of photons emitted into free space per second # of photons emitted from active region per second

The extraction efficiency is definitely not 100%. In practice, reflection at the interface between two materials with different refraction indices is unavoidable. This reflection can be considered as a loss (also known as Fresnel loss) mechanism in the LED. However, the main loss channel in the LED is caused by the total internal reflection (TIR). TIR is an optical phenomenon that occurs when the light enters from an optically dense medium to a less optically dense medium, such as when the light exits from GaN and enters the air. As the angle of incidence is greater than the critical angle, no light can be transmitted at the

> 1 2 *n n <sup>s</sup>* sin sin *air*

where ns and nair are the refractive indices of GaN and air, respectively. θ1 is the angle of incidence and θ2 is the refraction angle. When the refraction angle is greater than 90°, which forbids photons from being transmitted at the interface, the corresponding angle of

Fig. 1. (Left) The critical angle for the total internal reflection (Right) The light emission cone

, as shown in Figure 1- Left.

interface and all light is reflected. Snell's law is used to determine the critical angle:

*s n n*

LEDs are made from GaN, the compound discussed in this paper.

**2.1 Analysis of the light extraction efficiency** 

called the light extraction efficiency,

*extraction*

incidence is the critical angle, -<sup>1</sup> sin ( ) *air <sup>c</sup>*

of an LED

For instance, the refraction index of GaN at room temperature is 2.4, corresponding to a critical angle of 24.6o. When the angle of incidence (regarding air) of the point light sources in the active region of the GaN LED is smaller than 24.6o, all emitted photons can be delivered to the free space.

In 3D space, the point light sources emit photons isotropically in spherical directions. The radiation area is the area of the sphere 4πr2, where r is the distance from the wavefront to the point light source. When the spherical wavefront reaches the GaN/air interface, only the photons within the conical region (using the critical angle as the solid angle) can escape the semiconductor (Figure1- Right). The radiation area of the escaping cone is:

$$A = \int dA = \int\_{\theta=0}^{\theta\_c} 2\pi r \sin\theta r d\theta = 2\pi r^2 (1 - \cos\theta\_c)$$

The emitted optical power per unit area of the escaping cone is equal to the power per unit area of the emission sphere of the point source:

$$\frac{P\_{\text{escape}}}{2\pi r^2 (1 - \cos\theta\_c)} = \frac{P\_{\text{source}}}{4\pi r^2}$$

$$P\_{\text{escape}} = P\_{\text{source}} \frac{2\pi r^2 (1 - \cos\theta\_c)}{4\pi r^2} = P\_{\text{source}} \frac{(1 - \cos\theta\_c)}{2}$$

$$\frac{P\_{\text{escape}}}{P\_{\text{source}}} = \frac{(1 - \cos\theta\_c)}{2}$$

By expanding the cosine term into power series, the following equation can be derived:

$$\frac{P\_{\text{escape}}}{P\_{\text{source}}} \approx \frac{1}{2} \left[ 1 - \left( 1 - \frac{\theta\_c^2}{2!} + \frac{\theta\_c^4}{4!} - \dotsb \right) \right]$$

For semiconductors with high refraction indices, the critical angle is small. Therefore, the high order terms in the above equation can be neglected, leading to the following:

$$\frac{P\_{\text{escape}}}{P\_{\text{source}}} \approx \frac{1}{4} \theta\_c^2$$

Using GaN as an example once more, the light extraction efficiency (calculated below) from the surface of the active region to the air is only 4.61%.

$$\frac{P\_{esape}}{P\_{source}} \approx \frac{1}{4} \theta\_c^2 = \frac{1}{4} \times (\frac{24.6}{180} \times \pi)^2 \approx 4.61\%$$

#### **2.2 Methods to improve the light extraction efficiency**

Numerous methods have been proposed to circumvent the poor light extraction efficiency imposed by the TIR effect. Modifying the geometry of the LED chip is one such method [1-3]. By shaping the sides of the LED chip into trapezoidal (or up-side-down trapezoidal)

Improving the Light-Emitting Efficiency of GaN LEDs Using Nanoimprint Lithography 177

Fig. 3. Possible path for the light traveling in the patterned sapphire substrates (PSS) LED In 1987, Eli Yablonovitch [12] and Sajeev John [13] discovered that electromagnetic waves transmission is disallowed in certain periodically patterned structures (now called photonic crystals). This phenomenon is similar to electrons in the forbidden band of an energy band diagram in solids. Much research has since focused on utilizing the photonic crystal effect. Early applications of the photonic crystal effect were in the optical communication, where the photonic bandgap was used to create a waveguide of extremely large bending angles. More recently, the photonic crystal effect has also been applied to the LED. The photonic crystal can be used to enhance the external quantum efficiency of the LED by (a) forming a photonic bandgap within the LED chip, thereby forcing the generated photons to exit the chip and/or (b) forming an efficient waveguide to couple photons to the free space. Except to enhance the light extraction efficiency, photonic crystals can also modify far-field light pattern of the LED. The far-field light pattern is related to the information transmission of traffic lights [14], fiber coupling efficiency of LED sources [15], and the color mixing homogeneity in LED displays[16], etc. Therefore, the combination of the photonic crystal

Based on the aforementioned analysis, the light-emitting efficiency of the LED can be improved by fabricating micro- or nanostructures inside or on the surface of the chip/substrate. The small dimension patterns can be fabricated via nanoimprint lithography. The advantages of nanoimprint lithography are fast production and the low cost, which satisfy industry production requirements. In addition, the method simplifies the complex optical lithography, freeing it from the diffraction limit. The nanoimprint lithography was regarded as one of the ten emerging technologies in 20003 with the

and the LED has widespread application.

potential to change the world.

