Abstract
Using a combination of semiconductor theory and experimental results from the scientific literature, we have compiled and plotted the key third-order nonlinear optical coefficients of bulk crystalline Si and Ge as a function of wavelength (1.5−6.7 μm for Si and 2–14.7 μm for Ge). The real part of third-order nonlinear dielectric susceptibility (χ ^{(3)}′), the two-photon absorption coefficient (β _{TPA}), and the Raman gain coefficient (g_{R} ), have been investigated. Theoretical predictions were used to curve-fit the experimental data. For a spectral range in which no experimental data exists, we estimate and fill in the missing knowledge. Generally, these coefficient-values appear quite useful for a host of device applications, both Si and Ge offer large χ ^{(3)}′ and g_{R} with Ge offering the stronger nonlinearity. In addition, we use the same theory to predict the third-order nonlinear optical coefficients of Si_{1−} _{x} Ge _{x} alloy. By alloying Si and Ge, device designers can gain flexibility in tuning desired optical coefficients in between the two fundamental components based upon their application requirements.
This work was supported in part by the Air Force Office of Scientific Research, Dr. Gernot Pomrenke, Program Manager, and by DARPA, Dr. Jag Shah, Program Manager.
I. INTRODUCTION
II. REAL PART OF THE THIRD-ORDER NONLINEAR SUSCEPTIBILITY
III. TWO-PHOTON ABSORPTION COEFFICIENT β _{TPA}
IV. RAMAN GAIN COEFFICIENT g_{R}
V. CONCLUSIONS
Key Topics
- Germanium
- 91.0
- Multiphoton processes
- 63.0
- Band gap
- 24.0
- Elemental semiconductors
- 20.0
- Band models
- 9.0
Figures
(Color online) (a) Experimental of Si in the literature as a function of wavelength. Data points are determined from Refs. 14–21 (circles), Ref. 22 (upward triangles), Ref. 23 (downward triangles), and Ref. 27 (diamonds). Theoretical dispersion curves discussed here are fitted to Ref. 22 (Bristow et al.) and to Ref. 23 (Lin et al.). Inset lists the and n_{2} at 1.55 μm (telecommunication wavelength), 2.2 μm (TPA cutoff), 3.39 μm (HeNe laser), and 4.26 μm (a CO_{2} absorption band). A resulting average dispersion curve is shown which act as a reference for designers dealing with Si (b) Experimental of Ge in the literature as a function of wavelength. Ge data points are determined from Refs. 30–33 (circles) and Ref. 34 (diamonds). Two theoretical dispersive curves are used to set the upper and lower bound of the experimental data points. A resulting average dispersion curve is shown which act as a reference for designers dealing with Ge. Inset lists the and n_{2} at 3.17 μm (TPA cutoff), 3.39 μm (HeNe laser), and 4.26 μm (a CO_{2} absorption band).
(Color online) (a) Experimental of Si in the literature as a function of wavelength. Data points are determined from Refs. 14–21 (circles), Ref. 22 (upward triangles), Ref. 23 (downward triangles), and Ref. 27 (diamonds). Theoretical dispersion curves discussed here are fitted to Ref. 22 (Bristow et al.) and to Ref. 23 (Lin et al.). Inset lists the and n_{2} at 1.55 μm (telecommunication wavelength), 2.2 μm (TPA cutoff), 3.39 μm (HeNe laser), and 4.26 μm (a CO_{2} absorption band). A resulting average dispersion curve is shown which act as a reference for designers dealing with Si (b) Experimental of Ge in the literature as a function of wavelength. Ge data points are determined from Refs. 30–33 (circles) and Ref. 34 (diamonds). Two theoretical dispersive curves are used to set the upper and lower bound of the experimental data points. A resulting average dispersion curve is shown which act as a reference for designers dealing with Ge. Inset lists the and n_{2} at 3.17 μm (TPA cutoff), 3.39 μm (HeNe laser), and 4.26 μm (a CO_{2} absorption band).
