method, predictions for the ISF width can vary widely among atomistic studies.17–20 

At the other extreme lies classical linear elastic continuum dislocation theory, which neglects the atomic details of the core altogether and applies to the elastic continuum region outside the core. Presuming a dissociated or split core structure a priori, the ISF width is estimated based on a force balance between the linear elastic repulsion between the two partial dislocations and the attraction associated with γisf. It has the advantage of providing a quick and insightful estimation of the core width, but it can significantly deviate from the results based on atomistic simulations.14 

More advanced continuum-based computational models, such as the generalized Peierls-Nabarro (GPN) model21–24 and the phase-field (PF) model,25–27 have also been employed to simulate the relaxation of a dislocation from its compact, perfect state into its low-energy dissociated structure. Two essential components included in these computational techniques are the linear elastic interactions between the Shockley partial dislocations, as in the classical model, and the nonlinear interactions between atoms across the glide plane. The latter is usually represented as a function relating the excess energy incurred as the two atomic planes are displaced relative to one another in the plane of glide. In practice, the possible shift vectors made available to the core calculation have varied, from solely those along the Burgers vector to those corresponding to all possible in-plane displacements. The energetic surface corresponding to all in-plane shift vectors is called the generalized stacking fault energy (GSFE) surface.

GPN-based and PF-based models have demonstrated that the relaxation process into a dissociated state is best predicted when the entire GSFE surface is taken into account. For instance, it was shown using GPN-based simulations that using two GSFE surfaces for Ni that have the same γisf but 16.7% difference in the unstable stacking fault energy (USFE), γusf, results in an 8.7% difference in the ISF width.28 Linear elasticity theory, which uses only γisf, tends to predicts a higher ISF width than GPN/PF models, especially when γisf is relatively small.26,29,30 Another recent study using PF-based dislocation models found that calculations of the ISF widths achieved much better agreement with those from atomistic calculations when the GSFE surface representation was accurate over the entire surface and not just at special local maxima or minima.31 

Over the years, direct calculation of the entire GSFE surface is usually accomplished via MS, rather than DFT. Yet, as we have already alluded earlier, MS-based GSFE surfaces are sensitive to the interatomic potential used.32 Values for certain points on the GSFE surface are found to disagree with DFT. In Al and Ni, for instance, γisf predicted by various embedded-atom method (EAM) potentials was shown to be one order of magnitude smaller than that based on DFT.33 DFT is used to only calculate selected portions of the surface, usually the GSFE curves along the 112 and/or 110 directions, but not the entire surface. DFT has been used to calculate GSFE curves for Ag,30,34,35 Al,30,35–39 Au,30,34,35 Cu,30,34,35,38,39 Ir,30,34,35,37 Ni,30,34,35,38 Pb,34,35 Pd,30,34–37 Pt,30,34,35,37 and Rh.30,34,37 As one exception, Lu et al.40 obtained 40 values on 1/12 of the entire GSFE surface for Al.

As an alternative to direct calculation of the entire surface, many works have elected to use DFT to calculate specific points on the surface and use them to build the entire GSFE surface via a continous function. In particular, Nfs-term Fourier-series functions, with Nfs=2 (Ref. 34), 6 (Ref. 23), 7 (Ref. 41), 8 (Ref. 42), or 11 (Ref. 31), have been used for pure FCC metals. In each function, the Nfs coefficients are fit to preselected Nfs GSFE values, usually the local minima or maxima. The Fourier-series functions represent those selected points reliably, but not necessarily the remaining points. Furthermore, unless the entire surface is given, it is unknown whether the preselected points correspond to the actual local maxima and minima.

In this paper, we use DFT to calculate the full GSFE surfaces of eight FCC transition metals, including Ag, Au, Cu, Ir, Ni, Pd, Pt, and Rh. We show that five metals can be described well by an 11-term Fourier-series function, for which we characterize and provide. We identify an anomalous local minimum in the GSFE surfaces in Ir and Rh, suggesting a possible high-stress-induced dissociated structure. The results reveal that the GSFE surface of Pt has a special characteristic of not exhibiting any well-defined maxima or minima, implying that its core would spread rather than dissociate. DFT-based GSFE surfaces for these common pure FCC metals can provide understanding of atomic bonding across crystallographic planes on which dislocations glide and enable continuum dislocation models to reliably predict dislocation core structures.43 

All DFT calculations conducted in this study use VASP.44 Within the framework of the projector augmented wave method,45,46 a pseudopotential is used for each metal to approximate the core electrons, and a plane-wave basis is adopted for the valence electrons. Table I summarizes the numbers of valence electrons per atom and the cutoff energies Ecut for the plane-wave basis set. The exchange-correlation energy functional is approximated by the Perdew-Burke-Ernzerhof (PBE) formulation of the generalized gradient approximation (GGA).47 

TABLE I.

A summary of two parameters for DFT calculations: the number of valence electrons per atom Nve and the cutoff energy for the plane-wave basis set Ecut (in eV).

AgAuCuIrNiNMNiPdPtRh
Nve 17 11 17 16 10 10 15 
Ecut 446.8 344.91 552.97 316.3 551.98 376.39 345.42 371.11 
AgAuCuIrNiNMNiPdPtRh
Nve 17 11 17 16 10 10 15 
Ecut 446.8 344.91 552.97 316.3 551.98 376.39 345.42 371.11 

For the electronic self-consistent loop, the residual minimization scheme with direct inversion in the iterative subspace is used, and the convergence is reached when the total free energy change and the band structure energy change between two steps are both smaller than 104 eV. The Brillouin zone is constructed by the Monkhorst-Pack scheme.49 For correct handling of the Fermi surface, the Methfessel-Paxton smearing method50 with a smearing width σ=0.2 eV is employed, unless stated otherwise. Among the eight metals, spin-polarization is considered only for Ni, where ferromagnetism is assumed. Results based on nonmagnetic Ni are denoted as NiNM.

For each metal, a periodic cell containing four atoms is used to determine the lattice parameter a0 and elastic constants C11, C12, and C44. The k-point mesh in these calculations is 11×11×11. For a0, the cell size is varied, and for each size, the free energy without ionic relaxation is calculated. With this approach, a0 corresponds to the cell size with the smallest free energy. The elastic constants are calculated via the energy-strain method provided in AELAS.51 The results are summarized in Table II.

TABLE II.

A summary of the lattice parameter a0 (in Å) and elastic constants C11, C12, and C44 (in GPa), calculated by this work (DFT). Experimental measurements (Exp)48 are also given for comparison.

MethodAgAuCuIrNiNMNiPdPtRh
a0 DFT 4.153 4.157 3.634 3.873 3.514 3.519 3.942 3.968 3.831 
Exp 4.086 4.078 3.615 3.839 3.524 3.89 3.923 3.803 
C11 DFT 100.5 151.99 174.96 583.65 259.22 278.87 202.29 299.51 405.34 
Exp 122 191 169 580 247 221 347 413 
C12 DFT 85.25 134.16 121.57 234.36 172.74 158.83 151.32 225.36 183.1 
Exp 92 162 122 242 153 171 251 194 
C44 DFT 39.16 27.69 76.45 254.35 110.3 132.18 61.52 59.85 187.47 
Exp 45.5 42.2 75.3 256 122 70.8 76.5 184 
MethodAgAuCuIrNiNMNiPdPtRh
a0 DFT 4.153 4.157 3.634 3.873 3.514 3.519 3.942 3.968 3.831 
Exp 4.086 4.078 3.615 3.839 3.524 3.89 3.923 3.803 
C11 DFT 100.5 151.99 174.96 583.65 259.22 278.87 202.29 299.51 405.34 
Exp 122 191 169 580 247 221 347 413 
C12 DFT 85.25 134.16 121.57 234.36 172.74 158.83 151.32 225.36 183.1 
Exp 92 162 122 242 153 171 251 194 
C44 DFT 39.16 27.69 76.45 254.35 110.3 132.18 61.52 59.85 187.47 
Exp 45.5 42.2 75.3 256 122 70.8 76.5 184 