**3. Discussions of the nanoimprint techniques** 

shapes, the TIR effect from the top and the sides of the chip can be released. However, the typical thickness of a blue LED chip (including the substrate) is only approximately 90 μm, rendering the shaping of the sides an extremely difficult task. Another common method to reduce the TIR effect is by roughening the sides and the surface of the LED chip[4,5], which can enhance the scattering effect. As shown in Figure 2, the light is extracted whenever the guided light reaches the roughened interface. However, the roughening technique also degrades the electrical characteristics of the LED, causing the forward voltage to rise. If the electrical issue can be resolved, the roughening technique could bea simple and reliable solution.

Lee et al. [6, 7] simulated the effects of the surface structures, including the size and shape effects of these structures under different packaging conditions, on the light extraction efficiency of the LED, using ray tracing. However, the ray tracing technique only applies when the size of the structures is much bigger than the wavelength of the light. For structures that are close to the size of the photon wavelength, diffraction occurs. Though the wave nature of the light can be simulated by the FDTD or the RCWA methods, the applicability of these methods to the particle-wave duality nature of the light (prominent for sub-wavelength structures) requires further evaluation.

The most commonly used substrate for growing GaN films is sapphire. However, a 16% lattice mismatch between the GaN and the sapphire exists, causing a large number of threading dislocations (109~1012 cm-2) at the interface. Theoretically, these threading dislocations could extend from the epitaxial interface to the top of the p-GaN surface, leading to undesirable outcomes for the quality and lifetime of the LED chip. An approach to solving this problem requires altering the crystal growth orientation of the epitaxial film grown on etching surface structures of sapphire substrates. The interrupted and/or bent dislocations can reduce the density of the threading dislocations. In addition, these surface structures can display effects similar to the roughened surface to enhance the light extraction efficiency for the multiple reflections at the GaN/sapphire interface as shown in Figure 3, to cause guided light escape to free space while suppressing the TIR. Such substrate patterning is known as the patterned sapphire substrate (PSS) technique [8-11]. PSS is a popular technique for enhancing LED efficiency. Typical PSS patterns are on the order of micrometers, and shrinking the size of the patterns into nanoscales (also known as NPSS) is believed to significantly improve efficiency.

shapes, the TIR effect from the top and the sides of the chip can be released. However, the typical thickness of a blue LED chip (including the substrate) is only approximately 90 μm, rendering the shaping of the sides an extremely difficult task. Another common method to reduce the TIR effect is by roughening the sides and the surface of the LED chip[4,5], which can enhance the scattering effect. As shown in Figure 2, the light is extracted whenever the guided light reaches the roughened interface. However, the roughening technique also degrades the electrical characteristics of the LED, causing the forward voltage to rise. If the electrical issue can be resolved, the roughening technique

Lee et al. [6, 7] simulated the effects of the surface structures, including the size and shape effects of these structures under different packaging conditions, on the light extraction efficiency of the LED, using ray tracing. However, the ray tracing technique only applies when the size of the structures is much bigger than the wavelength of the light. For structures that are close to the size of the photon wavelength, diffraction occurs. Though the wave nature of the light can be simulated by the FDTD or the RCWA methods, the applicability of these methods to the particle-wave duality nature of the light (prominent for

The most commonly used substrate for growing GaN films is sapphire. However, a 16% lattice mismatch between the GaN and the sapphire exists, causing a large number of threading dislocations (109~1012 cm-2) at the interface. Theoretically, these threading dislocations could extend from the epitaxial interface to the top of the p-GaN surface, leading to undesirable outcomes for the quality and lifetime of the LED chip. An approach to solving this problem requires altering the crystal growth orientation of the epitaxial film grown on etching surface structures of sapphire substrates. The interrupted and/or bent dislocations can reduce the density of the threading dislocations. In addition, these surface structures can display effects similar to the roughened surface to enhance the light extraction efficiency for the multiple reflections at the GaN/sapphire interface as shown in Figure 3, to cause guided light escape to free space while suppressing the TIR. Such substrate patterning is known as the patterned sapphire substrate (PSS) technique [8-11]. PSS is a popular technique for enhancing LED efficiency. Typical PSS patterns are on the order of micrometers, and shrinking the size of the patterns into nanoscales (also known as

could bea simple and reliable solution.

Fig. 2. Reducing the TIR by scattering

sub-wavelength structures) requires further evaluation.

NPSS) is believed to significantly improve efficiency.

Fig. 3. Possible path for the light traveling in the patterned sapphire substrates (PSS) LED

In 1987, Eli Yablonovitch [12] and Sajeev John [13] discovered that electromagnetic waves transmission is disallowed in certain periodically patterned structures (now called photonic crystals). This phenomenon is similar to electrons in the forbidden band of an energy band diagram in solids. Much research has since focused on utilizing the photonic crystal effect. Early applications of the photonic crystal effect were in the optical communication, where the photonic bandgap was used to create a waveguide of extremely large bending angles. More recently, the photonic crystal effect has also been applied to the LED. The photonic crystal can be used to enhance the external quantum efficiency of the LED by (a) forming a photonic bandgap within the LED chip, thereby forcing the generated photons to exit the chip and/or (b) forming an efficient waveguide to couple photons to the free space. Except to enhance the light extraction efficiency, photonic crystals can also modify far-field light pattern of the LED. The far-field light pattern is related to the information transmission of traffic lights [14], fiber coupling efficiency of LED sources [15], and the color mixing homogeneity in LED displays[16], etc. Therefore, the combination of the photonic crystal and the LED has widespread application.
