1. Introduction

Nowadays, in order to obtain high brightness in GaN-based LEDs, many methods have been widely adopted, including photonic crystal, surface texturing, PSS, flip-chip and vertical structure^[1]. The PSS method has been attracting considerable attention due to its high luminous efficiency^[2]. This is because it is an effective and popular method to enhance both internal quantum efficiency (IQE), by reducing threading dislocations (TDs) density and improving the crystalline quality^[3-5], and light extraction efficiency (LEE), simultaneously^[6]. Due to the substantial difference in refractive index between the epitaxial GaN film and air, the relatively small total internal reflection angle leads to poor LEE^[7]. Hence, the patterns can serve as scattering centers for redirecting the guided light into a randomized distribution of angles that will enable multiple entries of photons into the escape cone^{[8, 9]}. This means that, in face-up LED structures, the light scattering effect from the patterns explains very well the enhanced top-surface light output power (LOP) and LEE^{[10, 11]}. Yet, limited research studies have been focused on the investigation of the change in optical parameters, such as reflection coefficient and transmission coefficient. The transmission coefficient is of great importance as it is the key feature to decide whether it is feasible to utilize the PSS technique in flip-chip structures, in order to obtain higher luminous efficiency.

In this study, the results of both experimental research and computer simulations reveal a great enhancement in LOP from either the top or bottom surface of PSS-LEDs, compared to CSS-LEDs. Following that, a theoretical analysis, which is based on two fundamental theories, i.e. the Fresnel Equations and geometrical optics, was performed in order to demonstrate the increase in effective reflection and transmission at the GaN-sapphire interface, which will explain the enhancement in LOP from both the top and bottom surface of PSS-LEDs. Here, it is necessary to clarify that the new term ``effective reflection'' is different from the ``total internal reflection''. The effectively reflected light refers to the amount of light that is reflected and then successfully escapes from the top-surface of the LED. This excludes the total internal reflection light, which cannot radiate out of the LED devices and is eventually absorbed by the LED^[12]. As an overall comment, due to the existence of PSS the total internal reflection light that is ultimately absorbed decreases, however, the effective reflection light and the transmission light increase. The increased effective reflection light and the transmission light that can radiate out from top-surface and bottom-surface, respectively, contribute to the enhanced top and bottom surface LOP. Furthermore, a mass of PSS-LED chips is used for separately measuring the top-surface and bottom-surface LOP. These experimental results testify that the latter has slightly superior light luminous performance to the former. It is mainly because the critical angle of GaN surface is smaller than the sapphire surface^[13]. Therefore, this study not only demonstrates how the PSS technology can improve the LOP of flip-chip and face-up structure LEDs, compared to CSS, but also that the LOP of flip-chip structure PSS-LEDs is superior to the face-up structure.

2. Simulation and results

To better demonstrate the improvement of PSS on the top and bottom LOP, the simulation based on Trace-Pro software is performed. The LED optical simulation models are presented in Figure 1. The models consist of a p-doped GaN layer with a thickness of 0.2 $\mu$m, an n-doped GaN layer with a thickness of 4 $\mu$m, and the sapphire substrate with a thickness of 150 $\mu$m. The interface is a well-ordered array of circular cones with a diameter of 2.58 $\mu$m, a height of 1.53 $\mu$m and a periodicity of 3 $\mu$m. The patterns shape and parameters were set according to the actual chips used in the aftermentioned experiment. Additionally, Figure 1(b) shows the CSS-LED which will serve as a reference in order to reveal the effect of PSS on LOP or LEE.

The total flux power of MQWs was set at 2 W and the absorption coefficient of the active layer was considered to be 1000 cm$^{-1}$^[14]. In Figure 1, the top monitor receives the optical waves that can be extracted from the top-surface and the bottom monitor 1 receives the optical waves that can be extracted from the bottom-surface of the LED. Therefore, the beam flux power absorbed by the top and bottom monitor 1 represent the LOP from the top-surface and bottom-surface of LED, respectively.

