Dispersion of nonresonant third-order nonlinearities in Silicon Carbide

In this paper we present a physical discussion of the indirect two-photon absorption (TPA) occuring in silicon carbide with either cubic or wurtzite structure. Phonon-electron interaction is analyzed by finding the phonon features involved in the process as depending upon the crystal symmetry. Consistent physical assumptions about the phonon-electron scattering mechanisms are proposed in order to give a mathematical formulation to predict the wavelength dispersion of TPA and the Kerr nonlinear refractive index n2. The TPA spectrum is investigated including the effects of band nonparabolicity and the influence of the continuum exciton. Moreover, a parametric analysis is presented in order to fit the experimental measurements. Finally, we have estimated the n2 in a large wavelength range spanning the visible to the mid-IR region.


Theory
Silicon carbide has been the subject of many theoretical studies. In this context, a variety of structural, electronic and optical properties in SiC have been examined theoretically by many research groups and the results have been well related to the experimental measurements. However, to the best of our knowledge, efforts have not yet been made in the literature to investigate the physical features of the TPA process and the wavelength dispersion of the TPA and Kerr coefficients. To solve that deficiency, we propose a physical discussion for cubic and hexagonal polytypes of SiC in order to theoretically predict the third-order nonlinearity in that spectral range where experimental data are not available. Moreover, as outlined in ref. 15, only 3C-SiC, 4H-SiC, and 6H-SiC are generally used for fabrication of photonic channel waveguides. Crystalline SiC waveguides, wurtzite as well as cubic, are ideal for realizing on-chip the third-order nonlinear applications discussed above.
Along with low loss, there is a requirement on indices for the nonlinear strip waveguides. To provide the needed refractive index contrast between the SiC waveguide core and the transparent lower cladding (substrate), the cladding should have an index less than ~2.0. In fact, there are several ways in which to construct these cubic or wurtzite SiC channels, specifically from starting wafers of SiC/SiO 2 /Si or SiC/Si 3 N 4 /Si or SiC/Al 2 O 3 /Si. A successful fabrication technique has proven to be the "smart cut" method, in which an H-implanted oxide-coated 4H SiC wafer is bonded to an oxidized Si wafer 15 . 3C SiC has also been grown epitaxially upon sapphire for waveguiding 19 . An acceptable propagation loss of 7 dB/cm has been measured for 4H SiC on silica upon silicon. From resonator Q, an implied loss of 12 dB/cm has been observed for 3C-SiC-on-insulator channels 18 . Small-area heteroepitaxy of 3C SiC on Si has been used to reduce defects 20 , and this wafer preparation for smart cut may yield a waveguide loss comparable to that found for wurtzite waveguides.
Due to the very large value of the direct bandgap, we guess that SiC materials do not suffer from any TPA effect induced by direct transitions as dominated by the allowed-forbidden (a-f) transitions (as will be demonstrated in the following section). Thus, based on our previous theoretical work 21 , we investigate the nonlinear absorption processes as induced "only" by indirect transitions involving the intermediate states with Γ symmetry in the conduction band as well as nonlinear absorption by phonon emission and absorption.
In Fig. 1(a) and (b) the schematic band diagrams are shown for 3C-SiC and 4H-SiC bulk materials, respectively. The plots indicate the fundamental symmetry points useful in applying the group-theory selection rules.
Actually, an electron makes a transition from the doubly degenerate valence bands at K = 0, v 1 (heavy hole) and v 2 (light hole), to the minimum conduction band c (at X 1 or M symmetry points for 3C-SiC and 4H-SiC, respectively) through the intermediate states, generally indicated as n and m 21 . Therefore, two photons at the frequencies ω 1 and ω 2 are absorbed to transit from the valence bands to the intermediate states, then a phonon of energy E ph = ħΩ is absorbed or emitted in order to complete the transition from one of the two intermediate states to the minimum conduction band.
According to the model detailed in our previous work 21 , under the hypothesis of parabolic bands the indirect TPA coefficient should be given by Eq. (1): , and the subscript (in) indicates the indirect transitions.
Generally speaking, the two-photon indirect absorption can also be influenced by the Coulomb interaction. In this sense, in Eq. (2), the term α is related to the continuum exciton effect. However, if we assume the nonparabolicity of both valence and conduction bands, the TPA coefficient, β ω ω ( , ) np in TPA , 1 2 , can be calculated as: np in TPA  p in  TPA  ,  1  2  ,  1  2  1  2 where the function R(ω 1 , ω 2 ) is defined as in Eqs (4)(5)(6):

Figure 1. Schematic band diagram for 3C-SiC (left) and 4H-SiC (right).
