Fitting the refractive indices of GaN at different conditions with MATLAB codes for optical simulations

Abstract

This mixed study presents the optical properties of gallium nitride material at different conditions and thicknesses over a very wide wavelength range for optical simulations. The availability of optical properties of gallium nitride over a very wide wavelength range facilitates the use of gallium nitride in different simulated optical devices such as laser diodes, LEDs, and other optical and electronic devices. Fitted equations, plotted figures, and MATLAB codes for refractive indices will be presented to reduce mistakes within simulations.

Introduction

The inherent benefits of optical and filters simulated sensors, including their tiny size, low weight, robustness, and resilience against electromagnetic interference, have attracted much attention^{1,2,3,4,5,6,7,8,9}. These optical simulations may save effort, money, and time by studying the optimum conditions and expected results for experimentalists. The refractive indices (RIs) of materials is the most important parametar for optical simulations^{10,11,12,13,14,15,16,17,18,19,20,21,22}. Providing a database for material RIs is crucial for enabling accurate simulations across various fields, such as optics, engineering design, and material sciences. A database containing the RIs of different materials helps in determining their optical properties, saving time and effort, improving simulation accuracy, and supporting research and studies.

The III-nitride compound semiconductors have been the subject of substantial research due to their enormous bulk moduli, RI, broadband gap, and high thermal conductivity²³. GaN can be used to prepare thin films, usually deposited by different techniques such as sputtering, molecular beam epitaxy (MBE)²⁴, or metalorganic vapor phase epitaxy (MOVPE). GaN with high-quality can be fabricated using hydride vapor phase epitaxy (HVPE)²⁵. GaN is a semiconductor compound that has gained immense popularity in recent years due to the improvements in its efficiency and longevity across various light-emitting diodes (LED)²⁶, laser diodes (LD)²⁷, and other optical and electronic devices^14,28. GaN’s 3.4 eV wide band gap provides it unique characteristic for optoelectronic, high-power applications²⁷. GaN exhibits a substantial transparency zone due to its wide bandgap encompassing the visible, near-infrared, and mid-infrared spectrums²⁹. GaN possesses numerous unique properties, including easily controllable carrier concentration, modifiable electrical conductivity, bandgap tunability, high carrier mobility, non-toxicity, and high thermal/chemical stability, making it an attractive functional film for photonic and electronic applications^{30,31,32,33,34,35,36}. $\:\text{G}\text{a}\text{N}$ can be stabilized with sapphire ($\:{Al}_{2}{O}_{3}$)^29,37,38,39, $\:AlN$²⁹, $\:{Al}_{0.4}{Ga}_{0.6}N$²⁹, GaAs⁴⁰, Si⁴¹, $\:{SiO}_{2}$⁴², SiC⁴², $\:{Ga}_{2}{O}_{3}$⁴³, etc.

Zincblende and wurtzite are two different crystal structures adopted by GaN. The lattice structures of wurtzite and zincblende GaN are hexagonal close-packed and cubic lattice structures, respectively. Wurtzite GaN is the most common and commonly used in optical devices. Though less frequent than wurtzite GaN, zincblende GaN has been investigated for possible uses in specific electronic devices. The distinct characteristics and behaviors of zincblende and wurtzite GaN may impact the functionality of GaN-fabricated devices. The particular needs of the gadget being produced determine the crystal structure to be used.

The most essential parameters in optical simulations are the extinction coefficient (EC) and index of refraction (RI). The RI of GaN may also be classified into ordinary and extraordinary RIs. Ordinary RI quantifies the degree to which light is bent through a material. The extraordinary RI quantifies the degree to which light is bent through a material in a specific direction. The EC of a material represents the absorption coefficient (α) of this material, where $\:{\upalpha\:}\left({\uplambda\:}\right)=4{\uppi\:}\text{k}\left({\uplambda}\right)/{\uplambda\:}$⁴⁴.

Porous GaN (P_GAN) is a sponge-like structure that has intentional pores. These intentional pores can be introduced into GaN with different techniques, such as photoelectrochemical or electrochemical etching, which introduce voids or pores. The fabrication procedure setup can cause variations in these voids’ shape, size, and distribution. P_GAN has been effectively employed to construct various applications because of its unique physical features, which include expanded surface area ratio mechanical, chemical, and thermal stability^45,46,47,48. The scattering and trapping of light within the P_GAN can improve the emission of light in LED devices^49,50,51.

Hayilesilassie et al.⁵² introduced defective $\:\text{Z}\text{r}{O}_{2}/GaN$ periodic structure to Gram-negative and negative bloodstream bacteria. Despite their calculations (λ in figures) are in nm, they wrote the RI formula of $\:\text{Z}\text{r}{O}_{2}$ as a function of $\:\lambda\:$ in µm (in calculations) without clarity on this for the readers in the text. Aly et al.⁵³ theoretically proposed a poliovirus sensor composed of AlN/GaN-defected photonic crystal. Despite the RI formula of GaN is a function of wavelength in µm (the value of the wavelength should be given to the MATLAB code in µm), they made their calculations in nm (they used the value of the wavelength in MATLAB code in nm). Besides, many experimental studies presented their RIs results as figures only⁵⁴. If these results are fitted to simple equations, it may help in different theoretical applications.

In this study, the RI of GaN will be fitted and studied at different conditions, such as temperature, pressure, and porosity. Fitting and grouping the rRIs of GaN over a wide range of wavelengths in the same paper may eliminate the mistakes in simulation studies. Presenting the fitted equation, MATLAB code (Supplementary Material), and the plotted curve with initial conditions helps researchers (maybe beginners) verify their codes and results step by step. We fitted the wavelength in all the equations in this manuscript to be microns. This connection between experimental and fitted RIs facilitates the fabrication of theoretical studies.

Basic equations, fitting, and discussion

Lin et al.⁴⁰ experimentally measured RIs of both wurtzite and zincblende$\:\:\text{G}\text{a}\text{N}$ with a thickness of about 6 nm over the 0.370–0.990 μm wavelength range ($\:\lambda\:$ in µm). We fitted the experiment data using the Sellmeier formula as follows:

For wurtzite $\:\text{G}\text{a}\text{N}$:

$$\:{n}^{2}=0.97344\:+\frac{\:3.80128{\lambda}^{2}}{{\lambda}^{2}-{0.07704}^{2}}+\frac{\:0.59917{\lambda}^{2}}{{\lambda}^{2}-{0.32445}^{2}}-\frac{\:3\times\:{10}^{10}{\lambda}^{2}}{{\lambda}^{2}-{(5\times{10}^{10})}^{2}}.$$

(1)

For zincblende $\:\text{G}\text{a}\text{N}$:

$$\:{n}^{2}=1.14365\:+\frac{\:3.50504{\lambda}^{2}}{{\lambda}^{2}-{(1.86\times\:{10}^{-5})}^{2}}+\frac{\:0.98235{\lambda}^{2}}{{\lambda}^{2}-{0.33219}^{2}}-\frac{\:5122.61849{\lambda}^{2}}{{\lambda}^{2}-{219.84241}^{2}}.$$

(2)

Figure 1 clearly shows the RI of wurtzite and zincblende $\:\text{G}\text{a}\text{N}$ as a function of wavelength according to the experimental measurements in Ref⁴⁰. and our fitted equations (Eqs. (1) and (2)).

Fig. 1

Experimental⁴⁰ and our fitted RI of wurtzite and zincblende $\:\text{G}\text{a}\text{N}$ versus the wavelength.

Watanabe et al.³⁸ investigated the RIs of $\:\text{G}\text{a}\text{N}\:$of thicknesses 5.18 μm and 10.6 μm at different temperatures (21 °C, 251 °C, and 515 °C) over the 0.367 –1.000 μm wavelength range ($\:\lambda\:$ in µm). The experimental data are fitted using the Sellmeier formula as follows:

At 21 °C:

$$\:{n}^{2}=1.11473+\frac{\:3.09869{\lambda}^{2}}{{\lambda}^{2}-{(3.2\times\:{10}^{-7})}^{2}}+\frac{\:1.035857{\lambda}^{2}}{{\lambda}^{2}-{0.298027}^{2}}-\frac{\:168.522387{\lambda}^{2}}{{\lambda}^{2}-{(-271.59957)}^{2}}.$$

(3)

At 251 °C:

$$\:{n}^{2}=0.94653+\frac{\:3.44998{\lambda}^{2}}{{\lambda}^{2}-{(4.57\times\:{10}^{-6})}^{2}}+\frac{\:0.93621{\lambda}^{2}}{{\lambda}^{2}-{0.309348}^{2}}+\frac{\:613.44183{\lambda}^{2}}{{\lambda}^{2}-{(-124.51323)}^{2}}.$$

(4)

At 515 °C:

$$\:{n}^{2}=1.17483+\frac{\:3.14409{\lambda}^{2}}{{\lambda}^{2}-{(7.06\times\:{10}^{-6})}^{2}}+\frac{\:1.05661{\lambda}^{2}}{{\lambda}^{2}-{0.31586}^{2}}-\frac{\:42.15556{\lambda}^{2}}{{\lambda}^{2}-{(-63.00241)}^{2}}.$$

(5)

The experimental³⁸ and our fitted RIs of $\:\text{G}\text{a}\text{N}$ at different temperatures (21 °C, 251 °C, and 515 °C) over the 0.367 –1.000 μm are plotted in Fig. 2.

Fig. 2

Experimental³⁸ and our fitted (Eqs. (3), (4), and (5)) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength.

Besides, Watanabe et al.³⁸ obtained a fitted equation for thermo-optic coefficients ($\:\frac{\partial\:n}{\partial\:T}$ in $\:{K}^{-1}$) of $\:\text{G}\text{a}\text{N}$ for temperature ranges from 21 °C to 515 °C as follows (Fig. 3):

$$\:\frac{\partial\:n}{\partial\:T}=4.247\times\:{10}^{-5}{\lambda}^{-3}-1.592\times\:{10}^{-4}{\lambda}^{-2}+2.187\times\:{10}^{-4}{\lambda}^{-1}-3.427\times\:{10}^{-5}.$$

(6)

Fig. 3

Thermo-optic coefficient of $\:\text{G}\text{a}\text{N}$ versus the wavelength according to Eq. (6)³⁸.

The RI of $\:\text{G}\text{a}\text{N}$ versus wavelength at different temperatures can be investigated as a function of the thermo-optic coefficient using the following equation:

$$\:n\left(T\right)={n}_{0}+\frac{\partial\:n}{\partial\:T}(T-21)$$

(7)

where $\:{n}_{0}$ is the RI at room temperature ($\:21$), and T is the temperature in °C. Figure 4 shows the match between the RI using Eq. (7) and the measured data. Near the bandgap of GaN (below 0.420 μm), the fitted data slightly deflects from the measured data. As clear in Fig. 4, Eq. (7) can represent the RI of GaN as a function of temperature for a wide range of wavelengths (from 0.367 μm to 1.00 μm).

Fig. 4

Experimental³⁸ and fitted (using Eq. 7) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength for temperature ranges from 21 °C to 515 °C.

In 2008, Pezzagna et al.²⁹ experimentally studied the RIs of ordinary and extraordinary GaN from wavelength 0.458 μm to 1.550 μm. The thickness of the measured film was 1.9 μm. The deduced experimental and fitted indices of refraction are presented in Fig. 5. According to Sellmeier formula²⁹, the ordinary ($\:{n}_{o}$) and extraordinary ($\:{n}_{e}$) RIs of GaN can be simulated (λ can be in nm or µm without any change in the equation) as follows.

For ordinary ($\:{n}_{o}$):

$$\:{n}_{o}^{2}=1+\frac{\:0.213{\times\:10}^{6}{\lambda}^{2}}{{{10}^{6}\times\:\lambda}^{2}-{350}^{2}}+\frac{\:3.988{\times\:10}^{6}{\lambda}^{2}}{{{10}^{6}\times\:\lambda}^{2}-{153}^{2}}$$

(8)

For extraordinary ($\:{n}_{e}$):

$$\:{n}_{e}^{2}=1+\frac{\:0.118{\times\:10}^{6}{\lambda}^{2}}{{{10}^{6}\times\:\lambda}^{2}-{350}^{2}}+\frac{\:4.201{\times\:10}^{6}{\lambda}^{2}}{{10}^{6}\times\:{\lambda}^{2}-{176.5}^{2}}$$

(9)

Fig. 5

Experimental and fitted (using Eqs. (8) and (9)) RIs of ordinary and extraordinary $\:\text{G}\text{a}\text{N}$ versus the wavelength²⁹.

Chowdhury et al.⁵⁵ measured the RIs of $\:\text{G}\text{a}\text{N}\:$of thickness 17.2 μm from wavelength 0.5 μm to 5.0 μm ($\:\lambda\:$ in µm). The experimental data are fitted using the Sellmeier formula as follows:

For ordinary ($\:{n}_{o}$):

$$\:{n}_{o}^{2}=0.637292+\frac{\:1.774337{\lambda}^{2}}{{\lambda}^{2}-{0.000267}^{2}}+\frac{\:2.762007{\lambda}^{2}}{{\lambda}^{2}-{0.213716}^{2}}+\frac{\:17.168186{\lambda}^{2}}{{\lambda}^{2}-{32.953550}^{2}}.$$

(10)

For extraordinary ($\:{n}_{e}$):

$$\:{n}_{e}^{2}=1.151455+\frac{\:3.125622{\lambda}^{2}}{{\lambda}^{2}-{(-4.551667\times\:{10}^{-6})}^{2}}+\frac{\:1.041621{\lambda}^{2}}{{\lambda}^{2}-{0.343057}^{2}}+\frac{\:16.637483{\lambda}^{2}}{{\lambda}^{2}-{25.507523}^{2}}.$$

(11)

The deduced experimental and fitted indices of refraction are presented in Fig. 6.

Fig. 6

Experimental⁵⁵ and our fitted (using Eqs. (10) and (11)) RIs of ordinary and extraordinary $\:\text{G}\text{a}\text{N}$ versus the wavelength.

In 2019, Lee et al.⁵⁶ experimentally measured the indices of refraction of nanoporous GaN from wavelength 0.40 μm to 1.00 μm. The thickness of the measured film was about 2.0 μm of n-type GaN. The deduced experimental and fitted indices of refraction are presented in Fig. 7. According to Bruggeman’s effective medium approximation (BEMA)^{57,58,59,60,61}, indices of refraction of nanoporous GaN can be simulated as a function of pores ratio as follows ($\:\lambda\:$ in µm):

$$\:{n}^{2}=1.11414+\frac{\:3.08320{\lambda}^{2}}{{\lambda}^{2}-{0.19165}^{2}}+\frac{\:1.051830{\lambda}^{2}}{{\lambda}^{2}-{0.188330}^{2}}+\frac{\:1166555.11758{\lambda}^{2}}{{\lambda}^{2}-{707395113.05074}^{2}}$$

(12)

$$\:{\text{n}}_{\text{P}\text{G}\text{a}\text{N}}^{\:}\:=0.5\sqrt{\mathcal{B}+\sqrt{{\mathcal{B}}^{2}+8\:{\text{n}}_{\text{G}\text{a}\text{N}}^{2}\:{\text{n}}_{0}^{2}}},\:\mathcal{B}=3\:\text{P}\:\left({\text{n}}_{0}^{2}-{\text{n}}_{\text{G}\text{a}\text{N}}^{2}\right)+\left(2\:{\text{n}}_{\text{G}\text{a}\text{N}}^{2}-{\text{n}}_{0}^{2}\right),$$

(13)

Fig. 7

Experimental⁵⁶ and fitted RIs (using BEMA) of $\:\text{G}\text{a}\text{N}$ versus the wavelength for porosities range from 0–38.7%.

where $\:{\text{n}}_{0}^{\:}$ is the pores’ RI.

