Nonlinear optical properties of 6H-SiC and 4H-SiC in an extensive spectral range

Silicon carbide (SiC), which is the leading representative of the third-generation of semiconductors, possesses many excellent physical properties. However, its advantages also incur difficulties in processing, which calls for special processing techniques, such as femtosecond laser machining. In addition, SiC has shown unprecedented potential for optoelectronic applications. Knowledge of the nonlinear absorption coefficient and the nonlinear refractive index coefficient of SiC is required in both of the fields. In this work, the nonlinear absorption coefficient and the nonlinear refractive index coefficient of semi-insulating (SI) 6H-SiC and SI 4H-SiC, the most pervasive SiC polytypes, are measured in an extensive spectral range from 400 nm to 1000 nm with the Z-scan technique. Besides, the spectral dependence of the nonlinear optical properties is analyzed, facilitated by linear absorption spectrum. Especially, two-photon absorption (2PA) and three-photon absorption (3PA) coefficients of SI 6H-SiC and SI 4H-SiC are characterized in the respective spectral ranges. From the characterization of SiC, we can observe self-focusing phenomenon for nonlinear refraction. In the end, we unravel the potential of SiC for ultrafast all-optical switching based on the measured nonlinear optical properties.


Nonlinear Optical Properties of 6H-SiC and 4H-SiC in an Extensive
Spectral Range:supplement 1

CALCULATION OF BAND STRUCTURE
Employing the density functional theory (DFT), the electronic band structures are calculated using Quantum Espresso package [1]. Generalized Gradient Approximation (GGA) [2] using Optimized Norm-Conserving Vanderbilt (ONCV) [3] pseudopotentials in our calculations treats the exchange and correlation (XC) functional. The lattice structures of SI 6H-SiC and SI 4H-SiC are first relaxed, in which the energy cutoff is set to be 80 Ry and the converging threshold for energy and atomic forces are set to be 10 −12 eV and 0.0001 eV/Å, respectively. The k k k points in the Brillouin zone are sampled with 10×10×3 for both SI 6H-SiC and SI 4H-SiC.
The electronic band structures of SI 6H-SiC and 4H-SiC along some high symmetry lines in the Brillouin zone are shown in Figs. S3(a) and S3(b), respectively. The highest occupied state of the valence band is at the Γ-point, and the conduction band minimum is at the M-point. The calculated band gaps agree well with our absorption spectral measurements.

NONLINEAR REFRACTION
Figs. S4 (SI 6H-SiC) and S5 (SI 4H-SiC) show the differences of fitting for nonlinear refraction by 2, 10 and 20 terms. Fitting by 2 and 10 terms shows obvious difference while fitting by 10 and 20 items are well consistent with each other. Therefore, we choose the fitting by 10 terms for the final results to guarantee convergence.

CROSS SECTIONS OF 2PA AND 3PA
The expressions of transition rate are given by [4,5] where σ 2PA is cross section of two-photon absorption (2PA), σ 3PA is cross section of three-photon absorption (3PA), β is 2PA coefficient, γ is 3PA coefficient, I indicates the laser intensity, N is the number density of absorbing species, hν is the photon energy and σ 2 , σ 3 are cross section coefficients of 2PA and 3PA, respectively. Cross sections can be calculated by Eqs. S1 and S2. Conventionally, absorption cross section is used to describe absorption capability of an individual absorbing unit, such as one dye molecule or one quantum dot. Such absorbing units are usually dispersed into solution or matrix for absorption coefficient measurement, and the obtained absorption coefficient should be converted into cross section of a single absorbing unit. However, semiconductors like SiC are in solid state condensed phase, which are different from dye molecules or quantum dots, so generally absorption coefficient is used to characterize such materials instead of cross section. We believe that cross section can be defined for semiconductors by replacing the number density of molecules N with population density of electrons, N 0 which can be expressed as where N A is the Avogadro constant, M is the molar mass, ρ is the density and "20" represents the number of electrons in SiC (Si-14, C-6). This is because electrons in semiconductors are the absorbing units. With our measured nonlinear absorption coefficients, the cross section or the cross section coefficient of SiC can be calculated based on Eqs. S1 and S2 if needed.