Scientific RepoRts | 7:40924 | DOI: 10.1038/srep40924  It is worth noting that Eq. (3) is rigorous in the absence of any continuum exciton influence. Moreover, the assumption of nonparabolicity applied only to the conduction band (see R(ω 1 , ω 2 ) defined in ref. 21) gives very good results for several semiconductors (i.e., Si, Ge and GeSiSn alloy), but it represents too large of an approximation in the case of the silicon carbide materials. Indeed, our investigations indicate that the nonparabolicity of both valence and conduction bands effect constitutes a fundamental aspect of SiC structures, confirmed by a very good matching between theoretical predictions and the experimental data. For this reason, the R(ω 1 , ω 2 ) function of Eqs (4)-(6) represents a generalization of that presented in ref. 21 by imposing the nonparabolicity effect on both conduction and valence bands.
In Eq. (1), E g,in , m v , M c indicate the indirect bandgap energy, the hole (heavy or light) effective mass, and the electron effective mass, respectively. In particular, since the absorption process involves the indirect conduction valleys, the electron effective mass may be approximated by , where m t , m l , and d c represent the transverse mass, the longitudinal mass, and the number of equivalent conduction band minima, respectively 22,23 . Moreover, the coefficients R y = 13.6 eV, Δ m , Δ n , and ε s represent the Rydberg energy, the energy of the intermediate states m and n, and the semiconductor static dielectric constant, respectively. The terms p mv (2) and p nm (1) are the transition matrix element for the optical transitions → . Now, ξ mn can assume a value of 0 or 1 if one or both of the transitions are allowed by the selection rules. Thus, the physical features differ when moving from 3C-SiC to 4H-SiC. In this context, it is convenient to give the selection rules using the general method of group theory in order to take into account the crystal symmetry 24 . In case of 3C-SiC, the schematic diagram in Fig. 1(a) shows that the top of the valence state, the lowest conduction state, and the nearest intermediate states, have a symmetry as Γ 15 , X 1 (indirect bandgap), Γ 1 , Γ 15 (direct bandgap), and Γ 12 , respectively. Since 3C-SiC has a zinc-blende structure, the dipole operator acts with a symmetry Γ 15 . Thus, under the group theory picture, the following relationship holds: In the case of 3C-SiC, Eq. (7) indicates that, the most favorable two-photon indirect transitions involve two equivalent steps:  (1)). Similarly, the group theory confirms Eqs (8) and (9): only transverse acoustic (TA) and transverse optical (TO) phonons are allowed. Consequently, we can conclude that all phonons are allowed in 3C-SiC except the longitudinal optical ones. Similar to all of the known SiC polytypes, 4H-SiC is an indirect band-gap semiconductor. Using different standard notations, the space group is C v 6 4 (Schoenflies notation) or P63mc (international notation), which is the character table at the Γ point given in ref. 25. According to a number of electronic band-structure calculations, the equivalent conduction-band minima are located at the point M of the Brillouin zone (see Fig. 1(b)). The maximum of the valence band is at the center of Γ . The symmetry of the conduction intermediate states, at the Γ point, will be Γ 6 and Γ 1 (in the BSW notation). The heavy hole and light hole bands will have symmetry Γ 6 . Furthermore, since the 4H-SiC semiconductor is an uniaxial crystal, the dipole operator can be decomposed in two components, parallel to the wurtzite unit cell optical axis (c-axis) with symmetry Γ 1 , and orthogonal to the c-axis with symmetry Γ 6 . We note that 4H-SiC and 6H-SiC are commercially available only as bulk crystalline wafers cut in on-axis or off-axis orientations. The on-axis orientation cut results in a wafer with c-axis perpendicular to its surface, with the ordinary, and extraordinary refractive index in the plane of the wafer, and perpendicular to the plane of the wafer, respectively. As evidenced clearly in ref. 15, this cut is ideal for photonic devices, since the TE polarization is aligned with the ordinary axes of the crystal and the TM polarization is aligned with the extraordinary axis, thereby preventing unwanted polarization rotation.