In 2018, Banerjee et al.⁶² measured the RI of polycrystalline GaN from wavelength 0.35 μm to 1.70 μm. Poly GaN has attracted more attention in electron and optical devices^63,64. The experimental and fitted indices of refraction of Poly GaN are presented in Fig. 8. The following new fitted equation can represent the RI of Poly GaN over the wavelength of concern:

$$\:{n}^{2}=0.21186+\frac{\:3.18946{\lambda}^{2}}{{\lambda}^{2}-{0.19289}^{2}}+\frac{\:1.26101{\lambda}^{2}}{{\lambda}^{2}-{0.19292}^{2}}+\frac{\:-985.55408{\lambda}^{2}}{{\lambda}^{2}-{(-155.25917)}^{2}}.$$

(14)

Fig. 8

Experimental⁶² and fitted RIs of Poly $\:\text{G}\text{a}\text{N}$ versus the wavelength.

The RI of epitaxial GaN thin film was measured from the reflectance spectra by El-Naggar et al.⁴¹ from wavelength 0.36 μm to 1.80 μm at an ambient temperature of 24 °C. The thickness of the film is about 608 nm ± 7 nm on Si (111) substrate. The measured and fitted RIs of epitaxial GaN are presented in Fig. 9. We fitted an equation to calculate the RI of epitaxial GaN over wavelength range 0.36 μm to 1.80 μm at an ambient temperature of 24 °C as follows ($\:\lambda\:$ in µm):

$$\:{n}^{2}=0.96222+\frac{\:3.05606{\lambda}^{2}}{{\lambda}^{2}-{\left(8.74345\times\:{10}^{-7}\right)}^{2}}+\frac{\:0.96866{\lambda}^{2}}{{\lambda}^{2}-{0.31817}^{2}}+\frac{\:-9.00019{\lambda}^{2}}{{\lambda}^{2}-{\left(-19.01746\right)}^{2}}.$$

(15)

Fig. 9

Measured⁴¹ and fitted RIs of epitaxial $\:\text{G}\text{a}\text{N}$ versus the wavelength.

El-Nahass et al.⁶⁵ fabricated a nanocrystalline GaN (stable phase of wurtzite structure) using metal-organic vapor phase epitaxy (MOVPE). The RI was measured from wavelength 0.2 –2.5 μm. The measured absorption index of wurtzite GaN over this wavelength range was very small and can be neglected (lower than 0.05). The experimental data over the 0.372 –2.5 μm wavelength range ($\:\lambda\:$ in µm) are fitted using the Sellmeier formula as follows (Fig. 10):

$$\:{n}^{2}={A}_{1}+\frac{\:{A}_{2}{\lambda}^{2}}{{\lambda}^{2}-{{A}_{3}}^{2}}+\frac{\:{A}_{4}{\lambda}^{2}}{{\lambda}^{2}-{{A}_{5}}^{2}}+\frac{\:{A}_{6}{\lambda}^{2}}{{\lambda}^{2}-{{A}_{7}}^{2}},$$

(16)

Fig. 10

Experimental⁶⁵ and fitted RIs of epitaxial $\:\text{G}\text{a}\text{N}$ versus the wavelength.

where $\:{A}_{1}=2.637\:+\:0.004\text{i}$; $\:{A}_{2}=-5.842\:+\:0.007\text{i}$; $\:{A}_{3}=0.004\:+\:2.176\text{i}$; $\:{A}_{4}=5.370\:-\:0.002\text{i}$; $\:{A}_{5}=0.327\:+\:1.314\times\:{10}^{-5}\text{i}$; $\:{A}_{6}=54264698.942\:-\:8537608.451\text{i}$; $\:{A}_{7}=5052995.106\:-\:829099.109\text{i}$.

In 2012, Watanabe et al.⁶⁶ investigated the RIs of $\:\text{G}\text{a}\text{N}\:$of thicknesses 5.18 μm at different temperatures (24 °C, 253 °C, and 506 °C) on (0001) oriented sapphire substrate over 0.370 –1.700 μm wavelength range ($\:\lambda\:$ in µm). We fitted the experimental data using the Sellmeier formula over the 0.390 –1.700 μm wavelength range ($\:\lambda\:$ in µm) as follows:

At T = 24 °C:

$$\:{n}^{2}=1.05392+\frac{\:3.14103{\lambda}^{2}}{{\lambda}^{2}-{0.06928}^{2}}+\frac{\:1.03126{\lambda}^{2}}{{\lambda}^{2}-{0.29534}^{2}}+\frac{\:70768530{\lambda}^{2}}{{\lambda}^{2}-{108407625}^{2}},$$

(17)

At T = 253 °C:

$$\:{n}^{2}=-1.28432+\frac{\:0.53946{\lambda}^{2}}{{\lambda}^{2}-{0.32822}^{2}}+\frac{\:6.01815{\lambda}^{2}}{{\lambda}^{2}-{0.10157}^{2}}+\frac{\:403348125{\lambda}^{2}}{{\lambda}^{2}-{(-623035599)}^{2}},$$