For the GSFE surface calculations, a cell with lattice orientations of x[1¯1¯2], y[111], and z[1¯10] is built containing 24 atoms, with two atoms on each (111) plane. In other words, the number of atomic layers, Nal, is 12. A vacuum region of 8 Å is added to each cell along the y direction, creating a series of noninteracting slabs. Unless stated otherwise, 13×1×17k-points are adopted. Note that the smearing width σ is related to the k-point number. To ensure convergence of the GSFEs, for σ=0.2 eV and Nal=12, different k-point numbers and Ecut values are tested. Then, with fixed k-point number and Ecut value, different values of σ and Nal are tested in selected cases, which will be presented in Sec. III. We remark that in previous work, (i) Nal=12 was found to be sufficiently large for selected GSFE values in ten FCC metals,34 and (ii) σ=0.2 eV was used to calculate selected GSFE values in nine FCC metals.30 

To obtain the GSFEs, the top six layers of atoms, in the y direction, are displaced with respect to the bottom six layers incrementally and independently along the x and z directions, until the displacements dx and dz reach the respective periodic lattice lengths dx0=6a0/2 and dz0=2a0/2. Following each displacement [dx,dz], the top two and bottom two layers of atoms are fixed, while the remaining eight layers are allowed to relax along the y direction. The ionic relaxation stops when the total energy between two steps is smaller than 103 eV/atom.39 The GSFE surface is represented by the energies associated with a fine 101×65 grid of the [dx,dz] displacements. Data for the GSFE surfaces in all eight metals are available in Materials Cloud.52 

As a way to check the DFT method and model, we first compare our results for Cu to those reported by prior DFT studies using the same PBE-GGA formulation of the exchange-correlation energy functional. Among the eight metals considered here, the fault energies for Cu are among the most extensively studied by DFT,30,34,35,38,39 and the structures of the dislocations in Cu have drawn considerable attention.53,54Figure 1 presents the GSFE surface of Cu from our DFT calculations. As mentioned, prior GSFE calculations in Cu have not aimed to produce the entire surface, but rather slices of the surface lying within the region AGA, which is illustrated in Fig. 1(d). This region contains key local minima and maxima related to the equilibrium core structure of dislocations in FCC systems. A perfect dislocation with Burgers vector AA dissociates into two Shockley partial dislocations AG and GA, i.e.,

a02[1¯10]=a06[2¯11]+a06[1¯21¯].
(1)
FIG. 1.

Results based on DFT calculations in Cu. (a) Relaxed GSFE curves along the [2¯11] direction, of which the portion between 0 and 0.4dx0 agrees with those of previous DFT calculations.30,34,38,39 (b) Illustrative atomic configurations for the four points (A, G2, G, and M) in (c) and (d); the dashed lines are the slip planes, with the atoms below fixed and the atoms above moved toward left as indicated by the arrow. (c) 3D relaxed GSFE surface on a (111) plane along both [1¯1¯2] and [1¯10] directions. (d) is the projection of a portion of (c) onto the (111) plane. Note that the arrows AG and GA point to the [2¯11] and [1¯21¯] directions, respectively.

FIG. 1.

Results based on DFT calculations in Cu. (a) Relaxed GSFE curves along the [2¯11] direction, of which the portion between 0 and 0.4dx0 agrees with those of previous DFT calculations.30,34,38,39 (b) Illustrative atomic configurations for the four points (A, G2, G, and M) in (c) and (d); the dashed lines are the slip planes, with the atoms below fixed and the atoms above moved toward left as indicated by the arrow. (c) 3D relaxed GSFE surface on a (111) plane along both [1¯1¯2] and [1¯10] directions. (d) is the projection of a portion of (c) onto the (111) plane. Note that the arrows AG and GA point to the [2¯11] and [1¯21¯] directions, respectively.

Close modal

Based on a hard-sphere model of the atomic positions during the displacements,5 a local minimum would ideally exist at dx0/3, where two adjacent hexagonal close-packed atomic layers form [see Fig. 1(b)]. The displacement is associated with the Shockley partial Burgers vector, and the corresponding GSFE value is the ISFE, γisf. In the displacement pathway along AG between 0 and dx0/3, there exists a local maximum, which is the USFE, γusf. The displacement for γusf, denoted as dusf, is ideally dx0/6, according to the hard-sphere model of an FCC lattice.