As clearly shown in Table 1, when considering either the top-surface or bottom-surface, the LOP of PSS-LED is higher than that of the CSS-LED. Thus, the simulation results indicate that both top and bottom LOP are increased due to the presence of PSS. Also, regardless of the use of either PSS or CSS model, the bottom LOP is always slightly higher than the top LOP. This also illustrates that the light extraction performance of the bottom-surface is slightly superior to the top-surface and that the flip-chip LED has many advantages over the face-up LED devices for obtaining superior luminous efficiency. Also, the bottom monitor 2 is embedded in the sapphire layer. It can receive directly the optical waves that can be transmitted from the GaN-sapphire interface. As can be seen in Table 1, the value of power recorded by monitor 1 is smaller than monitor 2. This is because the transmission light from the GaN-sapphire interface cannot be completely extracted from the sapphire surface because of total internal reflection at the surface. However, the value of power recorded by monitor 2 in the PSS model is larger than in the CSS model. Thus, the result also demonstrates the increased transmission light at PSS interface.

3. Experimental results and discussion

Regarding the experimental procedure, a mass of PSS-LED and CSS-LED chips measured the LOP of the top-surface and bottom-surface, by implementing the two different packaging methods. As presented in Figure 2, the PSS patterns are circular cone shaped, with 2.58 $\mu$m in diameter, 1.53 $\mu$m in height and 3 $\mu$m in period, and featuring a hexagonal array. Additionally, according to the cross-sectional SEM images, the inclination angle of patterns was calculated:

$$\sin\alpha =\frac{1.53}{1.99}\approx 0.7688, \quad\alpha \approx 50^\circ.$$

Regarding micro-PSS, this kind of pattern with the above feature has been widely used for configuration. Many studies demonstrate that cone-shaped PSS show superior output performance over any other shape and that the geometric parameters used are more effective for LEE in terms of the geometry optics^[15].

Figure 3 shows the top and bottom LOPs as a function of the injection current, for the LEDs grown on PSS and CSS. Under an injection current value of 20 mA, the bottom and top LOP of the PSS-LEDs and CSS-LEDs were 11.28, 9.11, 6.22, and 5.14 mW, respectively. Compared to CSS, the bottom and top LOP improvement of PSS-LEDs was 81.4% and 77.2%, respectively. In addition, compared to the top surface of the PSS-LED and CSS-LED, the LOP improvement from the bottom surface was 23.8% and 23.5%, respectively. The experimental results demonstrate the LOP values of PSS-LED compared to CSS-LED and the superior light output performance of the bottom-surface compared to the top-surface.

4. Discussion

When a beam of light penetrates an interface, the power is divided into the one that causes reflection and transmission of the light, as illustrated in Figure 4.

The amount of the two competing power values is decided by the Fresnel Equations^[16]. The value of the reflection coefficient is a function of the refractive index, $n$, and the angle of incidence, $\theta_{\rm i}$, based on the Fresnel formula:

$$\begin{gathered} R = \frac{1}{2}\left\{ {{{\left[ {\frac{{{n_1}\cos {\theta _{\text{i}}} - {n_2}\sqrt {1 - {{\left( {\frac{{{n_1}}}{{{n_2}}}\sin {\theta _{\text{i}}}} \right)}^2}} }}{{{n_1}\cos {\theta _{\text{i}}} + {n_2}\sqrt {1 - {{\left( {\frac{{{n_1}}}{{{n_2}}}\sin {\theta _{\text{i}}}} \right)}^2}} }}} \right]}^2}} \right. fanxiexiantihuan] {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \left. {{{\left[ {\frac{{{n_1}\sqrt {1 - {{\left( {\frac{{{n_1}}}{{{n_2}}}\sin {\theta _{\text{i}}}} \right)}^2}} - {n_2}\cos {\theta _i}}}{{{n_1}\sqrt {1 - {{\left( {\frac{{{n_1}}}{{{n_2}}}\sin {\theta _{\text{i}}}} \right)}^2}} + {n_2}\cos {\theta _i}}}} \right]}^2}} \right\}. \end{gathered}$$