(18)

At T = 506 °C:

$$\:{n}^{2}=1.13886+\frac{\:3.21301{\lambda}^{2}}{{\lambda}^{2}-{(-7.41927\times\:{10}^{-6})}^{2}}+\frac{\:1.05198{\lambda}^{2}}{{\lambda}^{2}-{0.31620}^{2}}+\frac{\:3349{\lambda}^{2}}{{\lambda}^{2}-{330}^{2}}.$$

(19)

The experimental⁶⁶ and our fitted RIs of $\:\text{G}\text{a}\text{N}$ at different temperatures (24 °C, 253 °C, and 506 °C) over the 0.390 –1.700 μm are plotted in Fig. 11.

Fig. 11

Experimental³⁸ and our fitted (Eqs. (17), (18), and (19)) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength.

Besides, Watanabe et al.⁶⁶ experimentally measured the thermo-optic coefficients ($\:\frac{\partial\:n}{\partial\:T}$ in $\:{K}^{-1}$) of $\:\text{G}\text{a}\text{N}$ for temperature ranges from 24 °C to 506 °C. We fitted the thermo-optic coefficients ($\:\frac{\partial\:n}{\partial\:T}$ in $\:{K}^{-1}$) over 0.390 –1.700 μm wavelength range ($\:\lambda\:$ in µm)as follows (Fig. 12):

$$\:\frac{\partial\:n}{\partial\:T}=-\:0.00530901{\lambda}^{7}+0.041193{\lambda}^{6}-0.13372{\lambda}^{5}+0.235114{\lambda}^{4}-0.241649{\lambda}^{3}+0.145215{\lambda}^{2}-0.0473364\lambda+0.00655608.$$

(20)

Fig. 12

Thermo-optic coefficient of $\:\text{G}\text{a}\text{N}$ versus the wavelength according to Eq. (20).

The RI of $\:\text{G}\text{a}\text{N}$ versus wavelength at different temperatures from 24 to 506 °C can be investigated as a function of the thermo-optic coefficient over 0.390–1.700 μm wavelength range using the following equation:

$$\:n\left(T\right)={n}_{0}+\frac{\partial\:n}{\partial\:T}(T-24),$$

(21)

where $\:{n}_{0}$ is the RI at room temperature ($\:24$), and T is the temperature in °C. Figure 13 clears the match between the RI using Eq. (21) and the measured data. As clear in Fig. 13, Eq. (21) is a good choice to represent the RI of GaN as a function of temperature for a wide range of wavelengths (from 0. 390 to 1.70 μm).

Fig. 13

Experimental⁶⁶ and fitted (using Eq. 21) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength for temperature ranges from 24 to 506 °C.

In 2018, Gungor et al.⁶⁷ evaluated the RIs of GaN at different thicknesses of 5.37 nm, 21.01 nm, 48.65 nm, and 81.40 nm over a wide range of wavelength (from 0.3 μm to 1.0 μm). They ensured that the RI of GaN film is significantly impacted by the film thickness. By increasing the thickness of GaN film from 5.37 nm to 21.01 nm, the RI significantly increases. The RI slightly changed (can be ignored) or saturated above this thickness, as clear in Fig. 14. We fitted the experimental data using the Sellmeier formula over 0.36 –1.00 μm wavelength range ($\:\lambda\:$ in µm) as follows:

Fig. 14

Experimental⁶⁷ and fitted (using Eqs. (22), (23), (24), and (25)) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength for different thicknesses of 5.37 nm, 21.01 nm, 48.65 nm, and 81.40 nm.

For thickness of 5.37 nm:

$$\:{n}^{2}=-1.798096+\frac{\:4.197311{\lambda}^{2}}{{\lambda}^{2}-{0.143908}^{2}}+\frac{\:2.071998{\lambda}^{2}}{{\lambda}^{2}-{0.139627}^{2}}+\frac{\:84{\lambda}^{2}}{{\lambda}^{2}-{(-46)}^{2}},$$

(22)

For thickness of 21.01 nm:

$$\:{n}^{2}=3.558494+\frac{\:26936628366{\lambda}^{2}}{{\lambda}^{2}-{(-6509902696)}^{2}}+\frac{\:1.426702{\lambda}^{2}}{{\lambda}^{2}-{0.263888}^{2}}+\frac{\:600270609{\lambda}^{2}}{{\lambda}^{2}-{(-56283058)}^{2}},$$

(23)

For thickness of 48.65 nm:

$$\:{n}^{2}=3.512247+\frac{\:103{\lambda}^{2}}{{\lambda}^{2}-{26}^{2}}+\frac{\:1.521155{\lambda}^{2}}{{\lambda}^{2}-{0.255582}^{2}}+\frac{\:594107724{\lambda}^{2}}{{\lambda}^{2}-{(-53272667)}^{2}},$$

(24)

For thickness of 81.40 nm:

$$\:{n}^{2}=1.970540+\frac{\:5892{\lambda}^{2}}{{\lambda}^{2}-{3608}^{2}}+\frac{\:2.904413{\lambda}^{2}}{{\lambda}^{2}-{0.219380}^{2}}+\frac{\:-2021676864{\lambda}^{2}}{{\lambda}^{2}-{189402418}^{2}}.$$