Previous DFT calculations reported the portion of the 112 GSFE curve between 0 and 0.4dx0, which spans γusf and γisf.30,34,38,39Figure 1(a) compares this part of the 112 GSFE curve from our calculations with those of others, showing excellent agreement. Common to the literature, the local maximum γusf is reached at 0.17dx0 in our DFT calculations, and the local minimum γisf occurs at 0.33dx0. Given the numerical resolution, ±0.005dx0, we consider these two displacements the same as the hard-sphere model prediction, dx0/6 and dx0/3, respectively.

In Fig. 1(a), we also show the energies from our calculations corresponding to 112 in-plane displacements larger than 0.4dx0 and up to the full periodic length dx0. This region of the 112 GSFE curve was considered inaccessible by dislocations and was not studied in prior DFT studies. In the ideal GSFE surface, the 112 curve is expected to achieve a maximum at point M [Fig. 1(d)] for a shift of 2dx0/3, when two adjacent layers of atoms are perfectly aligned, wherein atoms lie directly on top of each other [Fig. 1(b)]. Hence, we term the GSFE value at M as the “aligned SFE”, denoted as γasf. Note that it was called by Lu and Kaxiras55 as “run-one SFE”. Our calculations show that for Cu, γasf sits at the global maximum, as shown in Fig. 1(c).

Having demonstrated consistency with other DFT calculations in the literature, we proceed to examine the full γ-surface for all eight metals, with particular focus on their 110 curves and 112 GSFE curves. Figure 2 presents the GSFE curves for all eight metals along the 110 and 112 directions, respectively. For the 110 curve, it is expected that the peak energy, γ110p, would correspond to a displacement at the halfway point, at dz0/2. Indeed, a maximum is reached in all metals at dz0/2, and the GSFE values are symmetric with respect to it. The peak value γ110p varies substantially, being highest for Ir and lowest for Au.

FIG. 2.

Relaxed GSFE curves along the 112 and 110 directions for all eight metals.

FIG. 2.

Relaxed GSFE curves along the 112 and 110 directions for all eight metals.

Close modal

More studied in the literature are the 112 curves. The pathway AG possesses the local minimum γisf, and the local maximum along the 112 path, γusf, is much smaller than γ110p. Thus, displacements along the 112 pathway are likely to determine the behavior and structure of the dislocation. In all metals, the local minimum γisf is achieved at dx0/3. Our results that Ni has higher γisf and γusf than NiNM are consistent with a prior DFT calculation.38 Two more findings are worth noting. First, the 112 curve for Pt bears no well-defined maximum or minimum over the entire periodic length. Second, the 112 displacement to reach γusf, dusf, differs substantially among the metals, and all metals (except Ag and Cu) deviate from the hard-sphere value of dx0/6. The values of γusf and dusf are summarized in Table III. The value for dusf for Ag and Cu is the closest to the ideal one and the smallest, while that for Pt is the largest.

TABLE III.

A summary of the USFE γusf (in mJ/m2), the displacement dusf (in dx0) for γusf, ISFE γisf (in mJ/m2), aligned SFE γasf (in mJ/m2), and the peak GSFE value along the 〈110〉 curve, γ110p (in mJ/m2).

AgAuCuIrNiNMNiPdPtRh
γusf 92.01 76.68 160.52 671.66 279.69 300.89 214.93 297.92 475.34 
dusf 0.17 0.18 0.17 0.19 0.19 0.19 0.2 0.25 0.18 
γisf 14.49 26.79 41.83 385.87 139.01 152.13 135.54 280.87 208.67 
γasf 499.26 380 891.28 1194.4 1157.75 1262.14 698.69 566.96 1077.54 
γ110p 307.39 246.57 519.05 1250.63 716.3 801.47 476.98 501.97 958.3 
AgAuCuIrNiNMNiPdPtRh
γusf 92.01 76.68 160.52 671.66 279.69 300.89 214.93 297.92 475.34 
dusf 0.17 0.18 0.17 0.19 0.19 0.19 0.2 0.25 0.18 
γisf 14.49 26.79 41.83 385.87 139.01 152.13 135.54 280.87 208.67 
γasf 499.26 380 891.28 1194.4 1157.75 1262.14 698.69 566.96 1077.54 
γ110p 307.39 246.57 519.05 1250.63 716.3 801.47 476.98 501.97 958.3 

