Phonon structure of titanium under shear deformation along $\left\{10\overline{1}2\right\}$ twinning mode

We investigated phonon behavior of hexagonal close-packed titanium under homogeneous shear deformation corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode using first-principles calculations and phonon calculations. By this deformation, we found that a phonon mode located at a point on the Brillouin zone boundary is drastically softened, increasing the shear, and finally it triggers a spontaneous structural transition by breaking the crystal symmetry toward twin from parent.

Figure 1

Unit cell structures of HCP titanium, (a) the conventional unit cell (two atoms), and (b) an extended unit cell for easy representation of the $\left\{10\overline{1}2\right\}$ twinning mode (eight atoms). ${\mathrm{K}}_1$ (red) and ${\mathrm{K}}_2$ (blue) planes and ${\eta }_1$ and ${\eta }_2$ directions characterize the twinning mode. The plane of shear is depicted by $\mathrm{P}$ (green).

Figure 2

Homogeneous shear corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode. The extended unit cell given by Eq. (1) is projected on the plane of shear $\mathrm{P}$. The atoms depicted by the same symbols (open or filled) are located on the same plane parallel to the plane of shear. The distance between these two planes for the atoms with the open and filled symbols are $0.5{\mathbf{b}}^{\prime }$. The lattice drawn by the dotted lines is the sheared lattice. Applying the shear, ${\mathbf{c}}^{\prime }$ is changed to ${\mathbf{c}}^{\prime }+t{\mathbf{a}}^{\prime }$ with $t$ as a representative parameter.

Figure 3

Calculated lattice parameters $a$ and $c$ of the HCP conventional unit cell with respect to $k$-point sampling mesh density and smearing width $\sigma$. The circle, triangle, square, and diamond symbols denote those calculated with $\sigma =0.1$, 0.2, 0.3, and 0.4 eV, respectively. The lines connecting the points are guides to the eye.

Figure 4

Phonon band structures calculated with two different parameter pairs for Brillouin zone integration of the electronic structure calculation, $8\times 8\times 6k$-point sampling mesh ($2\times 2\times 2$ mesh for supercell) and 0.4 eV smearing width (solid curve), and $20\times 20\times 15k$-point sampling mesh ($5\times 5\times 5$ mesh for supercell) and 0.2 eV smearing width (dashed curve). The coordinates of the wave-vector labels are given in Ref. [21].

Figure 5

Crystal structures of (a) parent, (b) sheared parent, and (c) twin. The space-group type of the parent and twin structures is $P{6}_3/mmc$. That of the sheared-parent structure is $C2/m$ at the general $s/s_{\mathrm{t}}$ and $Cmcm$ at $s/s_{\mathrm{t}}=1$. See the caption of Fig. 2 for the different symbols (open and filled) of atoms. Due to the periodicity of the extended unit cell, some atoms are displayed by moving to the other side of the boundary after the structure optimization. To clarify the correspondence of the atoms, the indices are assigned to the atoms.

Figure 6

Displacement distances $\left|\mathbf{u}\right(s/s_{\mathrm{t}}\left)\right|$ of Eq. (11) at the homogeneous shears corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode. The lines connecting the open circles are guides to the eye. The sheared-parent structure at $s/s_{\mathrm{t}}=1$ before the optimization is shown in the inset. The arrows depict approximate directions of the displacements. The structure after the optimization is shown in Fig. 5. See the caption of Fig. 2 for the different symbols of atoms.

Figure 7

Increase of electronic total energy of the sheared-parent structure with respect to the homogeneous shear corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode. The lines connecting the open circles are guides to the eye. The dashed-dotted curve is the mirror image of the solid curve, which represents the energy increase of the sheared-twin structure. The arrows with the vertical dashed lines show the energies that are released by the transformation from the sheared-parent structures to the sheared-twin structures at general $s/s_{\mathrm{t}}$ and to the twin structure at $s/s_{\mathrm{t}}=1$ by the shuffling. The inset shows the sheared-parent structure at $s/s_{\mathrm{t}}=1$. The arrows show the atomic displacement directions indicated by the eigenvector of the imaginary phonon mode at $\mathbf{q}=(1/2,0,0)$. See the caption of Fig. 2 for the different symbols of atoms.

Figure 8

Phonon band structures at the homogeneous shears corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode. Those at $s/s_t=$ 0, 0.2, 0.4, 0.6, 0.8, and 1 are drawn by the different line styles from solid to shorter dashed lines. Phonon frequency below zero means imaginary frequency. A similar set of labels of points in Brillouin zones to that in Fig. 4 is used for easy comparison, although the homogeneous shear breaks the crystal symmetry. The explicit coordinates of these points are ${\mathrm{A}}^{\prime }(0,0,1/2), {\mathrm{H}}^{\prime }(1/3,1/3,1/2), {\mathrm{K}}^{\prime }(1/3,1/3,0), \mathrm{\Gamma }(0,0,0), {\mathrm{M}}^{\prime }(1/2,0,0)$, and ${\mathrm{L}}^{\prime }(1/2,0,1/2)$. Since the shapes of the Brillouin zones are different for the different shears, the positions of the reciprocal points in the Cartesian coordinates measured from the $\mathrm{\Gamma }$ points slightly disagree. The filled circles at the ${\mathrm{M}}^{\prime }$ points indicate the soft phonon modes that have the eigenvector shown in the inset of Fig. 7.

Figure 9

Squared frequencies of the characteristic phonon mode at $\mathbf{q}=(1/2,0,0)$ at the homogeneous shears corresponding to the $\left\{10\overline{1}2\right\}$ twinning mode. The lines connecting the open circles are guides to the eye.

Figure 10

The small (black) arrows depict atomic displacements induced by the structural optimization from the sheared-parent structure [Fig. 5] to the twin structure [Fig. 5]. The circular (red) arrows show how the displacements are arranged as structural units.