Symmetry-broken self-interstitial defects in chromium, molybdenum, and tungsten

For several decades, the striking contradiction between the Huang diffuse scattering experiments, resistivity recovery data, and predictions derived from density functional theory (DFT) remained one of the mysteries of defect physics in molybdenum. Since the 1970s, observations of Huang x-ray diffuse scattering appeared to indicate that a self-interstitial atom (SIA) defect in Mo adopts a $\langle 110\rangle$ dumbbell configuration. However, the low temperature defect diffusion data supported the DFT prediction of a different, highly mobile $\langle 111\rangle$ SIA defect structure in the same metal. Using DFT simulations, we show that an SIA adopts a symmetry-broken configuration in all the group 6 metals: chromium, molybdenum, and tungsten. The symmetry-broken defect structure, a $\langle 11\xi \rangle$ dumbbell, where $\xi$ is an irrational number, agrees with nudged elastic band analyses of $\langle 110\rangle$ to $\langle 111\rangle$ transformations. Direct simulations of Huang diffuse scattering by symmetry-broken defect configurations predicted by DFT explain why no zero intensity lines were observed in experiment and resolve the long outstanding question about the structure of defects in Mo and similar metals. A $\langle 11\xi \rangle$ defect migrates on average one dimensionally through a sequence of three-dimensional nonplanar $\left[11\xi \right]$ to $\left[\xi 11\right]$ or $\left[1\xi 1\right]$ transitions. Barriers for defect migration in nonmagnetic Cr, antiferromagnetic Cr, Mo, and W derived from DFT calculations, 0.052, 0.075, 0.064, and 0.040 eV are well correlated with the onset of defect migration temperatures observed experimentally.

Figure 1

Sketch illustrating a two-dimensional transition pathway of migration of a $\langle 110\rangle$ dumbbell proposed by Jacques and Robrock [17].

Figure 2

Results of nudged elastic band calculations illustrating how the formation energy of an SIA dumbbell varies as a function of the orientation of the axis of the defect. The orientation changes from being collinear to the [110] direction (the left edge of the diagram) to being collinear to the [111] direction (the right edge of the diagram). The curves were computed taking into account the elastic correction associated with the use of periodic boundary conditions [13, 33, 34]. Note that the only curve in the figure that is monotonic refers to Nb, which is a group 5 metal.

Figure 3

The elastic correction $E_{\text{el}}^{\text{corr}}$ part of the formation energy of defects corresponding to the curves shown in Fig. 2.

Figure 4

Atomic structure of a symmetry-broken $\langle 11\xi \rangle$ SIA dumbbell defect in Mo, simulated using a supercell containing $5\times 5\times 5$ bcc unit cells. Symmetry breaking in the core of the defect gives rise to buckling of the central [111] atomic string containing an extra atom. Buckling can occurs in one of the three $\left\{1\overline{1}0\right\}$ atomic planes equivalent by symmetry with respect to the straight linear $\langle 111\rangle$ configuration of the defect.

Figure 5

The local density of states (DOS) for one of the two atoms at the center of $\langle 111\rangle$ and $\langle 11\xi \rangle$ dumbbell configurations in Mo. A plot showing the DOS computed for perfect bcc lattice is shown for comparison. The position of the Fermi energy for all the three structures corresponds to the origin of the horizontal axis and is indicated by a dashed vertical line.

Figure 6

Distances between successive atoms $i$ and $j=i+1$ in a $\langle 111\rangle$ atomic string containing an extra atom. (Top) Data for a $\langle 111\rangle$ dumbbell and (bottom) data for a $\langle 11\xi \rangle$ configuration in Cr/AFM, Cr/NM, Mo, and W. Positions of atoms were computed using the GGA-PBE functional.

Figure 7

Voronoi volumes of atoms belonging to a $\langle 111\rangle$ atomic string containing an extra atom. (Top) Data for a $\langle 111\rangle$ dumbbell and (bottom) data for a $\langle 11\xi \rangle$ configuration of a defect in Cr/AFM, Cr/NM, Mo, and W. All the calculations were performed using the GGA-PBE functional.

Figure 8

Two-dimensional plots of electron charge density difference computed for a $\langle 11\xi \rangle$ dumbbell configuration in the $\left(1\overline{1}0\right)$ plane in Cr/AFM, Cr/NM, Mo, and W. Calculations were performed using the GGA-PBE functional. Electron charge density difference is defined as the self-consistent electron density minus a superposition of atomic charge densities.

Figure 9

Two-dimensional plots of electron charge density difference computed for a $\langle 111\rangle$ dumbbell configuration in the $\left(1\overline{1}0\right)$ plane in Cr/AFM, Cr/NM, Mo, and W. Calculations were performed using the GGA-PBE functional. Electron charge density difference is defined as the self-consistent electron density minus a superposition of atomic charge densities.

Figure 10

Schematic diagram illustrating the migration pathway of an SIA defect from a $\left[11\xi \right]$ to a $\left[\xi 11\right]$ dumbbell configuration. Values of parameter $\xi$ for group 6 metals are listed in Tables 2 and 3. Note that the initial and final configurations of the defect are associated with symmetry breaking occurring in two different $\left\{110\right\}$ planes.

Figure 11

Migration energy barriers computed using the nudged elastic band method, and corresponding to the trajectory of migration of a symmetry-broken $\left[11\xi \right]$ SIA dumbbell to an adjacent cell, as illustrated in Fig. 10. Calculations were performed using the GGA-PBE exchange-correlation functional.

Figure 12

Energy contribution due to the elastic correction $E_{\text{el}}^{\text{corr}}$ corresponding to the trajectory of migration shown in Figs. 10 and 11.

Figure 13

Patterns of Huang diffuse scattering produced by an ensemble of randomly distributed and average over equivalent crystallographic orientations $\langle 100\rangle$ dumbbells.

Figure 14

Patterns of Huang diffuse scattering produced by randomly distributed and average over equivalent crystallographic orientations $\langle 111\rangle$ dumbbells.

Figure 15

Patterns of Huang diffuse scattering produced by randomly distributed and average over equivalent crystallographic orientations $\langle 110\rangle$ dumbbells.

Figure 16

Patterns of Huang diffuse scattering produced by randomly distributed and average over equivalent crystallographic orientations $\langle 11\xi \rangle$ SIA dumbbells.

Figure 17

Patterns of Huang diffuse scattering by $\langle 111\rangle$, $\langle 110\rangle$, and $\langle 11\xi \rangle$ randomly distributed and average over the equivalent crystallographic orientations dumbbell configurations computed for a [022] reflection in the (100) plane in reciprocal space. Only the $\langle 111\rangle$ dumbbell configurations produce zero intensity lines parallel to the $\left[0\overline{1}1\right]$ direction in reciprocal space.