For the 112 curve, the GSFE values corresponding to displacements larger than 0.4dx0 are significantly larger than those in region AGA. We find that, in addition to Cu [Fig. 1(c)], γasf in four other FCC metals (Ag, Au, Ni, and Pd) correspond to the global maximum, as shown in Fig. 3, achieved at 2dx0/3. Pt, as shown in Fig. 4, does not exhibit well-defined extrema, particularly at γasf. In the two remaining metals (Ir and Rh), we observe an interesting finding that γasf sits at a local minimum. To better highlight this energy well, Figs. 5(a) and 5(b) present the entire GSFE surface for Ir and Rh.

FIG. 3.

Relaxed GSFE surfaces in Ag, Au, Ni, and Pd, for which γasf, denoted by the black filled circles, sits at a local maximum, which is also the global maximum.

FIG. 3.

Relaxed GSFE surfaces in Ag, Au, Ni, and Pd, for which γasf, denoted by the black filled circles, sits at a local maximum, which is also the global maximum.

Close modal
FIG. 4.

Relaxed GSFE surface in Pt for which γasf is denoted by the black filled circles. There are no well-defined extrema.

FIG. 4.

Relaxed GSFE surface in Pt for which γasf is denoted by the black filled circles. There are no well-defined extrema.

Close modal
FIG. 5.

(a) and (b) Relaxed GSFE surfaces in Ir and Rh for which γasf is denoted by the black filled circles and sits at a local minimum. (c) is the projection of a portion of (a) onto the (111) plane.

FIG. 5.

(a) and (b) Relaxed GSFE surfaces in Ir and Rh for which γasf is denoted by the black filled circles and sits at a local minimum. (c) is the projection of a portion of (a) onto the (111) plane.

Close modal

In this section, we consider parameterizations of these fully calculated GSFE surfaces via an 11-term Fourier-series continuous function.31 This type of functions can be directly used in continuum models for static and dynamic dislocation studies and can offer insight and understanding of dislocation energetics. The Fourier series function for the GSFE surface takes the form

γgsf=c0+c1[cos(2pdx)+cos(pdx+qdz)+cos(pyqz)]+c2[cos(2qdz)+cos(3pdx+qdz)+cos(3pdxqdz)]+c3[cos(4pdx)+cos(2pdx+2qdz)+cos(2pdx2qdz)]+c4[cos(pdx+3qdz)+cos(pdx3qdz)+cos(4pdx+2qdz)+cos(4pdx2qdz)+cos(5pdx+qdz)+cos(5pdxqdz)]+c5[sin(2pdx)sin(pdx+qdz)sin(pdxqdz)]+c6[sin(4pdx)sin(2pdx+2qdz)sin(2pdx2qdz)]+c7[cos(6pdx)+cos(3pdx+3qdz)+cos(3pdx3qdz)]+c8[cos(8pdx)+cos(4pdx+4qdz)+cos(4pdx4qdz)]+c9[cos(4qdz)+cos(6pdx+2qdz)+cos(6pdx2qdz)]+c10[cos(6qdz)+cos(9pdx+3qdz)+cos(9pdx3qdz)],
(2)

where p=(26/3)π/a0 and q=22π/a0 are magnitudes of the reciprocal lattice vectors. The 11 coefficients ci are presented in Table IV by fitting the equation above to the 11 GSFE values as described in Ref. 31.

TABLE IV.

A summary of the 11 coefficients ci, in units of mJ/m2, in Eq. (2).

