1. Introduction

In the transparent region of wurtzite GaN (w-GaN) the injection of electrons and/or holes into the intrinsic semiconductor results in a perturbation of its complex refractive index (n + ik). The resulting index is n + Δn + i(k + Δk), where k is very small because the photon energy is sub-bandgap. The changes Δn and Δk are roughly proportional to the concentration of free carriers that are injected [1, 2]. The purpose of this paper is to estimate both Δn and Δk based on a physically valid model using a realistic value of carrier injection. We investigate these free carrier effects in a wavelength range that begins at 1 um, covers the 1.5 µm telecom zone and extends out to the 5 µm mid infrared.

The III-Nitride semiconductors that include w-GaN are proving to be a new family of semiconductors for on-chip photonic integrated circuits (PICs) that contain both active and passive photonic components [3–8]. The wide gap semiconductor w-GaN can be combined readily in single-crystal alloys such as AlGaN and InGaN to have a bandgap that is tunable by choice of alloy composition, in order to tailor the active devices to the wavelength of interest [8]. Due to its 3.4 eV gap, the GaN material is transparent over a broad wavelength range from the visible to mid-wave infrared spectrum [9, 10]. This opens up a wider range for integrated photonics than that given by the popular Si material. The wide gap of w-GaN is also favorable for avoiding two-photon absorption effects in nonlinear optical applications of this material at the 1.55-µm telecom band.

Looking at the suite of possible integrated photonics components, the electro-optical (EO) modulators and 2 × 2 EO switches are highly desired components, and as we shall show here, the free carrier EO effects of electro-refraction (ER) and the associated electro-absorption (EA) have practical values for those components. The ER and EA can be induced by PIN and PN diode structures constructed laterally on the GaN waveguide, and the carrier induced effects can be attained by electron-hole injection or depletion.

In recent years, variety of GaN waveguiding structures have been developed on different material substrates [3–6] to provide optical confinement as well as fabrication compatibility. One of the promising structures includes the growth of c-plane w-GaN upon a sapphire substrate [3, 7]. By etching the GaN layer into rectangular strip or rib structures, waveguides can be made.

To implement efficient EO devices in GaN waveguides using PN or PIN, two important factors are: 1- having a GaN layer with high-quality electronic properties and minimal defects similar to those of the alternative technologies such as Si and GaAs, 2- A mature fabrication for GaN photonic devices with a low propagation loss in the range of other competing integrated photonic technologies. Efforts to improve the electronic qualities of GaN-on-sapphire have been promising and are still in progress [8]. Since the material growth technology of GaN is still young, there is a large variance in the reported electronic properties of GaN in the literature [11–13]. Like many other wide-bandgap semiconductors, GaN is a difficult material to etch. Nevertheless, with the ongoing work on fabricating GaN photonic devices [3–8] and with progress in plasma etching technology, achieving low-loss GaN photonic devices is anticipated.

In this paper, we determine the free-carrier ER and EA properties of GaN and compare the resulting free-carrier responses with the ER and EA found in two other practical integrated-photonic technologies, i.e. Si and GaAs. In the past there has been some studies on the refractive index and absorption change of GaN at very high electron concentrations (10¹⁸ −10²⁰ cm⁻³) [14]. Our work considers the effect of both electrons and holes at moderate concentrations (< 10¹⁸ cm⁻³) for applications at near- to mid-infrared wavelengths (1 to 5 μm). As mentioned, we examine the injection of electrons and/or holes into the undoped GaN waveguide for applications in electro-optical modulation and switching. Since the photon-energy range under discussion here is much less than the bandgap of GaN, the effects of band filling [1] and bandgap shrinkage [1] are negligible and the “Drude plasma” model plays the main role [1, 2]. Also, knowing that GaN is a polar material, the interaction of longitudinal optical phonons with plasmons (LOPP) is quite strong at longer infrared wavelengths [12, 15, 16]. We study the effect of this LOPP coupling and show that its effect is negligible in the 1-to-5 μm wavelength range for carrier concentrations below 1 x 10¹⁸ cm⁻³. Although, w-GaN is a uniaxial anisotropic crystal, we focus only upon the in- plane refractive index of GaN where the refractive index is isotropic. In addition, for many waveguide devices, the polarization of interest is transverse electric (TE), meaning that the infrared electric field is in-plane.