(Color online) (a) Calculated bandgap energy of E_{Γ} _{Δ} , E_{Γ} _{L} , and E_{g} of Si_{1−} _{x} Ge _{x} alloy with respect to the atomic fraction of Ge. (b) Calculated of Si_{1−} _{x} Ge _{x} alloy at the 6 μm wavelength with respect to the atomic fraction of Ge using the direct and indirect bandgap models.
(Color online) (a) Calculated bandgap energy of E_{Γ} _{Δ} , E_{Γ} _{L} , and E_{g} of Si_{1−} _{x} Ge _{x} alloy with respect to the atomic fraction of Ge. (b) Calculated of Si_{1−} _{x} Ge _{x} alloy at the 6 μm wavelength with respect to the atomic fraction of Ge using the direct and indirect bandgap models.
(Color online) Theoretical dispersion curves of Si_{1−} _{x} Ge _{x} alloy with x = 0.2, 0.4, 0.8, and 0.9.
(Color online) Theoretical dispersion curves of Si_{1−} _{x} Ge _{x} alloy with x = 0.2, 0.4, 0.8, and 0.9.
(Color online) (a) Experimental β _{TPA} of Si in the literature as a function of wavelength. Data points are determined from Refs. 14–16, 20, 30, 40, and 41 (circles), Ref. 22 (upward triangles), Ref. 23 (downward triangles), Ref. 27 (diamonds), and Ref. 42 (squares). One theoretical dispersion curve is fitted to Ref. 22 (Bristow et al.) and another to Ref. 23 (Lin et al.). (b) Experimental β _{TPA} of Ge in the literature as a function of wavelength. Data points are determined from Refs. 44 (diamond), 45 (triangles), and 47 (circles). A theoretical dispersion curve is fitted to the data points.
(Color online) (a) Experimental β _{TPA} of Si in the literature as a function of wavelength. Data points are determined from Refs. 14–16, 20, 30, 40, and 41 (circles), Ref. 22 (upward triangles), Ref. 23 (downward triangles), Ref. 27 (diamonds), and Ref. 42 (squares). One theoretical dispersion curve is fitted to Ref. 22 (Bristow et al.) and another to Ref. 23 (Lin et al.). (b) Experimental β _{TPA} of Ge in the literature as a function of wavelength. Data points are determined from Refs. 44 (diamond), 45 (triangles), and 47 (circles). A theoretical dispersion curve is fitted to the data points.
(Color online) Theoretical β _{TPA} dispersion curves of Si_{1−} _{x} Ge _{x} alloy with (a) x = 0.2, 0.4, and 0.6 using the indirect bandgap model and (b) x = 0.8, 0.9, 0.95, and 1 using the direct bandgap model.
(Color online) Theoretical β _{TPA} dispersion curves of Si_{1−} _{x} Ge _{x} alloy with (a) x = 0.2, 0.4, and 0.6 using the indirect bandgap model and (b) x = 0.8, 0.9, 0.95, and 1 using the direct bandgap model.
(Color online). Calculated TPA cut-off wavelength of Si_{1−} _{x} Ge _{x} alloy with respect to the atomic fraction of Ge.
(Color online). Calculated TPA cut-off wavelength of Si_{1−} _{x} Ge _{x} alloy with respect to the atomic fraction of Ge.
(Color online) (a) Experimental g_{R} of Si in literature as a function of the Stoke’s wavelength. Data points are determined from Refs. 39 and 50–55 (circles) and Ref. 10 (diamonds). A theoretical dispersion curve is fitted to the data points for Si. An estimated dispersion curve for Ge is shown. (b) The estimated g_{R} of Si_{1−} _{x} Ge _{x} alloy at the 6 μm Stoke’s wavelength with respect to the atomic fraction of Ge.
(Color online) (a) Experimental g_{R} of Si in literature as a function of the Stoke’s wavelength. Data points are determined from Refs. 39 and 50–55 (circles) and Ref. 10 (diamonds). A theoretical dispersion curve is fitted to the data points for Si. An estimated dispersion curve for Ge is shown. (b) The estimated g_{R} of Si_{1−} _{x} Ge _{x} alloy at the 6 μm Stoke’s wavelength with respect to the atomic fraction of Ge.
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