# Light-Emitting Diodes (LEDs)

This is a continuation from the previous tutorial - **lateral structures of semiconductor junctions**

LEDs are simple, but important, solid-state light sources that have a wide range of applications.

LEDs that emit light in the visible spectral region are widely used in displays and in fiber-optic illumination. Infrared LEDs are useful for fiber-optic communications in those systems where the coherence, high power, and high speed of semiconductor lasers are not needed.

Recent breakthroughs have resulted in LEDs of very high performance, in terms of efficiency and brightness, and have extended the spectral range of these high-brightness LEDs to the blue, violet and ultraviolet regions.

As the luminous performance of LEDs exceeds that of traditional incandescent lamps, LEDs become competitive in various lighting applications. Solid-state white light sources also become available by mixing the emission of red, green, and blue LEDs. These advances have created many new possibilities for the applications of LEDs.

Commercially available LEDs today cover the spectral range from the near ultraviolet to the near infrared, with optical wavelengths ranging from about 370 nm to 1.65 μm. These commercial LEDs are made of III-V compound semiconductors.

Blue LEDs based on SiC have been developed, but they have very low efficiencies and are not practically useful. Blue and green LEDs based on II-VI compounds such as ZnTeSe and ZnCdSe have also been developed, but their commercial usefulness is limited.

Organic LEDs based on polymers hold great promise, but they are still in the early stage of development. The main characteristics of LEDs based on III-V semiconductors are listed in Table 13-1.

**Table 13-1**. Basic characteristics of III-V semiconductor LEDsa D, direct gap; I, indirect gapb External quantum efficiency. High, greater than 10%; medium, between 1 and 10%; low, less than 1%c Efficiency varies with wavelength, being higher in the blue and green spectral regions and lower toward both ends of the spectral range

The light output from an LED is the spontaneous emission generated by radiative recombination of electrons and holes in the active region of the diode under forward bias.

From the discussions in the spontaneous emission in semiconductors tutorial, we learn that a semiconductor emits spontaneous photons no matter whether its electron and hole populations are in thermal equilibrium, characterized by a common Fermi level, or in quasi-equilibrium, characterized by separate quasi-Fermi levels.

Spontaneous emission occurs in both direct-gap and indirect-gap semiconductors though a direct-gap semiconductor generally has a much larger radiative recombination rate and thus a much higher spontaneous emission efficiency than an indirect-gap semiconductor.

As can be seen from Table 13-1, many LEDs are made of indirect-gap semiconductors doped with impurities that form isoelectronic centers to improve their luminescence efficiencies. Such LEDs typically have low quantum efficiencies compared with LEDs made of direct-gap semiconductors.

Unlike a laser, an LED emits incoherent and unpolarized spontaneous photons that are not amplified by stimulated emission. Therefore, no optical gain is needed, and the condition for population inversion given in (13-35) [refer to the optical gain in semiconductors tutorial] is not required for the operation of an LED.

No resonant cavity is needed for an LED, either. As a result, the emission from an LED does not have the coherence, or the directionality, of the emission from a laser. Unlike a laser, an LED does not have a threshold, either. It starts emitting light as soon as a forward bias voltage is applied to its junction.

## **LED Efficiency**

The **_power conversion efficiency_** of an LED is defined in the same manner as that of a laser given in (11-89) [refer to the laser power tutorial]:

\\[\tag{13-62}\eta_\text{c}=\frac{P_\text{out}}{P_\text{p}}\\]

where \\(P_\text{out}\\) is the optical output power of the LED and \\(P_\text{p}\\) is the electric pump power supplied by the injection current.

Because an LED has no threshold, its **_external quantum efficiency_** is defined as

\\[\tag{13-63}\eta_\text{e}=\frac{\Phi_\text{out}}{\Phi_\text{p}}\\]

where \\(\Phi_\text{out}\\) is the output photon flux of the LED and \\(\Phi_\text{p}\\) is the pump electron flux.

For an LED that emits at an optical frequency \\(\nu\\), the output photon flux is simply \\(\Phi_\text{out}=P_\text{out}/h\nu\\). If the LED is injected with a current \\(I\\) at a forward bias voltage \\(V\\), then the pump power is \\(P_\text{p}=IV\\) and the pump electron flux is \\(\Phi_\text{e}=I/e=P_\text{p}/(eV)\\), where \\(e\\) is the electronic charge.