2. Theoretical modeling and results

According to Drude model the change in the real and the imaginary part of the refractive index due to free carriers can be expressed as [2]:

We simulated the ER and EA effect using Eq. (1) and (2) for GaN, Si, and GaAs by employing the experimental parameters shown in Table 1. For the value of n in Eq. (1) and (2), we used the Sellmeier expressions presented in the literature for these semiconductors. Throughout this paper, the level of free-carrier injection into undoped semiconductor was selected to be 5 × 10¹⁷ cm⁻³ in a tradeoff against waveguide propagation loss and energy consumed during modulation [17]. In addition, this carrier concentration is large enough to produce practical modulation in real devices but is not so large as to require large applied voltages or currents.

Table 1. Conductivity effective mass and mobility of electrons and holes (in cm²V⁻¹s⁻¹ unit) at the carrier concentration of 5 × 10¹⁷ cm⁻³. m₀ is the free-electron mass. For each data, the reference is shown in its front.

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Figure 1 shows the results for -Δn, -Δn/Δk, and EA absorption loss Δα expressed in cm⁻¹. The first and second columns of Fig. 1 show the effect of electrons only, and holes only, respectively, while the third column shows the combined effect of electrons and holes. The results from Fig. 1 can tell a designer what P and N dopant densities to use in order to optimize the ER and EA effect in the selected semiconductor material. For electrons, we find that the w-GaN ER is larger than the Si ER and smaller than the GaAs ER. For holes, the w-GaN ER is smaller than that of GaAs and Si. Looking at dual e + h injection from a PIN, we see that the w-GaN ER is larger than the Si ER and smaller than the GaAs ER. When we compare ER and EA using the predicted Δn/Δk ratio of a particular material, we find that w-GaN is slightly lossier than the other two materials (this is true for electrons, holes and e + h). These results shows a lot of promise for implementing carrier-induced ER and EA modulators and routing switches in a GaN integrated photonic platform.

 

Fig. 1 Comparison of free-carrier electro-refraction and electro-absorption effects in GaN (green), Si (red), and GaAs (blue) with parameters given in Table 1 for a carrier injection of 5 × 10¹⁷ cm⁻³. The plots show the variation of (Δn), (Δn/ Δk), and loss Δα versus wavelength when the free carriers are electrons only ((a), (b), (c)), holes only ((d), (e), (f)), and both electrons and holes ((g), (h), (i)). The plots for Si and GaAs start from wavelength of 1.5 microns.

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To get an idea of what Δn and Δk changes are required for a practical GaN modulator we shall consider operation at 1.55 µm and at 5 µm in Fig. 1. At the 1.55 µm wavelength where ER is dominant to EA in both GaN and Si, Fig. 1 shows that the Δn of GaN for electron-hole injection is 0.0049 which is 25% larger than the Δn of Si; and the Δα of GaN corresponding to that injection is 75% larger than that of Si. This GaN-Si comparison suggests that the performance of the GaN modulator devices would be quite similar to those for well-known practical Si modulators at 1.55 µm. Turning to the 5 µm wavelength where EA is indeed dominant and ER is secondary, if we again consider PIN injection, then the Δα ≈100 cm⁻¹ for GaN at 5 µm (Fig. 1(i)) translates into 0.043 dB per micron attenuation (loss modulation) within a straight piece of GaN waveguide. Taking a PIN length of 200 µm, the active region in the intrinsic waveguide, we project a high on/off extinction ratio of 8.6 dB.