(1)

With the transmission coefficient it is:

$$\begin{equation} \label{eq2} T = 1 - R, \end{equation}$$

(2)

where $n_{1}$ is the index of refraction of substrate, and $n_{2}$ is the refractive index of the incident region. When the light beam arrives at the interface of GaN ($n_{1}=2.4$) and sapphire ($n_{2}=1.7$), the $R$-$\theta_{\rm i}$ curve that is developed is presented in the Figure 5.

The $R$-$\theta_{\rm i}$ curve shows that, in the range of 0$^\circ$-30$^\circ$, the reflection coefficient increases gradually with the increase of the incident angle. Remarkably, the reflection coefficient curve hikes from 0.02 to nearly 1, when the incident angle is about to reach the critical angle value of the total internal reflection, about 45$^\circ$. Therefore, the larger the incident angle $\theta_{\rm i}$, the larger the reflection coefficient and the smaller the transmission coefficient.

According to Snell's law, the critical angle of total internal reflection is approximately 25$^\circ$ and 45$^\circ$ for the GaN/air and GaN/sapphire interface, respectively^[17]. Hence, the light-emitting ray a$_{1}$ that propagates upwards from the MQWs, with a deflection angle over 25$^\circ$, is totally reflected by the top-surface. In the case of CSS-LEDs, these photons cannot still radiate outside the top-surface since they are completely reflected by the planar sapphire substrates and are eventually absorbed by the MQWs. However, due to the presence of the cone-shaped patterns layer, part of the reflected upwards photons could be scattered and redirected into a randomized distribution of angles, which enables multiple entries of photons into the top escape cone^{[18, 19]}. Therefore, the analysis described above explains the increase of the effective reflection at the expense of the total internal reflection.

Also, when the ray propagates downwards, with the deflection angle $\theta$ over 25$^\circ$, it will reach the GaN/PSS interface layer and the incidence angle will change from the original $\theta$ to 50$^\circ$-$\theta$, as ray b$_{1}$ shown in Figure 6. Because the angle $\theta$ is over 25$^\circ$, the angle 50$^\circ$-$\theta$ is smaller than the angle $\theta$, namely, the incident angle decreases. According to the $R-\theta_{\rm i}$ curve, the transmission coefficient increases as incident angle decreases. Thus, the realization of the increased transmission photons also comes at the expense of absorption photons that were originally completely reflected by the CSS-LEDs chip. Especially for the light beams with a deflection angle over 45$^\circ$, the total internal reflection will occur not only on the GaN/air interface but also on the GaN/CSS interface^[20] without the use of patterning. Therefore, these photons are less likely to radiate outside the surface and will be eventually absorbed. However, when this part of the ray is incident on the inclined planes of the patterns, the incident angle is significantly reduced. Hence, the part of the total internal reflection ray that was originally absorbed is transformed into large amounts of transmitted light.

5. Conclusions

The ray that arrives at the GaN/sapphire interface layer is divided into three categories, i.e. the effective reflection light that can radiate outside the top-surface, the total internal reflection light that will mainly be absorbed, and the transmission light that can radiate outside the bottom-surface. Due to the existence of patterned sapphire substrates, the absorbed light decreases and is mainly transformed into transmission light and light with more effective reflection. Thus, the enhancement of top and bottom LOP is a fact, due to the increase in effective reflection and transmission of GaN/sapphire interface. In addition, with either the PSS-LEDs or CSS-LEDs, the bottom-surface shows superior luminous performance, when compared to the top-surface. Therefore, the LEE can be further enhanced by integrating the method of patterned sapphire substrates and flip-chip structure design.