**2. Limits and enhancements of the light-extraction efficiency of GaN LEDs**

No single semiconductor material alone is capable of emitting white light. A white light LED typically consists of an appropriate mixture of (a) red, blue, and green LEDs, or (b) blue

Improving the Light-Emitting Efficiency of GaN LEDs Using Nanoimprint Lithography 175

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

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

2 sin 2 (1 cos ) *<sup>c</sup>*

2 (1 cos ) (1 cos ) 4 2

   

> 

semiconductor (Figure1- Right). The radiation area of the escaping cone is:

2

*sourse*

high order terms in the above equation can be neglected, leading to the following:

*P <sup>P</sup>*

*c*

*escape*

*sourse*

*<sup>P</sup>*

*<sup>r</sup> P P <sup>P</sup>*

*P P*

*sourse*

*P P*

the surface of the active region to the air is only 4.61%.

*escape*

*P*

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

*source*

area of the emission sphere of the point source:

. - <sup>2</sup> 0

*A dA r rd r <sup>c</sup>*

The emitted optical power per unit area of the escaping cone is equal to the power per unit


2

*r*

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

*c c escape source source*

*escape c*

2 4 <sup>1</sup> 1 1 2 2! 4! *escape c c*

For semiconductors with high refraction indices, the critical angle is small. Therefore, the

Using GaN as an example once more, the light extraction efficiency (calculated below) from

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)

(1 cos ) 2

 

> 1 <sup>2</sup> 4

1 1 24.6 2 2 ( ) 4.61% 4 4 180

 

*c*

 *r r*

.

delivered to the free space.

and yellow lights, where the blue LED stimulating yellow phosphor produces the yellow light. In either case, the blue LED is the main constituent of a white light LED. Most blue LEDs are made from GaN, the compound discussed in this paper.

#### **2.1 Analysis of the light extraction efficiency**

Excluding the substrate, a typical LED structure is only a few micrometers thick. The region capable of emitting light is called the active region, composed of multiple quantum wells (MQWs) of less than 1 μm in thickness. The active region can be regarded as a thin film containing a large number of point light sources radiating photons in all directions. The ratio of the number of these emitted photons to the number of electrons injected to excite the quantum wells constitutes an essential figure(also known as internal quantum efficiency) for judging the performance of an LED. Another significant figure of value concerns the number of photons out of the active region that can be extracted to the free space. This is called the light extraction efficiency, *extraction* , defined as follows:

$$
\eta\_{\text{extraction}} = \frac{\text{\# of photons emitted into free space per second}}{\text{\# of photons emitted from active region per second}}
$$

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

$$n\_s \sin \theta\_1 = n\_{air} \sin \theta\_2$$

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 incidence is the critical angle, -<sup>1</sup> sin ( ) *air <sup>c</sup> s n n* , as shown in Figure 1- Left.

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

and yellow lights, where the blue LED stimulating yellow phosphor produces the yellow light. In either case, the blue LED is the main constituent of a white light LED. Most blue

Excluding the substrate, a typical LED structure is only a few micrometers thick. The region capable of emitting light is called the active region, composed of multiple quantum wells (MQWs) of less than 1 μm in thickness. The active region can be regarded as a thin film containing a large number of point light sources radiating photons in all directions. The ratio of the number of these emitted photons to the number of electrons injected to excite the quantum wells constitutes an essential figure(also known as internal quantum efficiency) for judging the performance of an LED. Another significant figure of value concerns the number of photons out of the active region that can be extracted to the free space. This is

*extraction* , defined as follows:
