Lattice dynamics and metastability of fcc metals in the hcp structure and the crucial role of spin-orbit coupling in platinum

We investigate the lattice dynamical properties of Ni, Cu, Rh, Pd, Ag, Ir, Pt, and Au in the nonequilibrium hcp structure by means of density-functional simulations, wherein spin-orbit coupling (SOC) was considered for Ir, Pt, and Au. The determined dynamical properties reveal that all eight elements possess a metastable hcp phase at zero temperature and pressure. The hcp Ni, Cu, Rh, Pd, and Au previously observed in nanostructures support this finding. We make evident that the inclusion of SOC is mandatory for an accurate description of the phonon dispersion relations and dynamical stability of hcp Pt. The underlying sensitivity of the interatomic force constants is ascribed to a SOC-induced splitting of degenerate band states accompanied by a pronounced reduction of electronic density of states at the Fermi level. To give further insight into the importance of SOC in Pt, we (i) focus on phase stability and examine a lattice transformation related to optical phonons in the hcp phase and (ii) focus on the generalized stacking fault energy (GSFE) of the fcc phase pertinent to crystal plasticity. We show that the intrinsic stable and unstable fault energies of the GSFE scale as in other common fcc metals, provided that the spin-orbit interaction is taken into account.

Figure 1

Illustration of the crystal configuration at different stages of the GSF (from left to right, perfect fcc lattice, lattice with ISF, and lattice with ESF) and repeated slab geometry for calculating the GSFE surface. The letters A, B, and C, as well the corresponding color code, denote the stacking sequence, and the arrows indicate the slip planes with slip vector ${\mathit{b}}_p$.

Figure 2

Effect of SOC on the phonon dispersion relations for hcp Pt employing the scalar-relativistic geometry (see Table 2). $-\left|\nu \right|$ is plotted when ${\nu }^2<0$. The arrows indicate the two optical normal modes at $\mathrm{\Gamma }$, whose polarization vectors lie in the (0001) plane.

Figure 3

The electronic structure of hcp Pt. (a) Total electronic DOS in the scalar-relativistic and relativistic modes with fplo. (b) fplo-SR band structure and the $E_{1g}$ band weights represented by gray ribbons. (c) fplo-FR band structure. The equilibrium geometries were employed (see Table 2).

Figure 4

(a) Sketch of the hcp to fco transformation at the initial ($\delta =0$), intermediate ($\delta =0.5$), and final ($\delta =1$) stages by means of displacing the two types of hcp (0001) planes alternating in the $\left[1\overline{1}00\right]$ and $\left[\overline{1}100\right]$ directions. Atoms residing in the two planes are colored differently. The basal plane of the fco structure with lattice parameters $a_{\text{fco}}$ and $b_{\text{fco}}$ is indicated at $\delta =1$. $c_{\text{fco}}$ (not depicted) measures the height of the fco unit cell perpendicular to the image's plane. (b) Total energy difference $E\left(\delta \right)-E_{\text{hcp}}$ for the transformation of Pt in comparison to Pd, Ir (energies rescaled by a factor of 0.1), and Au. The inset shows the energies of Pt for small $\delta$.

Figure 5

(a) GSFE surface $\gamma$ of Pt with fault pathway $u=\left|\mathit{u}\right|$ given in fractions of the magnitude of the partial Burgers vector $b_p=\left|{\mathit{b}}_p\right|$. (b) Scaling plot for stable and unstable fault energies of the GSFE with our values for Pt computed with and without invoking SOC, DFT data taken from Jin et al. (neglecting SOC) [64], and the scaling law (5) suggested in Ref. [64].