Core energy and Peierls stress of a screw dislocation in bcc molybdenum: A periodic-cell tight-binding study

Using a formulation based on anisotropic elasticity we determine the core energy and Peierls stress of the $a_0∕2\left[111\right]$ screw dislocation in bcc molybdenum at $T=0$. We show that a proper definition of the core energy necessarily involves choosing a reference direction $\widehat{\mathbf{a}}$ and a reference radius $r_0$ in order to describe dislocation dipole rotation and dilatation respectively in the asymptotic expansion of the total energy. The core energy is extracted from atomistic calculations for supercells containing a single dislocation dipole with periodic boundary conditions in a manner that treats fully consistently the effects of image interactions, such that the core energy extracted is invariant with respect to the supercell size and shape, image-sum aspect ratio, and dislocation dipole distance and orientation. Using an environment-dependent tight-binding model we obtain $0.371\mathrm{eV∕\AA }$ at $\widehat{\mathbf{a}}=⟨11\overline{2}⟩$ and $r_0=b$ and $3.8\mathrm{GPa}$ for the energy of a core with zero polarity and Peierls stress for simple shear in $\left(\overline{1}10\right)⟨111⟩$, respectively, to be compared to $0.300\mathrm{eV}∕\AA$ and $2.4\mathrm{GPa}$ obtained using an empirical many-body potential for a polarized core. Our results suggest that the large Peierls stress of screw dislocation in $\mathrm{Mo}$ is due to the transition from nonplanar to planar core, rather than a direct effect of the equilibrium core polarity.

Figure 1

Relaxed $\left(\overline{1}10\right)⟨111⟩$ shear affine stress-strain response in $\mathrm{Mo}$ calculated using the Finnis-Sinclair potential (Ref. 22). The cusp in the response is due to the finite potential cutoff between the second- and third-nearest neighbors.

Figure 2

(Color online) (a) The angular function $A\left(\theta \right)$ of $a_0∕2\left[1\overline{1}0\right]$ shuffle-set screw dislocation in Stillinger-Weber potential $\mathrm{Si}$ (Ref. 20), with $⟨11\overline{2}⟩$ as the zero-angle reference axis $\widehat{\mathbf{a}}$. The corresponding core energy is computed to be $0.502\mathrm{eV}∕\mathrm{\AA }$ for $r_0=\mid \mathbf{b}\mid$. In a separate calculation,[20] with $⟨111⟩$ as the zero-angle reference axis, the core energy was computed to be $0.526\mathrm{eV}∕\mathrm{\AA }$. The $0.024\mathrm{eV}∕\mathrm{\AA }$ difference is verified to be exactly $A(\theta =\pi ∕2)$, as shown above in the circle. (b) The angular function $A\left(\theta \right)$ for $\mathrm{Mo}$ $a_0∕2\left[111\right]$ screw dislocation using the Finnis-Sinclair potential (dashed line) and the tight-binding potential (solid line), both with $\widehat{\mathbf{a}}$ chosen to be $⟨11\overline{2}⟩$. There is $A\left(\theta \right)=A(\theta +\pi ∕3)$ due to crystal symmetry.

Figure 3

(Color online) Differential displacement map of $\mathrm{Mo}$ screw dislocation using the Finnis-Sinclair potential. (i) ${\mathbf{h}}_1=7{\mathbf{e}}_1$, ${\mathbf{h}}_2=3.5{\mathbf{e}}_1+5.5{\mathbf{e}}_2+0.5{\mathbf{e}}_3$, ${\mathbf{h}}_3={\mathbf{e}}_3$ cell. (ii) ${\mathbf{h}}_1=8{\mathbf{e}}_1$, ${\mathbf{h}}_2=16{\mathbf{e}}_2+0.5{\mathbf{e}}_3$, ${\mathbf{h}}_3={\mathbf{e}}_3$ cell.

Figure 4

(Color online) Differential displacement map of $\mathrm{Mo}$ screw dislocation using the tight-binding potential. (a) ${\tau }_1={\tau }_2=0$, (b) ${\tau }_1={\tau }_2\approx 3.8\mathrm{GPa}$, (c) transient configuration during the stress-driven instability.