According to the group theory, the symmetry of the dipole operator acts on the valence bands as: By inspecting Eqs (10) and (11), one recognizes that single optical transitions are allowed into Γ c 1 or Γ c 6 conduction bands, for light polarized parallel (TM polarization), or orthogonal (TE polarization) to the c-axis, respectively. However, in the frame of our TPA physical model, two allowed-allowed transitions involving two different intermediate states (see ref. 20 for detail) must be considered in order to apply Eq. (1). In this context, Eqs (10) and (11) indicate that the most favorable two-photon transitions for TE polarization are given by: Thus, by referring to Eq. (1), the exchange among Γ c 1 , and Γ c 6 is forbidden, resulting in ξ mn = 0. Conversely, when the light polarization is parallel to the c-axis (TM polarization), the optical transition 1 is forbidden. As a result, Eq. (1) is not directly applied. Thus, we guess that in the case of TM polarization the indirect TPA effect is characterized by intraband transitions in Γ v 6 , or Γ c 6 , respectively. The following transitions: control the indirect TPA process with light polarization parallel to the c-axis. In this context, the β ω ω ( , )  Table 1 26 .
According to the irreducible representation and the phonon dispersion calculation provided in ref. 26 Consequently, we can conclude that in 4H-SiC all phonons with M 2 and M 4 symmetry are allowed in the indirect TPA process for TE polarization. Similar considerations hold for 6H-SiC material, in which the minima of conduction band occur in both M and L points.
Having attained the TPA coefficient, we now apply the Kramers-Kronig relationship in order to derive the Kerr nonlinear coefficient (n 2 ) as: However, the functional expression of β ω ω ( , ) np in TPA , 1 2 , in Eqs (1)-(6), does not lead us to derive a numerical solution for predicting the wavelength dispersion of n 2 . To get that solution, we adopt a number of approximations as follows. In particular, we neglect both the continuum exciton effect and photon energy with respect to the energies Δ m , and Δ n . Moreover, although the non-degenerate TPA coefficient is well defined by Eq. (1), the approximation 22 of a degenerate TPA function with the substitution ω → (ω 1 + ω 2 )/2 has been used. This approximation becomes progressively less accurate as the photon energy increases well above the half-bandgap, but it can give results reasonably close to the experimental values since it avoids mathematical and unphysical divergences at zero photon-energy. Under these assumptions, the integral of Eq. (13) admits of the solution given in Eq. (14), where the term C np is a fitting coefficient depending on the explained approximations.

Results
The goal of this section is to evaluate the theoretical wavelength dispersion for the third-order absorption occurring in 3C, 4H and 6H SiC polytypes. For that purpose, we apply the Kramers-Kronig relationship in order to estimate the nonlinear Kerr refractive index. The SiC physical parameters used in our simulation are listed in Table 2. Furthermore, Sellmeier's index equations for 3C-SiC 27 , 4H-SiC and 6H-SiC 28 have been used in simulations to take into account the index dispersion of the material. Since Δ m and Δ n are not available in literature for the 6H-SiC polytype, we have assumed as references the values given for 4H-SiC. This assumption is in general not rigorous, however our parametric investigations indicate that small changes in Δ m , and Δ n do not influence significantly the TPA dispersion, confirming the consistency of our approximation.
By referring to Eq. (1), knowledge of the p mv , and |Q cn | 2 parameters is required to obtain numerical values to be compared with experimental results. Generally speaking, these values can be derived from both electronic structure and electron-phonon scattering theoretical calculations. Additionally, experimental measurements could be used to better set the values of p mv , and |Q cn | 2 , in order to improve the model predictions of the third order nonlinearity in the wavelength range where experimental data do not exist. Although theoretical calculations of |Q cn | 2 are difficult and unreliable, some preliminary numerical estimations can be derived by considering the acoustic phonon deformation potential and the intervalley phonon deformation potential scattering (see Eqs (15) and (16) in the Method section).
Due to the lack of a reasonable number of TPA coefficient measurements, we have carried out a set of parametric simulations in order to estimate, in a consistent way, the TPA wavelength dispersion. In this context, we adopt as a starting point the work proposed in ref. 17, in which the nonlinear absorption coefficient of 0.064 cm/ GW has been measured at 780 nm for a semi-insulating 6H-SiC crystal.  Tables 2 and 3, and then the contribution for photon emission and absorption is added for the case of acoustic and intervalley deformation potential scattering (see Eqs (15) and (16)).
