B.1. The Basis Set
TheFHI-aimscode employs numeric atom-centered basis sets; basic descrip-tions of their mathematical form and properties as introduced in Sec.2.5 or in Ref. [32]. TheFHI-aimsbasis sets are defined by numerically determ-ined radial functions Eq.2.28corresponding to different angula momentum channels. Each radial function is obtained by solving a radial Schrödinger equation and is subject to a confinement potential vcut(r) in Eq. 2.28. It ensures that the radial function is strictly zero beyond the confining radius
C Si
minimal [He]+2s2p [Ne]+3s3p tier 1 H(2p, 1.7) H(3d, 4.2)
H(3d, 6.0) H(2p, 1.4) H(2s,4.9) H(4f, 6.2)
Si2+(3s) tier 2 H(4f, 9.8) H(3d, 9.0)
H(3p, 5.2) H(5g, 9.4) H(3s, 4.3) H(4p, 4.0) H(5g, 14.4) H(1s, 0.65)
H(3d, 6.2)
. . . .
Table B.1.: Radial functions used for C and Si. The first line (“minimal”) denotes the radial functions of the occupied orbitals of spherically symmetric free atoms as computed inDFT-LDAor -PBE(noble-gas configuration of the core and quantum numbers of the additional valence radial functions). “H(nl, z)” denotes a hydrogen-like basis function for the bare Coulomb potentialz/r, including its radial and angular momentum quantum numbers,nandl. X2+(nl) denotes an,lradial function of a doubly positive free ion of element X. See also Ref. Blumet al.[32] for notational details.
rcutand decays smoothly.
What is important for the present purposes is to demonstrate the accurate convergence, up to a few meV at most, of our calculated surface energies with respect to the basis set used. As is typical of atom-centered basis sets (Gaussian-type, Slater-type, numerically tabulated etc.), variational flexibility is achieved by successively adding radial functions for individual angular momentum channels, until convergence is achieved. In practice, the basis functions for individual elements inFHI-aimsare grouped in so-called “tiers” or “levels”: tier 1, tier 2, and so forth. In the present work, basis functions for Si and C up to tier 2were used. The pertinent radial functions are summarised in Table (B.1), using the exact same notation that was established in Ref. [32].
Figure B.1.: Effect of increasing the basis set size on the surface energy of the√
3 approx-imant to theZLGphase at the chemical potential limit of bulk graphite. The PBE+vdW functional was used. In the plot we use T1 (T2) as abbrevation oftier 1(tier 2) , respectively
The convergence of the calculated surface energies in this work with basis size is exemplified for the (√
3×√
3)-R30◦ small unit cell approximant [334,214] to theZLGphase in Fig. (B.1) including 6 SiC-bilayer. The bottom carbon atoms were saturated by hydrogen. Its geometry was first fully relaxed with FHI-aimstight grid settings, a tier 1+dg basis set for Si and a tier 2 basis set for C basis settings. This geometry was then kept fixed
for the convergence tests shown here. What is shown in Fig. (B.1) is the development of the surface energy (Si face and H-terminated C face) with increasing basis size for both C and Si were calculated using
γSi face+γC face= 1 A
Eslab−NSiµSi−NCµC
. (B.1)
where NSiandNCdenote the number of Si and C atoms in the slab, respect-ively, andAis the chosen area. The computed surface energy is shown per 1×1 surface area as in all surface energies given in the main text.
The notation in the figure is as follows:
• “T1” and “T2” abbreviate the set of radial functions included intier 1 andtier 2, respectively (see TableB.1).
• “Si T1-f” denotes the Sitier 1basis set, but with the f radial function omitted.
• “C T2-g denotes the set of radial functions for C up to tier 2, but omitting theg-type radial function oftier 2.
• “Si T1+dg” denotes the radial functions included up totier 1of Si, and additionally thedandgradial functions that are part oftier 2. This is also the predefined default basis set forFHI-aims’tight’ settings for Si.
• “C T2” denotes the radial functions of C up totier 2and is the default choice for ’tight’ settings inFHI-aims.
In short, the plot indicates the required convergence of the surface energy to a few meV/(1×1) surface area if the default FHI-aims’tight’ settings are used. It is evident that the high-l g-type component for C contributes noticeably to the surface energy.
B.2. Slab Thickness
The convergence of our surface calculations with respect to the number of SiC bilayers is shown in Fig. (B.2) by considering surface energies for the (√
3×√
3)-R30◦ small unit cell approximant [334,214]. The zero reference
energy is an unreconstructed six bilayer 1x1 SiC surface. A six bilayer slab is sufficient to accurately represent bulk effects.
Figure B.2.: Slab thickness dependence of the surface energy of the (√ 3×√
3)-R30◦ ap-proximant to theZLGphase.
B.3. k-Space Integration Grids
We demonstrate the accuracy of the 2D Brillouin zone (BZ) integrals for our graphene-like surface phases by comparing different k-space integration grids in Table (B.2). The ZLGand mono-layer graphene (MLG) surface energies in the full (6√
3×6√
3)-R30◦cell are compared using theΓ-point only and using a 2x2x1k-space grid. Thesek-mesh tests were performed in a four bilayer slab, using thePBEincluding van-der-Waals effects [326]
(PBE+vdW) functional, lightreal space integration grids, a tier1basis set without the f functions for Si andtier1for C. The geometries were kept fixed at the configuration relaxed with a Γ-point only k-space grid. In Table (B.2), the surface energies relative to the unreconstructed 1x1 surface in the graphite limit are listed for the silicon rich√
3x√
3 reconstruction, the ZLGandMLGphases. Table (B.2) clearly shows that the surface energies are well converged using theΓ-point only. Hence in this work, all surface energies for the (6√
3×6√
3)-R30◦phases were calculated using thisk-space integration grid. For all bulk reference energies as well as for the two silicon rich surface reconstructions, the convergence with respect to the k-mesh size has been tested and found to be well converged for grids equivalent to theZLGphase.
system k-grid
6x6x1 12x12x1 24x24x1
√3x√
3 -0.439 -0.435 -0.436 1x1x1 2x2x1
ZLG -0.426 -0.426 MLG -0.454 -0.455
Table B.2.: Surface energies in [eV/SiC(1x1)] relative to the unreconstructed3C-SiC(1x1) surface for the chemical potential limit of graphite, four-bilayer SiC slabs. The silicon-rich
√3x√
3 reconstruction, ZLGandMLGphases using differentk-grids are shown. The PBE+vdW exchange-correlation functional was used.
B.4. The Heyd-Scuseria-Ernzerhof Hybrid Functional Family for 3C-SiC
In this work, we used theHSE[175] for calculating the electronic structure and validating surface energies. InHSE06the amount of exact exchange is set to α = 0.25 and the range-separation parameter ω = 0.2Å−1. We calculated the band gap and valence band width of3C-SiC for different values ofα. We used with ’tight’ numerical settings as introduced in Sec.B.1 with an 12×12×12 off-Γk-grid.
As can be seen in Fig. B.3, for fixed (ω) the band gap and valence band width depend practically linearly on the exchange parameterα. We tested the HSE06 with respect to the band gap and band width, the latter being a measure for the cohesive properties of a crystal [257]. The defaultHSE06 value ofα =0.25 captures both the band gap and the band width well and we therefore adopt it for our calculations.
Figure B.3.: The Kohn-Sham band-gap (blue circles) and valence band width alongΓtoX (red circles) of 3C-SiC as a function ofα. The HSE06 valueα=0.25 is marked by a vertical line. The experimental valence band width is shown as horizontal line at 3.6 eV[143] and the exp. band gap at 2.42 eV[148].