AgAuCuIrNiNMNiPdPtRh
c0 222.77 177.08 383.24 932.83 530.09 589.67 360.73 385.07 708.48 
c1 −60.11 −48.06 −107.47 −185.45 −146.6 −161.16 −93.69 −93.85 −151.06 
c2 −16.8 −13.39 −22.86 −131 −36.91 −42.21 −27.91 −34.63 −90.14 
c3 3.21 3.42 3.82 11.95 3.97 5.07 1.74 1.71 9.2 
c4 0.01 −0.47 0.07 −0.77 −0.01 −0.18 −0.14 −0.59 −0.3 
c5 −99.9 −75.8 −172.98 −189.38 −208.12 −229.7 −117.38 −73.57 −189.92 
c6 −6.6 −7.83 −9.5 −33.78 −12.06 −16.07 −9 −18.51 −22.71 
c7 0.41 0.91 −0.04 5.95 5.63 4.29 2.82 4.48 3.04 
c8 0.02 0.38 −0.12 −0.53 −0.52 0.03 −0.45 −0.82 −0.43 
c9 −0.89 −1.18 −0.95 −9.72 −2.1 −2.08 −2.31 −3.82 −5.38 
c10 −0.12 −0.16 −0.27 −0.6 −0.15 −0.12 −0.18 −0.24 −0.79 
AgAuCuIrNiNMNiPdPtRh
c0 222.77 177.08 383.24 932.83 530.09 589.67 360.73 385.07 708.48 
c1 −60.11 −48.06 −107.47 −185.45 −146.6 −161.16 −93.69 −93.85 −151.06 
c2 −16.8 −13.39 −22.86 −131 −36.91 −42.21 −27.91 −34.63 −90.14 
c3 3.21 3.42 3.82 11.95 3.97 5.07 1.74 1.71 9.2 
c4 0.01 −0.47 0.07 −0.77 −0.01 −0.18 −0.14 −0.59 −0.3 
c5 −99.9 −75.8 −172.98 −189.38 −208.12 −229.7 −117.38 −73.57 −189.92 
c6 −6.6 −7.83 −9.5 −33.78 −12.06 −16.07 −9 −18.51 −22.71 
c7 0.41 0.91 −0.04 5.95 5.63 4.29 2.82 4.48 3.04 
c8 0.02 0.38 −0.12 −0.53 −0.52 0.03 −0.45 −0.82 −0.43 
c9 −0.89 −1.18 −0.95 −9.72 −2.1 −2.08 −2.31 −3.82 −5.38 
c10 −0.12 −0.16 −0.27 −0.6 −0.15 −0.12 −0.18 −0.24 −0.79 

The 11-term Fourier-series function is found to provide an accurate representation for the GSFE surfaces for five metals: Ag, Au, Cu, Ni, and Pd. For Pt, the fit is not as perfect but reasonable, as will be discussed shortly. For Ir and Rh, it fails to capture the local minimum at point M, the origin of which we examine further below. Thus, for these three transition FCC metals, Pt, Ir, and Rh, direct DFT calculations of the GSFE curves are preferred over analytic approximations.

The unique energetic landscape of the entire GSFE surface of Pt shows that Pt has a distinct and more complex bonding structure than other metals. To assess the effects of the smearing width σ and the number of atomic layers Nal, we vary σ from 0.05 eV to 0.4 eV and Nal from 6 to 15. Results, presented in Fig. 6, demonstrate that use of σ=0.2 eV and Nal=12 in these calculations is appropriate for Pt.

FIG. 6.

Relaxed GSFE curves along the 112 direction for Pt, with (a) Nal=12 but different smearing width σ and (b) σ=0.2 eV but different number of atomic layers Nal.

FIG. 6.

Relaxed GSFE curves along the 112 direction for Pt, with (a) Nal=12 but different smearing width σ and (b) σ=0.2 eV but different number of atomic layers Nal.

Close modal

To further validate the result, we sought another method, in which the exchange-correlation functional in the DFT calculations is approximated by GGA+U56 rather than GGA. Compared with GGA, GGA+U is considered to more accurately describe the localization of d orbitals of transition metals. It does require, however, at least one additional energy term that is usually unknown a priori.57 Here, for the GGA+U calculation, we aim to characterize a single effective Hubbard correction, Ueff, for the on-site Coulomb interactions of localized electrons in the d orbitals. For each metal, we determine its value by fitting the GGA+U-based lattice parameter a0 to that measured in experiments.48 As shown in Fig. 7(a), Ueff=3.357 eV provides a good fit for Pt. The two calculations for Pt are compared in Fig. 7(b), where it is seen that for Pt, γasf from GGA+U provides generally different fault energies and even shallower maxima and minima, compared to the 112 GSFE curve based on GGA.

FIG. 7.

(a) Lattice parameter, a0, predicted by DFT calculations with the exchange-correlation energy functional approximated by GGA+U, with respect to Ueff. Experimental values of a0, as summarized in Table II, are shown as horizontal dashed lines. (b) Relaxed GSFE curves along the 112 direction for Ir, Pt, and Rh, while the exchange-correlation energy functional approximated by GGA (symbols) or GGA+U (solid line).

FIG. 7.

(a) Lattice parameter, a0, predicted by DFT calculations with the exchange-correlation energy functional approximated by GGA+U, with respect to Ueff. Experimental values of a0, as summarized in Table II, are shown as horizontal dashed lines. (b) Relaxed GSFE curves along the 112 direction for Ir, Pt, and Rh, while the exchange-correlation energy functional approximated by GGA (symbols) or GGA+U (solid line).

Close modal

Figure 8 presents the GSFE curves in Pt along the 112 and 110 directions using DFT, the 11-term Fourier-series representation,31 and several EAM potentials.58–62 It is found that (i) all EAM potentials predict γisf less than 50% of the DFT-calculated value; (ii) while one EAM potential61 gives a reasonably good γusf value, none correctly predict the large dusf; and (iii) although the 11-term Fourier-series function captures the general shape of the DFT-calculated curve, it predicts a larger convexity at 2dx0/3 along the 112 direction and an incorrect local minimum at dz0/2 on the 110 curve.

FIG. 8.

Relaxed GSFE curves along the 112 and 110 directions for Pt using DFT, DFT-informed 11-term Fourier-series function [Eq. (2)], and several EAM potentials.58–62 

FIG. 8.

Relaxed GSFE curves along the 112 and 110 directions for Pt using DFT, DFT-informed 11-term Fourier-series function [Eq. (2)], and several EAM potentials.58–62 

Close modal

Some of these anomalies in the 112 GSFE curve of Pt, namely, the unusually large dusf=dx0/4 and the lack of a well-defined local minimum at γisf, have been reported in prior DFT calculations.30,34,35 These oddities were attributed to a unique change in the interatomic bonding in the ISF, which were revealed by a valence charge density difference (VCDD) analysis.30 Furthermore, the lack of a deep local minimum in the 112 GSFE curve, and the entire surface, suggests that, unlike the other metals, dislocation dissociation is not favorable. Incorporating the DFT-informed GSFE surface into the energy functional, a PF-based model indicated that the equilibrium dislocation core did not involve a dissociation but rather a spreading within its {111} glide plane.30 Taken together, to date, for Pt, direct calculation of the entire GSFE surface using DFT is preferred over using semiempirical interatomic potentials or Fourier-series representations.

As mentioned, in Ir and Rh, γasf sits at a local minimum. To validate the result, we again vary σ from 0.05 to 0.4 eV and Nal from 6 to 15. Figure 9 shows that our choices of σ=0.2 eV and Nal=12 are appropriate for Ir. As another check, for Ir and Rh, we recalculated the entire 112 GSFE curves using DFT with the exchange-correlation functional approximated by GGA+U.56 Using the same method for characterizing Ueff already described, we determine that Ueff=2.786 and 3.375 eV, respectively, for Ir and Rh. We observe in Fig. 7(b) that with GGA+U, γasf still sits at a local minimum of the 112 GSFE curve. Agreement in this particular feature suggests that the anomalous local minimum at γasf for Ir and Rh is not the result of an inaccurate description of the d-orbital localization via GGA.

FIG. 9.

Relaxed GSFE curves along the 112 direction for Ir, with (a) Nal=12 but different smearing width σ and (b) σ=0.2 eV but different number of atomic layers Nal.

FIG. 9.

Relaxed GSFE curves along the 112 direction for Ir, with (a) Nal=12 but different smearing width σ and (b) σ=0.2 eV but different number of atomic layers Nal.

