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Appendix D From *γ*-surface to Peierls Stress

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Appendix D From *γ*-surface to Peierls Stress

The concept of generalized stacking fault energy surface (or *γ*-*surface*), which was introduced by Vitek (1968), provides a simple bypass to full DFT-based dislocation core calculations (see Vitek and Paidar 2008 for references to various materials). A perfect crystal is cut along a crystallographic plane (Fig. D.1). The upper half is sheared rigidly with respect to the lower half by an arbitrary fault vector $\mathbf{u}$ of the cut plane. The fault that is generated is called a *generalized stacking fault* (GSF).

When $\mathbf{u}$ sweeps all possible directions in the unit cell of the cut plane, the surface giving the surplus energy per unit area, $\gamma (\mathbf{u})$, is the *γ*-surface for the plane considered. An interfacial restoring force per unit surface (i.e., a stress) $\mathit{\tau}(\mathbf{u})$, is associated to each fault vector. It is given by

The GSF energy surface is minimum at the origin ($\mathit{\tau}=0$), where the stacking is perfect. It has the symmetries and periodicities of the cut plane.

The *γ*-surface is currently computed using atomistic simulations or *ab initio* methods based on the DFT framework. Relaxation is authorized along the direction normal to the cut plane, but not in directions parallel to it. The surface that is generated may have metastable minima for intermediate values of the shear vector, ${\mathbf{u}}_{s}$. At such points, the restoring stress $\mathit{\tau}$ vanishes and the perfect dislocation can split into two partials separated by a metastable stacking fault of energy $\gamma ({\mathbf{u}}_{s})$.

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This situation is illustrated in Fig. D.2 *a* by a $(\stackrel{\u02c9}{1}11)$ *γ*-surface in fcc copper computed using a phenomenological many-body potential (Aslanides and Pontikis, 1998). The high-energy peaks correspond to shears leading to stacking faults such that two identical atomic layers are on top of each other. The metastable positions corresponding to intrinsic stacking faults are reached via a minimum energy path consisting of a first shear towards a metastable position, $\mathbf{O}\mathbf{A}=\frac{a}{6}[211]$ in Fig. D.2 *b*, which forms a first Shockley partial. The second shear $\mathbf{A}\mathbf{B}=\frac{a}{6}[1\stackrel{\u02c9}{1}2]$ forms the second Shockley partial. Fig. D.2 *a* can be compared to a first-principles calculation of the same *γ*-surface that was performed by Lu *et al*. (2000).

In bcc metals, the *γ*-surfaces exhibit no local minima in the planes where the core of screw dislocations spreads out (see Fig 3.5 and Vitek and Paidar 2008). In such cases, however, a reduction of the total core energy can be obtained by defining a generalized type of planar dissociation (Vitek 1968; Duesbery and Vitek, 1998). The latter consists of particular types of partial dislocations called fractional dislocations. The Burgers vectors of these dislocations can be irrational and they are separated by ribbons of constant but unstable SFE. The main difference with respect to classical dissociations is that the restoring force being non-zero, the stacking fault ribbon is not stress-free. It exerts on the bordering partials a force that depends on the fault width.

Care must be taken when using *γ*-surfaces in covalent crystals, for instance in silicon. Indeed, dangling bonds and core reconstruction processes (Section 3.4.3)
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are not accounted for during the rigid shearing of the two halves of the crystal with respect to each other.

Two quantities are of general interest in sections of *γ*-surfaces along a given fault direction. One is the steepest positive slope, which corresponds to a maximum absolute value of the restoring force per unit area. This quantity is the maximum stress required for shearing the crystal in a block-like manner, that is, the *theoretical strength* along the considered direction of the slip plane. The second quantity is the maximum energy, or *unstable stacking fault energy*, ${\gamma}_{us}$, which is involved in the model for dislocation nucleation at crack tips developed by Rice and co-workers (see Rice and Beltz 1994). In Fig. D.2 *a*, this maximum is met along the [$12\stackrel{\u02c9}{1}$] direction, which is perpendicular to the Burgers vector; its energy is about 0.8 $\mathrm{m}\mathrm{J}{\mathrm{m}}^{-2}$.

GSF energy surfaces furnish simple means for comparing the responses of *ab initio* calculations to those of semi-empirical potentials, for instance in bcc metals (Frederiksen and Jacobsen 2003; Ventelon and Willaime, 2010) and fcc metals (Boyer *et al*., 2004). The concept of *γ*-surface can also be extended to generalized planar fault energies, for instance twin-like shears in consecutive or non-consecutive planes (Van Swygenhoven *et al*., 2004); (Ogata *et al*., 2005); (Li *et al*., 2009).