(25)

Gungor et al.⁶⁸ measured the RIs of GaN at different thicknesses of 6.56 nm, 20.93 nm, 52.01 nm, and 84.35 nm over a wide range of wavelength (from 0.3 μm to 1.0 μm). They found that the RI of GaN film is significantly impacted by the film thickness. By increasing the thickness of GaN film up to 52.01 nm, the RI increases and is saturated gradually (slightly decreases) above this thickness, as clear in Fig. 15. This behavior is due to the significant decrease of the voids ratio by increasing the thickness of the GaN film from 6.57 to 52.01 nm. For thicknesses higher than 52.01 nm, the RI becomes nearly steady. It has also been discovered that increasing the thickness of GaN film decreases the roughened overlayer thickness on the surfaces of GaN films. We fitted the experimental data using the Sellmeier formula over the 0.36 –1.00 μm wavelength range ($\:\lambda\:$ in µm) as follows:

Fig. 15

Experimental⁶⁸ and fitted (using Eqs. (26), (27), (28), and (29)) RIs of $\:\text{G}\text{a}\text{N}$ versus the wavelength for different thicknesses of 6.56 nm, 20.93 nm, 52.01 nm, and 84.35 nm.

For thickness of 6.57 nm:

$$\:{n}^{2}=0.392697+\frac{\:2.423788{\lambda}^{2}}{{\lambda}^{2}-{0.101481}^{2}}+\frac{\:1.135063{\lambda}^{2}}{{\lambda}^{2}-{0.240199}^{2}}+\frac{\:-997698{\lambda}^{2}}{{\lambda}^{2}-{(-30649647)}^{2}},$$

(26)

For the thickness of 20.93 nm:

$$\:{n}^{2}=1.886147+\frac{\:2.036566{\lambda}^{2}}{{\lambda}^{2}-{0.224662}^{2}}+\frac{\:0.626150{\lambda}^{2}}{{\lambda}^{2}-{0.224661}^{2}}+\frac{\:-981{\lambda}^{2}}{{\lambda}^{2}-{(-131)}^{2}},$$

(27)

For the thickness of 52.01 nm:

$$\:{n}^{2}=0.415881+\frac{\:2.428052{\lambda}^{2}}{{\lambda}^{2}-{(-3.009347\times\:{10}^{-5})}^{2}}+\frac{\:1.983056{\lambda}^{2}}{{\lambda}^{2}-{0.244739}^{2}}+\frac{\:30{\lambda}^{2}}{{\lambda}^{2}-{(-22)}^{2}},$$

(28)

For the thickness of 84.35 nm:

$$\:{n}^{2}=0.345015+\frac{\:1.971097{\lambda}^{2}}{{\lambda}^{2}-{(3.279298\times\:{10}^{-5})}^{2}}+\frac{\:2.384852\:{\lambda}^{2}}{{\lambda}^{2}-{0.233322}^{2}}+\frac{\:64{\lambda}^{2}}{{\lambda}^{2}-{42}^{2}}.$$

(29)

In 2016, Alevli et al.⁵⁴ deposited trimethylgallium (TMG) and triethylgallium (TEG) on Si (100) substrates. The substrate temperature was 200 °C during the growing process. The TMG thickness of the bulk layer is about 56.88 nm, and 11.22 nm for the rough layer. For TEG, the thickness of the bulk layer is 49 nm. Figure 16 clearly shows the RIs of TEG and TMG from wavelength 0.3 μm to 1.0 μm. We fitted the experimental data using the Sellmeier formula over the 0.355 –1.000 μm wavelength range ($\:\lambda\:$ in µm) as follows:

Fig. 16

Experimental⁵⁴ and fitted (using Eqs. (30), and (31)) RIs of TEG and TMG $\:\text{G}\text{a}\text{N}$ versus the wavelength.

For TEG:

$$\:{n}^{2}=0.588807+\frac{\:2.841507{\lambda}^{2}}{{\lambda\:}^{2}-{0.209579}^{2}}+\frac{\:0.856046{\lambda}^{2}}{{\lambda}^{2}-{0.212284}^{2}}+\frac{\:35835127{\lambda}^{2}}{{\lambda}^{2}-{(-47303997104)}^{2}},$$

(30)

For TMG:

$$\:{n}^{2}=3.013836+\frac{\:12539293{\lambda}^{2}}{{\lambda}^{2}-{208093734}^{2}}+\frac{\:1.984806{\lambda}^{2}}{{\lambda}^{2}-{0.241192}^{2}}+\frac{\:1426051620{\lambda}^{2}}{{\lambda}^{2}-{(-137831263)}^{2}}.$$

(31)

Besides, Alevli et al.⁴⁴ studied the impact of substrate temperature (200 °C and 450 °C) on the RI of TEG on Si (100) substrates. The thickness of deposited GaN film at 450 °C was a bulk layer of 54.1 nm and a rough layer of 14.8 nm. For deposited GaN film at 200 °C, it was a bulk layer of 49.0 nm. They observed that the crystalline quality TEG was improved by increasing the substrate temperature. The rough layer formation increases with the increase of substrate temperature. Figure 17 clearly shows that the RI of TEG strongly depends on substrate temperature, which may be due to the crystalline orientation variation, improved crystallinity, or surface morphology. We fitted the experimental data using the Sellmeier formula from 0.44 μm to 1.80 μm wavelength range (λ in µm) as follows:

Fig. 17

Experimental⁴⁴ and fitted (using Eqs. (32) and (33)) RIs of TEG versus the wavelength at substrate temperature of 200–450 °C.

