Multiscale modeling of crowdion and vacancy defects in body-centered-cubic transition metals

We investigate the structure and mobility of single self-interstitial atom and vacancy defects in body-centered-cubic transition metals forming groups 5B (vanadium, niobium, and tantalum) and 6B (chromium, molybdenum, and tungsten) of the Periodic Table. Density-functional calculations show that in all these metals the axially symmetric ⟨111⟩ self-interstitial atom configuration has the lowest formation energy. In chromium, the difference between the energies of the ⟨111⟩ and the ⟨110⟩ self-interstitial configurations is very small, making the two structures almost degenerate. Local densities of states for the atoms forming the core of crowdion configurations exhibit systematic widening of the “local” $d$ band and an upward shift of the antibonding peak. Using the information provided by electronic structure calculations, we derive a family of Finnis-Sinclair-type interatomic potentials for vanadium, niobium, tantalum, molybdenum, and tungsten. Using these potentials, we investigate the thermally activated migration of self-interstitial atom defects in tungsten. We rationalize the results of simulations using analytical solutions of the multistring Frenkel-Kontorova model describing nonlinear elastic interactions between a defect and phonon excitations. We find that the discreteness of the crystal lattice plays a dominant part in the picture of mobility of defects. We are also able to explain the origin of the non-Arrhenius diffusion of crowdions and to show that at elevated temperatures the diffusion coefficient varies linearly as a function of absolute temperature.

Figure 1

(Color online) Schematic view of the vacancy and the five high-symmetry self-interstitial atom configurations in a bcc metal. A 111 crowdion configuration shown in this figure is formed by inserting an extra atom halfway between the nearest neighbors and by relaxing the resulting configuration at $T=0$. A tetrahedral configuration is formed if an interstitial atom is placed at $(a∕4,a∕2,0)$, where $a$ is the parameter of the crystal lattice. An octahedral configuration is formed if an extra atom is placed at the center of the face of the cell, for example, at $(a∕2,a∕2,0)$.

Figure 2

(Color online) Atomic displacement fields in the central string of the ⟨111⟩ crowdion self-interstitial atom configuration evaluated using density-functional theory for all the bcc transition metals. The solid and the dashed lines show the analytical soliton solutions of the sine-Gordon model [Eq. (33)] plotted for $\mathcal{N}=1.4$ and $\mathcal{N}=1.9$.

Figure 3

(Color online) The local electronic density of states (LDOS) evaluated for the five atoms located in the central ⟨111⟩ string of the crowdion configuration for each of the bcc metals forming groups V and IV of the Periodic Table. Calculations were performed using a $4\times 4\times 4$ $128+1$ atom supercell. Atom 1 is situated in the center of the crowdion, and atom 5 is the furthest away from the center of the crowdion along the ⟨111⟩ string.

Figure 4

(Color online) The local electronic density of states (LDOS) evaluated for the three atoms situated in one of the six nearest-neighbor strings (adjacent to the central ⟨111⟩ string) of the crowdion configuration for each of the bcc metals forming groups V and IV of the Periodic Table. Atom labeled NN-1 is the closest to the center of crowdion.

Figure 5

(Color online) The local electronic density of states (LDOS) evaluated for the five atoms forming the central ⟨111⟩ string of the crowdion defect in bcc iron, and the three atoms in one of the NN strings of the defect’s configuration. Calculations were performed both for the spin-polarized and the nonmagnetic electronic configurations. The fixed atomic positions used in the nonmagnetic calculation are taken from the relaxed magnetic calculation. Contributions of $s$, $p$, and $d$ orbitals are shown, with $d$ orbitals giving the dominant contribution to the total LDOS.

Figure 6

(Color online) The local density of states for the 1NN and 2NN atoms of a vacancy evaluated for the six nonmagnetic bcc metals. Calculations were performed using a $4\times 4\times 4$ $128-1$ atom supercell. The uppermost figure for each element shows the total density of states averaged over all the 127 atoms in the cell.

Figure 7

(Color online) Same as in Fig. 6 but for bcc Fe. The figure shows results of both magnetic and nonmagnetic calculations of the LDOS for a monovacancy. The fixed atomic positions used in the nonmagnetic calculations are taken from the relaxed magnetic calculations.

Figure 8

(Color online) [(a)–(e)] Plot of pairwise density and repulsive potential radial functions of optimized fits. Note that the pairwise density is multiplied by a factor of 10. (f) Plots the DFT-derived second moments of the density of states as a function of the product of the empirical potential parameter $A$ and the square root of the effective density ${\rho }_0(=1)$ corresponding to the equilibrium lattice constant for bcc crystal lattice.

Figure 9

(Color online) (a) Plot of predicted first nearest and second nearest divacancy binding energies and (b) vacancy migration energies using the current empirical potentials and DFT data obtained from Table .

Figure 10

The trajectory of the center of a ⟨111⟩ crowdion in tungsten at $T=100\mathrm{K}$ simulated using the semiempirical many-body potential described in the Appendix.

Figure 11

Same as in Fig. 10 but for $T=400\mathrm{K}$.

Figure 12

Same as in Fig. 10 but for $T=800\mathrm{K}$.

Figure 13

Same as in Fig. 10 but for $T=1100\mathrm{K}$.

Figure 14

The frequency of the $⟨111⟩\leftrightarrow ⟨110⟩$ transformation observed in molecular dynamics simulations and plotted as a function of absolute temperature $T$. The dashed line corresponds to ${\nu }_r\left(T\right)=6.59\times {10}^{12}\exp (-\left[0.385\mathrm{eV}\right]∕k_{B}T)∕\mathrm{s}$. The Debye temperature $T_D$ of tungsten is $T_D=310\mathrm{K}$. This corresponds to $1∕k_B{T}_D=37.43{\mathrm{eV}}^{-1}$.

Figure 15

Coefficient of diffusion of crowdions found using molecular dynamics simulations and plotted as a function of the absolute temperature $T$. The dashed line is given by an analytical formula $D\left(T\right)=(1.9\times {10}^{-4})T\exp (-\left[0.013\mathrm{eV}\right]∕k_{B}T){\mathrm{m}}^2∕\mathrm{s}$. The Debye temperature $T_D$ of tungsten is $T_D=310\mathrm{K}$.

Figure 16

The coefficient of dissipative thermal friction of a crowdion $\gamma =k_{B}T∕D$ plotted as a function of absolute temperature. The dashed curve corresponds to the analytical formula for the diffusion coefficient shown by the dashed line in Fig. 15. The coefficient of thermal friction is almost independent of temperature for $T⩾T_D$, where thermal displacements of atoms in the lattice can be approximately treated as uncorrelated.