A last application of *γ*-surfaces is concerned with the search for improved 2D solutions of the classical *Peierls–Nabarro model* (Peierls 1940; Nabarro, 1947). This semi-continuum model treats dislocation cores confined in a single slip plane and extended in only one direction. It provides analytical values for their equilibrium width and the Peierls stress. The model was developed during a time span of fifty years, reviewed several times (Nabarro 1997*a*; Joós and Duesbery, 1997) and further reviewed in the context of its extension to 2D models (Lu, 2005); (Bulatov and Cai, 2006); (Schoeck, 2005).

The basic idea of the Peierls−Nabarro (PN) models is illustrated by Figure D.3, which shows the planar core of an edge dislocation. The total energy of the defected crystal is split into two parts. The core energy is estimated from the definition of a continuous distribution of the relative displacements between the two half-crystals, $\mathbf{u}(x,y)$ in 2D. The energy of the two semi-infinite regions above and below the core region also depends on $\mathbf{u}$ and is treated by linear elasticity. Taken alone, the core contribution to the total energy favours narrow cores, which minimize large misfits between the two half-crystals, whereas the elastic energy favours extended cores that limit large elastic distortions. The solution $\mathbf{u}$ to the problem is obtained using minimization procedures for the total energy in 2D, taking into account whenever possible that the displacements in the elastic and core regions must match at their interface. In 1D, $u(x)$ is given by an integro-differential equation and the restoring stress profile $u(x)=-du/dx$ is assumed to be sinusoidal with period *b*. An analytical solution was derived by Peierls (1940) and further
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developed by Nabarro (1947), which yielded simple expressions for the core width and the Peierls stress.

In spite of several improvements, the 1D PN model is fundamentally handicapped by its fully continuum description of the core. Indeed, the effect of lattice periodicity only appears in the phenomenological restoring stress, which induces severe limitations (Lu 2005; Schoeck 2005). The model is not really predictive and becomes all the more inaccurate as dislocation cores are narrow. Nevertheless, it keeps a heuristic value. Calculated Peierls stresses for various materials were compiled by Nabarro (1997*b*), Wang (1996) and Takeuchi and Suzuki (1988). In some cases, they reasonably agree with expectation, which may seem surprising for such a simple model.

The development of a generalized 2D PN model became possible when *ab initio* *γ*-surfaces could be calculated and employed for estimating the misfit energy. This allowed the influence of the lattice periodicity on the misfit energy to be better taken into account (see Schoeck 2005 for a simple, efficient implementation and Bulatov and Cai 2006).

The core configurations obtained by full atomistic or *ab initio* simulations correspond well to those obtained by the 2D PN model provided that the corresponding *γ*-surface is properly implemented and use is made of anisotropic elasticity. However, the obtained configurations do not depend only on the SFE, they are also markedly influenced by the fine structure of the *γ*-surface (Schoeck, 2006). Thus, it is preferable to employ DFT *γ*-surfaces, especially for narrow cores. This is exemplified by the cases of aluminium (Schoeck, 2002) and palladium (Schoeck, 2001), where the estimated widths compare well with the result of full DFT calculations. Hence, the question of planar dissociations widths is basically solved.

The question of expanding the PN model to curved dissociated dislocations has been the subject of several attempts because of its relevance to the problem of homogeneous or heterogeneous nucleation of dislocation loops. A convincing
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implementation of *ab initio* *γ*-surfaces in the generalized PN model, using anisotropic elasticity, was performed by Yang *et al*. (2008) for dislocation loops in Cu and Al. It paves the way for further developments.

As far as Peierls stresses are concerned, the generalized PN model has limitations, which were recalled by Schoeck (2005). As already mentioned, the cores of strongly covalent materials cannot be adequately described by a *γ*-surface. For metallic materials, the core energy is treated as a local function of the displacement and this approximation is no longer valid for the strong distortions and large gradients involved in narrow cores. Nevertheless, the generalized PN model allows obtaining useful information in complex materials that are not suited for full DFT calculations. It was applied to minerals of the earth mantle, where the Peierls stresses in potentially active slip systems are poorly known (Carrez *et al*. 2007; Ferré *et al*. 2008). In the last two studies, the method employed consists in coupling the PN model to an element-free FE Galerkin model (Denoual 2007). In this last reference, the flexibility of the method is illustrated by a calculation of the Peierls stress in tantalum, using an *ab initio* *γ*-surface to describe the faulted layers of the extended core. As is usual for calculations performed in bcc metals, the obtained Peierls stress is substantially larger than the experimental one (Section 3.2.3).