For substrate temperature of 200 °C:

$$\:{n}^{2}=-205.823612+\frac{\:210.282727{\lambda}^{2}}{{\lambda}^{2}-{0.039944}^{2}}+\frac{\:232917{\lambda}^{2}}{{\lambda}^{2}-{(-1639255)}^{2}}+\frac{\:936948{\lambda}^{2}}{{\lambda}^{2}-{(-4719)}^{2}}.$$

(32)

For substrate temperature of 450 °C:

$$\:{n}^{2}=-168.477338+\frac{\:271785{\lambda}^{2}}{{\lambda}^{2}-{5036}^{2}}+\frac{\:173.822753{\lambda}^{2}}{{\lambda}^{2}-{0.043871}^{2}}+\frac{\:6130670{\lambda}^{2}}{{\lambda}^{2}-{7761892}^{2}}.$$

(33)

In 2018, Bowman et al.⁶⁹ experimentally measured the indices of refraction of ordinary and extraordinary single crystal GaN from wavelength 0.46 μm to 1.91 μm at 25 °C. The deduced experimental and fitted indices of refraction are presented in Fig. 18. According to Sellmeier formula, the ordinary ($\:{n}_{o}$) and extraordinary ($\:{n}_{e}$) RIs of GaN can be simulated (λ in µm) as follows.

Fig. 18

Experimental⁶⁹ and fitted (using Eqs. (34) and (35)) RIs of ordinary and extraordinary $\:\text{G}\text{a}\text{N}$ versus the wavelength.

For extraordinary GaN:

$$\:{n}_{e}^{2}=2.30309+\frac{\:2.80472{\lambda}^{2}}{{\lambda}^{2}-{0.19614}^{2}}+\frac{\:0.23021{\lambda}^{2}}{{\lambda}^{2}-{0.33811}^{2}}+\frac{\:0.08844{\lambda}^{2}}{{\lambda}^{2}-{3.22201}^{2}},$$

(34)

For ordinary GaN:

$$\:{n}_{o}^{2}=3.48159+\frac{\:257815547{\lambda}^{2}}{{\lambda}^{2}-{136566}^{2}}+\frac{\:1.71698{\lambda}^{2}}{{\lambda}^{2}-{0.26011}^{2}}+\frac{\:18545092{\lambda}^{2}}{{\lambda}^{2}-{(-379587)}^{2}}.$$

(35)

Bowman et al.⁷⁰ measured and fitted the RIs of free-standing samples of ordinary and extraordinary $\:\text{G}\text{a}\text{N}$ (Bulk GaN) from wavelength 0.50 μm to 5.10 μm. The samples had a thickness of millimeters at room temperature. Figure 19 clear the experimental and fitted indices of refraction according to the following equations (λ in µm):

Fig. 19

Experimental and fitted (using Eqs. (36) and (37)) RIs of ordinary and extraordinary $\:\text{G}\text{a}\text{N}$ versus the wavelength⁷⁰.

For extraordinary GaN:

$$\:{n}_{e}^{2}=1+\frac{\:4.347{\lambda}^{2}}{{\lambda}^{2}-{0.1781}^{2}}+\frac{\:2.964{\lambda}^{2}}{{\lambda}^{2}-{15.23}^{2}},$$

(36)

For ordinary GaN:

$$\:{n}_{o}^{2}=1+\frac{\:4.199{\lambda}^{2}}{{\lambda}^{2}-{0.1753}^{2}}+\frac{\:3.625{\lambda}^{2}}{{\lambda}^{2}-{17.05}^{2}}.$$

(37)

In 2011, Usman et al.²³ calculated the RIs of cubic $\:\text{G}\text{a}\text{N}$ (2 atoms per primitive cell) using the Material Studio Software environment. The RIs were calculated at normal (0 GPa) and hydrostatic pressure (10 GPa, 20 GPa, 30 GPa, and 40 GPa), as clear in Fig. 20. By increasing the hydrostatic pressure from 0 GPa to 40 GPa, the RI of cubic $\:\text{G}\text{a}\text{N}$ decreases. According to the Sellmeier formula, the RI of cubic $\:\text{G}\text{a}\text{N}$ from wavelength 0.85 μm to 467 μm can be simulated (λ in µm) as follows:

Fig. 20

Calculated²³ and fitted (using Eqs. (38), (39), (40), (41), and (42)) RIs of cubic $\:\text{G}\text{a}\text{N}$ versus the wavelength at normal and hydrostatic pressure.