The plot of Fig. 2(a) reveals that, for D ac = 11 eV, the β np in TPA , coefficient at 780 nm changes from 0.026 cm/GW to 0.15 cm/GW, with D 0 ranging from 2 × 10 11 eV/m to 6 × 10 11 eV/m. Similarly, Fig. 2(b) shows that the TPA coefficient at 780 nm ranges from 0.026 cm/GW to 0.05 cm/GW, if D ac is changed from 11 to 20 eV, with D 0 = 2 × 10 11 eV/m. Moreover, our simulations indicate that the indirect TPA process is mainly influenced by the intervalley potential deformation. Indeed, by considering a 10% change in both D 0 and D ac parameters, we have   Fig. 3 shows the 4H-SiC indirect TPA coefficient spectra for different values of the product |p mv | 2 · |p nm | 2 , and assuming two different combinations between the parameters D 0 and D ac (see Table 3). For the case |p mv | 2 · |p nm | 2 = 6.618 × 10 −100 Kg 2 ·J 2 , the plot shows that changing the wavelength in the range [500-810 nm] the β np in TPA , coefficient ranges from 0.416 cm/GW to 0.045cm/GW, and from 0.151 to 0.017 cm/GW for the parameter set D 0 = 3.7 × 10 11 eV/m, D ac = 21 eV, and D 0 = 2.3 × 10 11 eV/m, D ac = 11.6 eV, respectively. Moreover, the TPA coefficient assumes values of 0.047 cm/GW and 0.0176 cm/GW at 780 nm. These numerical evaluations are consistent with the experimental values of β np in TPA , = 0.064 cm/GW at 780 nm 17 . Therefore, we believe that by    assuming |p mv | 2 = 1.1 × 10 −48 Kg·J and |p nm | 2 = 6.0 × 10 −52 Kg·J as for the cubic silicon carbide, consistent information about the indirect TPA process in the 4H-SiC polytype can be achieved. A comparison of TPA for 4H-SiC, 6H-SiC, and 3C-SiC is shown in Fig. 4, where the indirect TPA spectrum has been simulated assuming the same dipole transition matrix elements (|p mv | 2 = 1.1 × 10 −48 Kg·J and |p nm | 2 = 6.0 × 10 −52 Kg·J) for all three materials. It is interesting to note that the 6H-SiC curve with D 0 = 2.1 × 10 11 eV/m and D ac = 11.2 eV presents β np in TPA , = 0.056 cm/GW at 780 nm, in good agreement with the experimental measurement given in ref. 17, and confirms that the selected parameters can be considered suitable to take into account the electron-phonon interactions in the indirect TPA process. Thus, the plot indicates that the TPA coefficient for 3C-SiC dominates the 6H-SiC one, as a result of the lower value of the indirect energy gap. Therefore, in order to hold the same trend, we guess that 4H-SiC could admit a parameter set such as D 0 = 2.3 × 10 11 eV/m and D ac = 11.6 eV to describe the phonon-assisted nonlinear absorption process.
In Fig. 5 the Kerr refractive index (n 2 ) spectra are sketched for 4H-SiC material and the TE polarization, showing a very good agreement with experimental data 15,33 . In the simulations we have assumed a fitting factor C np = 4.3 × 10 3 (see Eq. [14]). Moreover, the assumption of nonparabolicity for both conduction and valence bands is demonstrated to be critical in order to match the experimental data. Indeed, our simulations indicate a considerable discrepancy between our theoretical n 2 dispersion and the measurements of refs 15 and 33, if the hypothesis of parabolicity or nonparabolicity only for the conduction band is adopted. Some problems that are linked to the parabolicity hypothesis are: (1) a reduction in the position (λ peak ) of the n 2 peak; (2) creation of a smaller tail in the n 2 shape at higher photon energy, and (3) a reduction in the mid-IR asymptotic value. Thus, the matching with experimental data distributed both at "low" and "high" photon energy would thus be definitely compromised. For example, our investigations indicate that λ peak assumes values of 665.7, 590.9, and 796.4 nm for parabolic, nonparabolic conduction band, and nonparabolic conduction and valence bands, respectively. The n 2 average spectrum induced by the direct transitions is also included in Fig. 5 for comparison. Indeed, generally speaking, 4H-SiC could suffer from the two photon absorption induced by direct transitions, because the direct TPA cut-off wavelength (λ d cut = 482.7 nm) is larger than the transparency wavelength (λ T = 395.1 nm). The direct transition-induced n 2 has been calculated according to the formula proposed in our previous work 21 and shows  how the typical peak shape 21 (see also ref. 34) totally disagrees with the experimental data. As a result, we can definitely conclude that the SiC material is dominated by the phonon-assisted two photon absorption.
