B. Jensen and A. Torabi, "Temperature and intensity dependence of the refractive index of a compound semiconductor," J. Opt. Soc. Am. B 2, 1395-1401 (1985)

An investigation of the effect of high-intensity radiation at frequencies below the fundamental absorption edge on the refractive index of a compound semiconductor must include consideration of effects due to generated carriers and effects due to heating. The latter include the change of band-gap energy with temperature and thermal expansion, both of which change the refractive index. A calculation of the refractive index of a semiconductor with the band structure of the Kane theory is given for the case of arbitrary spin–orbit splitting energy. Theoretical results are expressed in terms of the following experimental parameters: band-gap energy, effective electron mass, effective hole masses of the three valence bands, the spin–orbit splitting energy, and the lattice constant. An expression for the thermo-optic coefficient dn/dT is given that makes possible the numerical evaluation of the refractive index as a function of temperature. The nonlinear intensity-dependent refractive index is calculated, assuming that the change in carrier concentration is generated by an applied radiation field of high intensity, and theoretical results are compared with experiment for InAs and InSb.

Mario Bertolotti, Victor Bogdanov, Aldo Ferrari, Andrei Jascow, Natalia Nazorova, Alexander Pikhtin, and Luigi Schirone J. Opt. Soc. Am. B 7(6) 918-922 (1990)

N. Peyghambarian and H. M. Gibbs J. Opt. Soc. Am. B 2(7) 1215-1227 (1985)

References

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Experimental Parameters Involved in the Calculation of dn/dT (K^{−1}) at T = 300 K12,13

Semiconductor

G (eV)

dG/dT (eV/K)

a (Å)

α_{T} (K^{−1})

III–V

InP

1.35

−4.6 × 10^{−4}

5.869

4.5 × 10^{−6}

GaAs

1.43

−5.0 × 10^{−4}

5.653

5.5 × 10^{−6}

II-VI

CdTe

1.50

−4.1 × 10^{−4}

6.477

–

ZnSe

2.58

−7.2 × 10^{−4}

5.667

7.7 × 10^{−6}

ZnTe

2.28

−5.0 × 10^{−4}

6.101

–

Table 2

Theoretical Quantities That Are Functions of Experimental Parameters Involved in the Calculation of dn/dT, Where −dn(0,0)/dT = [mac_{o}/n(0, 0)] [α_{T} − α_{T}^{o}(0, 0)] (K^{−1})

Estimated using the value of α_{T} for ZnSe listed in Table 1.

Table 3

Theoretical Values of dn(0,0)/dT = [mac_{o}/(1 + 2c_{o}y_{B})^{1/2}][α_{T}^{o}(0,0) − α_{T}] for the III–V and II–VI Compounds and Experimental Values of α_{T} and dn/dT^{a}

For the covalently bonded group IV semiconductors, dn/dT is positive and large compared with α_{T}, whereas for the ionically bonded I–VII alkali halides, dn/dT is negative and of the same order of magnitude as α_{T}. The group IV semiconductors and the alkali halides exhibit behavior predicted by the theoretical expression in the limits α_{T}^{o}(0,0) ≫ α_{T} and α_{T}^{o}(0,0) ≪ α_{T}, respectively.

Table 4

A Comparison of Values of
$$\begin{array}{l}{\frac{\text{d}n(z,0)}{\text{d}T}|}_{z}=\frac{M(z)}{n(z,0)}[{{\alpha}_{T}}^{o}(z,0)-{\alpha}_{T}]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[\begin{array}{l}z={(1-x)}^{1/2}\hfill \\ x=\hslash w/G\hfill \\ x\mathrm{\lambda}=0.4805\hspace{0.17em}\mu \text{m}\hfill \end{array}\\ n(z,0)={\{1+2{c}_{o}[{y}_{B}-z\hspace{0.17em}{\text{tan}}^{-1}({y}_{B}/z)]\}}^{1/2}\end{array}$$ Calculated from Eqs. (34)–(38) for ZnSe at T = 300 K with the Recommended Values for These Quantities from Reference 14 at T = 293 K^{a}

The recommended values are based on a survey of experimental data. The experimental parameters for ZnSe listed in Table 1 have been used in the calculation. A pure material (y_{F} = 0) is assumed.
Theory.
Recommended experimental value from Ref. 14.