Close modal

The reason for the anomaly at γasf may lie in a change in the electronic or atomic structure in the aligned faulted region that arises for these two metals but not the others. To expose the change in the electronic structure, we calculated the VCDD in the aligned fault at point M for all eight metals. The VCDD maps between the unfaulted and faulted regions produced are similar; that is, no unique change in the symmetry and directionality of the charge transfer from the unfaulted to the faulted region emerged for Ir and Rh. Therefore, alterations in the charge transfer at point M do not explain the anomalous local minimum seen in these two metals.

Last, we examine the atomic displacement associated with point M. Recall that the GSFE surfaces are obtained after energy minimization during which atoms were allowed to move along the y direction. To quantify the atomic displacements during the relaxation, we revisit the unrelaxed 112 GSFE curves, which are shown in Fig. 10(a). Significantly, we find that a global maximum lies at point M for all metals. The difference between the relaxed and unrelaxed curves, particularly at point M, indicates that additional local atomic displacements took place to accommodate fault formation at point M. Here, to determine whether a special set of displacements occurrs in forming the fault at point M, we further study in Fig. 10(b) the atomic displacements involved during relaxation at point M in the nearest four atomic planes for all eight metals. The analysis shows that, in Ir and Rh, the two atomic layers that are immediately adjacent to the glide plane have been significantly displaced compared to the third and fourth layers from the glide plane. Unlike all other metals, large, local atomic displacements occurred in Ir and Rh to accommodate the alignment of two adjacent {111} planes at point M.

FIG. 10.

(a) Unrelaxed GSFE curves along the 112 direction for all eight metals. (b) The atomic displacement hasf at point M during relaxation along the glide plane normal.

FIG. 10.

(a) Unrelaxed GSFE curves along the 112 direction for all eight metals. (b) The atomic displacement hasf at point M during relaxation along the glide plane normal.

Close modal

The sufficiently deep local minimum at M indicates a possible dissociation reaction to an extended dislocation structure that would take place in Ir and Rh, but not in the other metals. A full dislocation AA would dissociate into two Shockley partial dislocations AM and MA, i.e.,

a02[1¯10]=a06[1¯21¯]+a06[2¯11].
(3)

The ISF width associated with this reaction would be much smaller than that associated with the creation of an ISF at γisf. For an edge dislocation in Rh, for instance, linear elasticity theory predicts that the ISF widths following Eqs. (1) and (3) are, respectively, 9b and 1.74b. Furthermore, the former reaction occurs under no stress, whereas the latter would only likely occur when the dislocation is subject to shear stresses that are sufficiently large to overcome the energy barriers between point A and point M.

In this paper, we use DFT to calculate the (111) GSFE surfaces of eight FCC transition metals, including Ag, Au, Cu, Ir, Ni, Pd, Pt, and Rh. Analysis of the 112 GSFE curves finds that the displacements corresponding to the USFE are larger than the ideal value for all eight metals except Ag and Cu, with Pt having the largest displacement. The GSFE surface of Pt does not possess well-defined local maxima and minima, suggesting that its dislocation core does not dissociate but spreads in the glide plane. In addition, at a 112 displacement 2dx0/3, at which two adjacent planes would geometrically align, an anomalous local minimum occurs for Ir and Rh, suggesting the existence of a novel dissociated dislocation structure. This anomaly is shown to be a result of special, relatively large localized atomic displacements, taking place to accommodate the alignment that does not occur in the other metals. Based on these surface calculations, a Fourier-series parameterization is tested and shown to offer an excellent continuous function representation for five of these metals, Ag, Au, Cu, Ni, and Pd. The results of these analyses and the GSFE surfaces can provide insight and benefit continuum modeling of dislocations, such as the GPN/PF-based modeling, which requires reliable GSFE surfaces as input.23,26–31,41

We thank Dr. Xiangguo Li for helpful discussions. The authors gratefully acknowledge support from the Office of Naval Research under contract ONR BRC Grant No. N00014-18-1-2392. S.X. also gratefully acknowledges support in part by the Elings Prize Fellowship in Science offered by the California NanoSystems Institute on the UC Santa Barbara campus. Use was made of computational facilities purchased with funds from the National Science Foundation (NSF) (No. CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 1720256) at UC Santa Barbara. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (Grant No. ACI-1053575).

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