In Fig. 6 the comparison among Kerr refractive indices for 4H-SiC, 6H-SiC, and 3C-SiC is shown. The plot predicts n 2 values in very good agreement with the data proposed in literature, as summarized in Table 3. Over the 500 to 5000 nm range in Fig. 6, n 2 for three carbides is within the range 0.5 to 6.3 × 10 −18 m 2 /W. It is important to compare this result with that for different materials used in nonlinear photonic applications. An immediate comparison can be made with crystal silicon over the 1500 to 5000 nm wavelength range. We find, by inspecting the red curve-fit-to-data in Fig. 1(a) of ref. 34, that n 2 dispersion for Si is in the range of 0.5 to 5.7 × 10 −18 m 2 /W. Recently, chalcogenide (ChG) glasses have been proposed in order to fabricate photonic integrated circuits (PICs). They offer unique optical properties for nonlinear optics with a strong Kerr nonlinearity with low two photon absorption and negligible free-carrier effects. These properties have been exploited in a review paper 35 , where recent progress in developing ChG PICs for ultrafast optical processing has been reported. At the same time, hydrogenated amorphous silicon (a-Si:H) has attracted a lot of attention as a platform for nonlinear optics, mainly because it has a larger third order nonlinearity, n 2 , compared with other common materials. As outlined in ref. 36, the values for n 2 that have been reported are 4~6 times those of crystalline silicon, 7~13 times those of As 2 S 3 glass, and 3~5 times those of Ge 11.5 As 24 Se 64.5 . However, a drawback of a-Si:H is that, like crystalline Si, it is reported to suffer from two photon absorption as well as TPA-induced free carrier absorption (FCA) and this ultimately will limit the efficiency of nonlinear devices. In this context, diamond, has recently emerged as a possible platform to combine the advantages of a relatively high nonlinear refractive index, and low nonlinear absorption losses within its large transmission window (from UV to mid-IR) 37 . In Table 4 we summarize the Kerr nonlinear refractive index for the platforms above mentioned.
We believe that the physical model presented here gives practical theoretical predictions and a comprehensive physical overview of the silicon carbide nonlinear nonresonant properties over a wide wavelength range, from visible to mid-IR. Additionally, although only the acoustic phonon deformation potential and the intervalley phonon deformation potential have been considered to describe the complex electron-phonon interactions (see 4H-SiC), the model predictions significantly give consistent results when compared with the number of measurements data available on the third order nonlinearity. Of course, a systematic set of experimental measurements would be useful to better select the values of physical parameters such as p mv , |Q cn | 2 which have been numerically estimated in this work.

Conclusions
In this paper, mathematical modeling based on a physical approach has been implemented to investigate the spectrum of two-photon absorption induced by indirect transitions in crystalline silicon carbide having either the cubic or the wurtzite structure. The proposed model has been validated by comparing our predictions with the experimental measurements presented in literature. A group theory analysis has been performed in order to describe the physical features of the phonon-assisted two-photon absorption process. The theoretical investigations have shown that all phonons, except the longitudinal optical phonons, are involved in the process for  the 3C-SiC crystal in which the indirect bandgap is induced by the X valley. Moreover, phonons with M 2 and M 4 symmetry located in the both acoustic and optical branches (12 phonons in total) are involved in the indirect TPA process for 4H-SiC and 6H-SiC in which the indirect bandgap is induced by M valleys. In order to perform numerical simulations, the complexity of the electron-phonon interactions has been reduced by considering only the acoustic phonon deformation potential and the intervalley phonon deformation potential. However, good agreement with experimental measurement has been achieved, demonstrating the consistency of the physical assumptions adopted. Finally, the Kerr refractive index has been calculated as a function of wavelength for 3C, 4H and 6H. As a result, good agreement between our numerical predictions and experimental measurement has been achieved, demonstrating that the 4H-SiC (6H-SiC) material is dominated by the TPA indirect process, although it could suffer from the direct TPA effect, too. From these results, the silicon carbide can be considered as a very good candidate for nonlinear optical applications since it can guarantee a Kerr effect as large as that of silicon, but without any TPA effect in both the NIR and mid-IR spectral regions.

Methods
Numerical estimations of |Q cn | 2 can be obtained by considering only two fundamental scattering mechanisms: acoustic phonon deformation potential scattering (LA phonons), and intervalley phonon deformation potential scattering. In this context, the matrix elements are given by Eqs (15) and (16), respectively: In Eqs (15) and (16), D ac and D 0 represent the effective acoustic and optical deformation potential, respectively. The coefficients ρ, and v s are the SiC density and the acoustic velocity, respectively. The ± signs correspond to phonon absorption and emission processes. Finally, E ph represents the acoustic phonon and the intervalley energy in Eqs (15) and (16), respectively. The relevant numerical values used in our simulations are listed in Table 5.
It is worth noting that Eqs (15) and (16) do not describe the exact electron-phonon interaction (especially for the case of 4H-SiC, where complex phonon dispersion occurs), but they still provide useful information about the order of magnitude of |Q cn | 2 .