Phase-field crystal model for a diamond-cubic structure

We present a structural phase-field crystal model [M. Greenwood et al., Phys. Rev. Lett. 105, 045702 (2010)] that yields a stable $dc$ structure. The stabilization of a $dc$ structure is accomplished by constructing a two-body direct correlation function (DCF) approximated by a combination of two Gaussian functions in Fourier space. A phase diagram containing a $dc$-liquid phase coexistence region is calculated for this model. We examine the energies of solid-liquid interfaces with normals along the [100], [110], and [111] directions. The dependence of the interfacial energy on a temperature parameter, which controls the heights of the peaks in the two-body DCF, is described by a Gaussian function. Furthermore, the dependence of the interfacial energy on the peak widths of the two-body DCF, which controls the excess energy associated with interfaces, defects, and strain, is described by an inverse power law. These relationships can be used to parametrize the phase-field crystal model for the $dc$ structure to match solid-liquid interfacial energies to those measured experimentally or calculated from atomistic simulations.

Figure 1

(a) Schematic of the free-energy density of a relaxed system as a function of the lattice spacing for a given $\overline{n}$. The point at which $\Delta f^{\alpha }(\overline{n},a)$ is minimized with respect to $a$ is marked by an X. (b) Schematic of the free-energy densities that satisfy $\partial \Delta f^{\alpha }(\overline{n},a)/\partial a=0$ at each $\overline{n}$ [as illustrated in (a)]. The solid curve shows the free-energy density for the crystalline phase, and the dashed curve shows the corresponding values for the liquid phase. An X denotes the comment-tangent points of the free-energy density curves, which satisfy Eqs. (9) and (10).

Figure 2

(a) Schematic of a unit cell of the $dc$ structure where the shift of $a_{fcc}/4$ in each direction between the lattice-site positions of two $fcc$ structures [one solid (blue) colored and the other colored (red) with white crosses overlain] is denoted by arrows. Schematics of the (b) (111) and (c) (220) crystallographic planes, where the lattice points that are intersected by the atomic planes are colored (red) overlain with white crosses. Each plane of the {111} and {220} families of planes intersects two atoms for the $dc$ structure.

Figure 3

(a) Two-body DCF for a $dc$ structure for $\sigma =0.0$, 0.2, and 0.4. The parameters used are ${\alpha }_1={\alpha }_2=1.0,{\lambda }_1=4/\sqrt{3}{\AA }^{-2},{\lambda }_2=4/\sqrt{2}{\AA }^{-2},{\beta }_1=8,{\beta }_2=12,k_1=2\pi \sqrt{3}{\AA }^{-1}$, and $k_2=2\pi \sqrt{8}{\AA }^{-1}$. (b) The isosurface of a 64-unit-cell $dc$ structure calculated for $\overline{n}=0.02$ and $\sigma =0.01$. (c) A small portion of Fig. (b) showing two overlapping $fcc$ lattices in a $dc$ structure. The black arrow denotes the shift of a lattice site from one $fcc$ lattice to the other. (d) Phase diagram containing body-centered-cubic $\left(bcc\right)$, diamond-cubic $\left(dc\right)$, and liquid (L) phases.

Figure 4

Phase diagram containing diamond-cubic $\left(dc\right)$ and liquid (L) phases. The parameters of the two-body DCF used to construct this phase diagram are ${\alpha }_1={\alpha }_2=1.0,{\lambda }_1=4/\sqrt{3}{\AA }^{-2},{\beta }_1=8,k_1=2\pi \sqrt{3}{\AA }^{-1},k_2=2\pi \sqrt{8}{\AA }^{-1}$, and ${\lambda }_2{\beta }_2=8/3{\lambda }_1{\beta }_1$.

Figure 5

Plots of the natural logarithms of ${\gamma }_{100}(0,{\alpha }_0,{\alpha }_0)$ [(blue) X's], ${\gamma }_{110}(0,{\alpha }_0,{\alpha }_0)$ [(green) circles], and ${\gamma }_{111}(0,{\alpha }_0,{\alpha }_0)$ [(red) squares] for the $dc$ free energy used to construct the phase diagram in Fig. 3 as a function of the natural logarithms of ${\alpha }_0$. Here, ${\alpha }_1={\alpha }_2={\alpha }_0$ and $\sigma =0$. Dashed lines are the best fits to the data in the form of Eq. (15).

Figure 6

(a) ${\gamma }_p(\sigma ,1.0,1.0)$ and (b) ${\gamma }_p(\sigma ,1.0,1.0)/{\gamma }_p(0,1.0,1.0)$ as a function of $\sigma$ for the (100) [(blue) X's], (110) [(green) circles], and (111) [(red) squares interfaces. Calculations are for ${\alpha }_0=1.0$ and dashed curves show best fits to the data in the form of Eq. (16).

Figure 7

(a) ${\gamma }_{100}(\sigma ,{\alpha }_0,{\alpha }_0)$ and (b) ${\gamma }_{100}(\sigma ,{\alpha }_0,{\alpha }_0)/{\gamma }_{100}(0,{\alpha }_0,{\alpha }_0)$ as a function of $\sigma$ for ${\alpha }_0=0.25$ [(blueX's], ${\alpha }_0=0.5$ [(green) circles], and ${\alpha }_0=1.0$ [(red) squares]. Dashed curves show best fits to the data in the form of Eq. (16).

Figure 8

Natural logarithms of ${\gamma }_{100}(0,{\alpha }_1,{\alpha }_2)$ [(blue) X's], ${\gamma }_{110}(0,{\alpha }_1,{\alpha }_2)$ [(green) circles], and ${\gamma }_{111}(0,{\alpha }_1,{\alpha }_2)$ [(red) squares] plotted as a function of the natural logarithms of the ratio ${\alpha }_2/{\alpha }_1$. In these calculations ${\alpha }_1=0.625$ and $\sigma =0$. Dashed lines show fits to Eq. (18) and the solid vertical line marks the position where ${\alpha }_2/{\alpha }_1=1$.