It is well known that due to the polar property of crystalline w-GaN, the interaction between LO phonons and plasmons can be strong at longer wavelengths and at higher free-carrier densities, particularly for electrons [12, 15, 16, 18]. This interaction occurs when the plasmon frequency of free carriers is close to the frequency of LO phonons. A consequence of this interaction is a stronger absorption near the LO phonon resonance. While the LO phonon wavelength is quite far from our 1-5 µm range, the tail of the “broadened” LOPP interaction may reach shorter wavelengths and affect the absorption in addition to Δα that the Drude model predicts. We have studied this LOPP interaction over wide wavelength and concentration ranges. The generalized Drude-LOPP model for the in-plane relative permittivity of w-GaN can be written as [15, 17]:

The LOPP interaction occurs at frequencies where ${\epsilon }_{\perp }(\omega )=0$ [12, 13, 16]. By putting Eq. (3) equal to zero and neglecting the damping factors, the two resonance frequencies at which LOPP coupling occurs can be found as [16, 18, 19]:

Figures 2(a) and 2(b) show the LOPP ± resonance wavelengths (i.e. $2\pi c/{\omega }_{LOPP\pm }$) for electrons and holes at different carrier densities, respectively. For the simulations, we used the values of ${\epsilon }_{\perp }=5.35$, ω_LO = 743 cm⁻¹, and ω_TO = 533 cm⁻¹ [16, 20, 21] and the parameters in Table 1 for electrons and holes in w-GaN. In Figs. 2(a) and 2(b) a gray-shaded zone is shown that corresponds to the wavelength range 1 to 5 microns and a free carrier range from low concentrations up to 5 × 10¹⁷ cm⁻³. The high frequency branch of LOPP (i.e. LOPP + ) is closer to the gray zone and needs to be investigated. Figure 2(c) shows a zoomed-in view of LOPP + for both electrons and holes. From this figure we see that the LOPP + resonance for holes has less variation versus carrier density and stays close to the original value (λ_LO). Experimental results from the literature also show that the LOPP interaction is very weak and negligible for holes even at higher concentrations above 10¹⁸ cm⁻³. Therefore, using Eq. (1) and (2) for holes is quite valid.

 

Fig. 2 The carrier-density dependence of coupled LOPP resonance wavelengths for (a) electrons and (b) holes in w-GaN. Upper and lower coupled resonances have been designated by LOPP + and LOPP-, respectively. The variation of plasmon wavelength in the absence of LOPP coupling is shown in each figure. (c) a zoomed-in view and comparison of the upper resonances (LOPP + ) of holes and electrons at different carrier densities.

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LOPP coupling for electrons is stronger when going to larger carrier densities according to Fig. 2(c). This is mainly due to a smaller effective mass for electrons which results in a larger plasma frequency in Eq. (5). Nevertheless, up to a carrier density of 10¹⁸ cm⁻³, we see that the LOPP₊ resonance of electrons is still very far from the gray zone shown in Fig. 2(b). There is some concern that LOPP tailing might reach the gray zone. However, experimental results from the literature show that the reaching does not happen [18, 19]. As a result, Eq. (1) and (2) are also valid for the electron concentration within the gray zone in Fig. 2(a).

We also compared the absorption loss Δα obtained from Eq. (2) for an electron density of 2 × 10¹⁸ cm⁻³ with the results from a more rigorous theoretical analysis from [15] for a wavelength of 6 microns which is the starting wavelength of interest in that work. Our comparison showed a very good agreement between full theory and Drude, meaning that the Drude expression in Eq. (1) and (2) can be still used for electron concentrations as high as 2 × 10¹⁸ cm⁻³.

3. Conclusions

In this paper we study carrier-induced electrorefraction and electroabsorption in wurtizite GaN for modulator and switch applications and compare the results to those of Si and GaAs. We find that the Drude model and Eqs. (1) and (2) are valid for predicting the ER and EA effects in w-GaN for carrier concentrations that range up to 10¹⁸ cm⁻³ and for the 1 to 5 μm wavelength range discussed in this paper. At 1.55-μm telecoms, the ER is dominant over EA in all three materials. At a given ΔN or ΔP in all three materials, increasing the wavelength from 1 to 5 μm brings with it a rather strong increase of EA with respect to ER, opening the door for EA intensity modulation in the mid infrared. Not only does w-GaN exhibit this λ-decrease of Δn/Δk, but Δn/Δk for GaN is lower than that for Si and GaAs.