Effects of thermal, elastic, and surface properties on the stability of SiC polytypes

SiC polytypes have been studied for decades, both experimentally and with atomistic simulations, yet no consensus has been reached on the factors that determine their stability and growth. Proposed governing factors are temperature-dependent differences in the bulk energy, biaxial strain induced through point defects, and surface properties. In this work, we investigate the thermodynamic stability of the 3C, 2H, 4H, and 6H polytypes with density functional theory (DFT) calculations. The small differences of the bulk energies between the polytypes can lead to intricate changes in their energetic ordering depending on the computational method. Therefore, we employ and compare various DFT codes, i.e., vasp, cp2k, and fhi-aims; exchange-correlation functionals, i.e., LDA, PBE, PBEsol, PW91, HSE06, SCAN, and RTPSS; and nine different van der Waals (vdW) corrections. At $T=0$ K, 4H-SiC is marginally more stable than 3C-SiC, and the stability further increases with temperature by including entropic effects from lattice vibrations. Neither the most advanced vdW corrections nor strain on the lattice have a significant effect on the relative polytype stability. We further investigate the energies of the (0001) polytype surfaces that are commonly exposed during epitaxial growth. For Si-terminated surfaces, we find 3C-SiC to be significantly more stable than 4H-SiC. We conclude that the difference in surface energy is likely the driving force for 3C-nucleation, whereas the difference in the bulk thermodynamic stability slightly favors the 4H and 6H polytypes. In order to describe the polytype stability during crystal growth correctly, it is thus crucial to take into account both of these effects.

Figure 1

The unit cells of the polytypes investigated in this work. The direction of the polyhedra, to the left (red) and right (blue), is directly related to the hexagonal $h$ sites and cubic $k$ sites of the Si and C atoms; when the color from one layer to the next remains unchanged, it indicates a $k$ site, whereas when the color changes it indicates an $h$ site.

Figure 2

Energy differences of the relaxed 3C, 6H, 4H, and 2H structures as a function of hexagonality for different DFT codes, XC functionals, and basis sets. fhi-aims t (rt) stands for tight (really tight). To improve visualization, we shifted the vasp data to the left and the fhi-aims and cp2k data to the right.

Figure 3

Lattice constants $a$ vs bulk hexagonality for the conventional DFT and van der Waals corrections. fhi-aims t (rt) stands for tight (really tight). Experimental x-ray diffraction results of Levinshtein et al. [58] and Harris et al. [59] are included.

Figure 4

Normalized ratio of lattice constants, $c/\left(2xa\right)$, vs bulk hexagonality for the conventional DFT and vdW dispersion methods. fhi-aims t (rt) stands for tight (really tight). Experimental x-ray diffraction results of Levinshtein et al. [58] and Harris et al. [59] are included.

Figure 5

Energy differences for 3C, 6H, 4H, and 2H-SiC as a function of hexagonality, calculated either with vasp-PBE and a vdW correction or with a vdW XC functional.

Figure 6

The phonon vibrational free energy of the $x\mathrm{H}$ polytypes relative to 3C, calculated for the LDA and PBE XC functional in the harmonic approximation. The inset shows the difference in vibrational entropy.

Figure 7

The change in elastic moduli relative to 3C vs bulk hexagonality.

Figure 8

Effects of (0001) biaxial strain on the relative stability of 3C and 4H-SiC at $T=0$ K for three XC functionals. The inset shows the energy-strain curves for 3C and 4H. Elastic energy difference between 3C and 4H is plotted vs biaxial strain as a percentage of the lattice constant $a$, where a positive difference means 3C is more stable.

Figure 9

The surface structures investigated in this work. The Si-terminated (C-terminated) surfaces are oriented towards the top (bottom). The hexagonality of the surfaces is shown in Table 4. Each unit cell has a 12 $\text{Å}$ vacuum, one dangling bond, and an H atom bound to the (no longer) dangling bond of the opposite surface.

Figure 10

Surface energy relative to the 3C polytype vs surface hexagonality, which is determined by the polytype and the termination of the surface.

Figure 11

Absolute energies of the Si- and C-terminated surfaces vs hexagonality for the non-spin-polarized and ferromagnetic ideal surface, and, in the case of Si termination, a $2\times 1$ (nonmagnetic) reconstruction. All results are obtained with vasp-PBE.

Figure 12

Comparison of the energy contributions studied in this work: the bulk internal energy $\mathrm{\Delta }{U}_0$ at $T=0$ K, the phonon vibrational energy $F_{\mathrm{vib}}$ at $T=1800$ K, the elastic contribution $\mathrm{\Delta }E$ at 1%, and the Si-face surface energy. The differences between 3C and $x\mathrm{H}$ polytypes are plotted for each contribution based on the results obtained by the vasp-PAW-PBE method. The surface energy differences are labeled according to Table 4.