At 0 GPa:

$$\:{n}^{2}=3.025038+\frac{\:-3.998069{\lambda}^{2}}{{\lambda}^{2}-{0.424907}^{2}}+\frac{\:6.494450{\lambda}^{2}}{{\lambda}^{2}-{0.404139}^{2}},$$

(38)

At 10 GPa:

$$\:{n}^{2}=-1.470803+\frac{\:7.072916{\lambda}^{2}}{{\lambda}^{2}-{0.264431}^{2}}+\frac{\:-0.199289{\lambda}^{2}}{{\lambda}^{2}-{0.642746}^{2}},$$

(39)

At 20 GPa:

$$\:{n}^{2}=2.212337+\frac{\:3.649775{\lambda}^{2}}{{\lambda}^{2}-{0.358214}^{2}}+\frac{\:-0.754690{\lambda}^{2}}{{\lambda}^{2}-{0.498287}^{2}},$$

(40)

At 30 GPa:

$$\:{n}^{2}=1.966181+\frac{\:1.608604{\lambda}^{2}}{{\lambda}^{2}-{(-0.252844)}^{2}}+\frac{\:1.332746{\lambda}^{2}}{{\lambda}^{2}-{0.266210}^{2}},$$

(41)

At 40 GPa:

$$\:{n}^{2}=2.618795+\frac{\:1.204564{\lambda}^{2}}{{\lambda}^{2}-{(-0.269502)}^{2}}+\frac{\:0.937258{\lambda}^{2}}{{\lambda}^{2}-{0.269691}^{2}}.$$

(42)

The extinction coefficient (imaginary part of RI) of cubic $\:\text{G}\text{a}\text{N}$ starts at about 2 eV (0. 620 μm)²³. As clear in Fig. 21, the extinction coefficient of cubic $\:\text{G}\text{a}\text{N}$ can be neglected (near zero) at wavelengths higher than 0.75 μm.

Fig. 21

Calculated extinction coefficient of cubic $\:\text{G}\text{a}\text{N}$ versus a wide range of wavelengths.

Gasmi et al.⁷¹ investigated the RIs and extinction coefficients of different phases of GaN (zinc blende and wurtzite) using the full-potential linearized augmented plane wave method. These calculations depended on generalized gradient approximation (GGA) and local density approximation (LDA). The extinction coefficients of these phases of GaN can be neglected from wavelength 0.716 μm to 766 μm. According to the Sellmeier formula, the RI of zinc blende and wurtzite $\:\text{G}\text{a}\text{N}$ from wavelength 0.716 μm to 766 μm can be fitted (λ in µm) as follows (Fig. 22):

Fig. 22

Theoritical⁷¹ and fitted (using Eqs. (43), (44), and (45)) RIs of zinc blende and wurtzite $\:\text{G}\text{a}\text{N}$ versus the wavelength.

For zinc-blende GaN:

$$\:{n}^{2}=4.22060+\frac{\:0.78217{\lambda}^{2}}{{\lambda}^{2}-{0.453395}^{2}}+\frac{\:0.772100{\lambda}^{2}}{{\lambda}^{2}-{0.45339}^{2}},$$

(43)

For wurtzite (100) GaN:

$$\:{n}^{2}=2.56720+\frac{\:0.94733{\lambda}^{2}}{{\lambda}^{2}-{(-0.42357)}^{2}}+\frac{0.95072{\lambda}^{2}}{{\lambda}^{2}-{(-0.42357)}^{2}},$$

(44)

For wurtzite (001) GaN:

$$\:{n}^{2}=2.25377+\frac{\:1.36026{\lambda}^{2}}{{\lambda}^{2}-{(-0.37788)}^{2}}+\frac{0.86835{\lambda}^{2}}{{\lambda}^{2}-{(-0.37788)}^{2}},$$

(45)

Singh et al.⁷² studied the optical properties of wurtzite GaN using the method of plane-wave pseudopotential depending on GGA and LDA at 0 GPa. According to the Sellmeier formula, the RI of $\:\text{G}\text{a}\text{N}$ using GGA and LDA from wavelength 0.6 μm to 3099.5 μm can be fitted (λ in µm) as follows (Fig. 23):

Fig. 23

Calculated⁷² and fitted (using Eqs. (46) and (47)) RIs of wurtzite $\:\text{G}\text{a}\text{N}$ using GGA and LDA versus the wavelength.

Using GGA:

$$\:{n}^{2}=-1.09416+\frac{\:-7.19965{\lambda}^{2}}{{\lambda}^{2}-{0.38243}^{2}}+\frac{\:14.35423{\lambda}^{2}}{{\lambda}^{2}-{0.32032}^{2}},$$

(46)

Using LDA:

$$\:{n}^{2}=0.95394+\frac{\:2.02687{\lambda}^{2}}{{\lambda}^{2}-{0.20530}^{2}}+\frac{\:2.82113{\lambda}^{2}}{{\lambda}^{2}-{0.21178}^{2}},$$

(47)

Conclusion

In this study, the RI of GaN has been studied a wide wavelength from 0.35 μm to 3099.5 μm. In visible spectra, the RI of GaN strongly decreases and slightly decreases in near IR spectra. From a wavelength of 3.0 μm to 3099.5 μm, the RI of GaN can be considered a constant value (ex. 2.41 for wurtzite GaN using LDA). Besides, the effect of different parameters such as the temperature from 21 to 515 °C, porosity from 0 to 38.7%, thickness from 5.37 nm to 17.2 μm, substrate temperature from 200 to 450 °C, and pressure from 0 GPa to 40 GPa are studied and fitted.