Table 5

The Experimental Parameters of InAs and Theoretical and Experimental Values of Nonlinear Index of Refraction n_{2} (cm^{2}/W) as a Function of Temperature

The Experimental Parameters of InSb and Theoretical and Experimental Values of Nonlinear Index of Refraction n_{2} (cm^{2}/W) as a Function of Photon Energy

Experimental Parameters Involved in the Calculation of dn/dT (K^{−1}) at T = 300 K12,13

Semiconductor

G (eV)

dG/dT (eV/K)

a (Å)

α_{T} (K^{−1})

III–V

InP

1.35

−4.6 × 10^{−4}

5.869

4.5 × 10^{−6}

GaAs

1.43

−5.0 × 10^{−4}

5.653

5.5 × 10^{−6}

II-VI

CdTe

1.50

−4.1 × 10^{−4}

6.477

–

ZnSe

2.58

−7.2 × 10^{−4}

5.667

7.7 × 10^{−6}

ZnTe

2.28

−5.0 × 10^{−4}

6.101

–

Table 2

Theoretical Quantities That Are Functions of Experimental Parameters Involved in the Calculation of dn/dT, Where −dn(0,0)/dT = [mac_{o}/n(0, 0)] [α_{T} − α_{T}^{o}(0, 0)] (K^{−1})

Estimated using the value of α_{T} for ZnSe listed in Table 1.

Table 3

Theoretical Values of dn(0,0)/dT = [mac_{o}/(1 + 2c_{o}y_{B})^{1/2}][α_{T}^{o}(0,0) − α_{T}] for the III–V and II–VI Compounds and Experimental Values of α_{T} and dn/dT^{a}

For the covalently bonded group IV semiconductors, dn/dT is positive and large compared with α_{T}, whereas for the ionically bonded I–VII alkali halides, dn/dT is negative and of the same order of magnitude as α_{T}. The group IV semiconductors and the alkali halides exhibit behavior predicted by the theoretical expression in the limits α_{T}^{o}(0,0) ≫ α_{T} and α_{T}^{o}(0,0) ≪ α_{T}, respectively.

Table 4

A Comparison of Values of
$$\begin{array}{l}{\frac{\text{d}n(z,0)}{\text{d}T}|}_{z}=\frac{M(z)}{n(z,0)}[{{\alpha}_{T}}^{o}(z,0)-{\alpha}_{T}]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}[\begin{array}{l}z={(1-x)}^{1/2}\hfill \\ x=\hslash w/G\hfill \\ x\mathrm{\lambda}=0.4805\hspace{0.17em}\mu \text{m}\hfill \end{array}\\ n(z,0)={\{1+2{c}_{o}[{y}_{B}-z\hspace{0.17em}{\text{tan}}^{-1}({y}_{B}/z)]\}}^{1/2}\end{array}$$ Calculated from Eqs. (34)–(38) for ZnSe at T = 300 K with the Recommended Values for These Quantities from Reference 14 at T = 293 K^{a}

The recommended values are based on a survey of experimental data. The experimental parameters for ZnSe listed in Table 1 have been used in the calculation. A pure material (y_{F} = 0) is assumed.
Theory.
Recommended experimental value from Ref. 14.

Table 5

The Experimental Parameters of InAs and Theoretical and Experimental Values of Nonlinear Index of Refraction n_{2} (cm^{2}/W) as a Function of Temperature

The Experimental Parameters of InSb and Theoretical and Experimental Values of Nonlinear Index of Refraction n_{2} (cm^{2}/W) as a Function of Photon Energy