Al4SiC4 w\"urtzite crystal: structural, optoelectronic, elastic and piezoelectric properties

New experimental results supported by theoretical analyses are proposed for aluminum silicon carbide (Al4SiC4). A state of the art implementation of the Density Functional Theory is used to analyze the experimental crystal structure, the Born charges, the elastic and piezoelectric properties. The Born charge tensor is correlated to the local bonding environment for each atom. The electronic band structure is computed including self-consistent many-body corrections. Al4SiC4 material properties are compared to other wide band gap W\"urtzite materials. From a comparison between an ellipsometry study of the optical properties and theoretical results, we conclude that the Al4SiC4 material has indirect and direct band gap energies of about 2.5eV and 3.2 eV respectively.

Al 4 SiC 4 is known for almost six decades since early optical and X-ray diffraction characterizations 9 . This material crystallizes in a yellow form belonging to the hexagonal system. The bonding characteristics, elastic stiffness, ideal strengths, and atomistic deformation modes of Al 4 SiC 4 were investigated by using Density functional theory (DFT) theory 10 .
The electronic band structure and optical properties were simulated at the same level of theory 11 , the Al 4 SiC 4 crystal being predicted to be a small gap semiconductor with an M g E   indirect band gap of 1.05 eV. No experimental results on the optical properties were available up to now to confirm these predictions.
In this paper, we perform an experimental and theoretical study of the structural, electronic, optical, elastic and piezoelectric properties of the Al 4 SiC 4 crystal. The crystal structure simulated at the DFT level is compared to available Xray diffraction data 1 . Density functional perturbation theory (DFPT) is used to study the Born dynamical effective charges, elastic and piezoelectric properties of the Al 4 SiC 4 crystal. The electronic band structure is calculated including self-consistent many-body (scGW) contributions 12,13 to correct the well-known underestimation of the band gaps computed DFT level. The electronic band structure is used to analyse new optical data from phase modulated spectroscopic ellipsometry.
DFT calculations are performed using the plane-wave projector augmented wave (PAW) method as implemented in the VASP code [14][15][16] . The local density approximation (LDA) is used for the exchange-correlation functional 17 10,11 . A plane-wave basis set with an energy cut-off of 950 eV is used to expand the electronic wave-functions. The reciprocal space integration is performed over a 18x18x3 Monkhorst-Pack grid 21 . Energy convergence is accurately reached with tolerance on the residual potential which stems from difference between the input and output potentials. The crystal structure has been relaxed until the forces acting on each atom are smaller than 10 -6 eV/Å. Al 4 SiC 4 single crystals have been obtained by high temperature synthesis, high purity silicon (9N) and aluminium (99.5) pieces were melted in a graphite crucible which acted both as a container for the melt and as carbon source. Al 4 SiC 4 single crystals were grown by maintaining the melt at high temperature (1800 o C) before cooling down at a very low and controlled rate. The crystal structure was confirmed by TEM and XRD to be Al4SiC4 phase in hexagonal structure (Space group P6 3 mc) with lattice parameters a=0.32812±0.00045nm and c=2.17042±0.00554 22 .
Ellipsometry yields the ratio, ρ, of the Fresnel reflection coefficient of the p-polarized (parallel to the plane of incidence of the linearly polarized light beam) and s-polarized (perpendicular to the plane of incidence) light reflected from the surface through the Ellipsometry angles  and  defined by the equation: i exp tan ρ and, hence,  and , are related to the material pseudodielectric function, 2 1 i      , through the equation: The Spectroscopic Ellipsometry data (,) have been measured on the Al 4 SiC 4 compound between 1eV and 5eV at 70 o angle of incidence. The pseudodielectric function for Al 4 SiC 4 has been deduced from the experimental data using a simple twophase model consisting of Al 4 SiC 4 /air.

 Crystal structure
We have performed a full optimization of the hexagonal structure (space group P6 3 mc) by minimizing the total energy with respect to the lattice constants a and c and the internal positions of each atoms in the unit cell. Figure 1 shows an overview of the crystal structure projected on the (a,c) and (a,b) planes. The DFT results agree rather well with experimental data 1 as reported in Table I : discrepancies are found less than 2%. The calculated ratio c/a=6.630 10 , c/a=6.646 11 , c/a=6.619 and c/a=6.610 respectively obtained at GGA ultrasoft, GGA, LDA and PAW LDA levels, differ only slightly from the experiment value 6.614.

 Born dynamical effective charges, Elastic and Piezoelectric constants
The knowledge of the local bonding structures and local charges allows investigating the correlation between these quantities. The Born dynamical effective charge tensor 23 is calculated by the second derivatives of the total energy E with respect to one atomic displacement u and one component of electric field. This tensor is then used to calculate the atomic relaxation contribution to both the elastic and the piezoelectric tensors.
To correlate the Born effective charges with the local bonding structures, nine building blocks have been isolated (Fig 2.): one SiC 4 block, four AlC 4 blocks, and also four other blocks each containing a carbon atom e.g. CSi 3 Al, CSiAl 4 , CAl 5 and CAl 6 from the Al 4 SiC 4 material. On Figure 2, the Bond lengths and bond Angles in each block have been drawn, with atomic notations according to Z. Inoue 1 . The SiC 4 block and the four AlC 4 blocks are found to be distorted tetrahedral structures, which is is consistent with previous results 11 . The four last blocks are completely different from the first 5 ones. In CSi 3 Al, the carbon atom (C4) is in coordination four which corresponds to a tetrahedral geometry. In CSiAl 4 and CAl 5, the carbon atoms (C2 and C3) are in coordination five which corresponds to a trigonal bipyramid geometry. In particular, the carbon atom C2 is bridging the SiC 4 and AlC 4 tetrahedral blocks. Finally, the Carbon atom C1 is in coordination six in CAl 6 and is located in a middle of a trigonal prism geometry. The carbon atoms C1 and C3 are in the middle of the AlC 4 tetrahedral entities. The variety of local environments can be characterized by the mean value of the Born charge , computed from the trace of the Born charge tensor (Table II) The Al 4 SiC 4 Würtzite crystal has five independent elastic constants (Table III), namely, C 11 , C 12 , C 13 , C 33 and C 44 , defined in Cartesian coordinates, assuming a symmetry axis along z. Due to the hexagonal symmetry, the C 66 constant depends on C 11 and C 12 : C 66 =(C 11 -C 12 )/2. In order to estimate the elastic constants, the second-order strain derivatives of total energy were computed within DFPT 23 . To obtain the piezoelectric coefficients, the second-order derivatives of total energy have been calculated with respect to the strain and the electric field components 23  In order to interpret the optical measurements, a first computation of the electronic band structure was performed exactly at the same DFT theory level than the previous work with the same k-grid (4x4x1) and plane-wave cutoff of 500eV 11