Therefore, the power conversion efficiency and the external quantum efficiency have the following relation:

\\[\tag{13-64}\eta_\text{c}=\eta_\text{e}\frac{h\nu}{eV}\\]

In general, \\(\eta_\text{e}\ge\eta_\text{c}\\) because the law of the conservation of energy requires that \\(eV\ge{h\nu}\\).

The power conversion efficiency of a typical LED that emits photons at an energy close to its bandgap energy is approximately equal to, though slightly less than, its external quantum efficiency.

For an LED that emits in the visible spectral region, a **_photometric efficiency_** , or **_luminous efficiency_** , \\(\eta_\text{l}\\), is also introduced to account for the spectral response of the human eye:

\\[\tag{13-65}\eta_\text{l}=K\displaystyle\int\limits{\frac{\text{d}\eta_\text{c}}{\text{d}\lambda}}V(\lambda)\text{d}\lambda\approx{K}\eta_\text{c}V(\lambda_0)\\]

where \\(K=683\text{ lm W}^{-1}\\), known as **_peak efficacy_** , is the **_photometric radiation equivalent_** for photopic vision; \\(V(\lambda)\\) is the **_normalized photopic spectral luminous efficiency_** that characterizes the relative spectral sensitivity of the human eye; and \\(\lambda_0\\) is the peak emission wavelength of the LED.

The luminous function \\(V(\lambda)\\), shown in Figure 13-21, has a peak value of \\(1\\) at the green spectral wavelength of \\(\lambda=555\text{ nm}\\) where the human eye is most responsive, and it drops to a value of \\(0.01\\) at the violet wavelength of \\(\lambda=414\text{ nm}\\) and at the red wavelength of \\(\lambda=687\text{ nm}\\) near the two edges of the visible region.

Therefore, a green LED appears much brighter than a blue or red LED of the same efficiency.

**Figure 13-21**. Photopic luminous efficiency function, \\(V(\lambda)\\), plotted in linear scale (solid curve referring to the left axis) and logarithmic scale (dashed curve referring to the right axis). This plot is based on CIE 1988 updated data.

The luminous efficiency is the luminous flux emitted by an LED per watt of electric pump power. It has the unit of lumen per watt, \\(\text{lm W}^{-1}\\).

The **_luminous flux_** , \\(\Phi_\text{l}\\), which is measured in lumens, of an LED that has an electric pump power of \\(P_\text{p}\\) and an optical output power of \\(P_\text{out}\\) is

\\[\tag{13-66}\Phi_\text{l}=\eta_\text{l}P_\text{p}=\frac{\eta_\text{l}}{\eta_\text{c}}P_\text{out}=KV(\lambda_0)P_\text{out}\\]

**_Example 13-10_**

A transparent substrate (TS) AlGaInP/GaP LED emits at 636 nm. It has an external quantum efficiency of \\(\eta_\text{e}=23.7\%\\) when operated at a forward voltage of 2.02 V with an injection current of 20 mA.

(a) Find its power conversion efficiency under the given operating conditions.

(b) Find its optical output power.

(c) Find its luminous efficiency and luminous flux.

**(a)**

The photon energy at 636 nm is

\\[h\nu=\frac{1239.8}{636}\text{ eV}\\]

using (13-64) with \\(V=2.02\text{ V}\\), we find the following power conversion efficiency:

\\[\eta_\text{c}=\eta_\text{e}\frac{h\nu}{eV}=23.7\%\times\frac{1239.8}{636}\times\frac{1}{2.02}=22.9\%\\]

**(b)**

The electric pump power is \\(P_\text{p}=IV=2.02\times20\text{ mW}=40.4\text{ mW}\\). Thus the optical output power is

\\[P_\text{out}=\eta_\text{c}P_\text{p}=22.9\%\times40.4\text{ mW}=9.25\text{ mW}\\]

**(c)**

From Figure 13-21, we find that \\(V(\lambda)=0.20816\\) for \\(\lambda=636\text{ nm}\\). Therefore, the luminous efficiency of this LED is

\\[\eta_\text{l}=K\eta_\text{c}V(\lambda_0)=683\times22.9\%\times0.20816\text{ lm W}^{-1}=32.6\text{ lm W}^{-1}\\]

The luminous flux is

\\[\Phi_\text{l}=\eta_\text{l}P_\text{p}=32.6\times40.4\times10^{-3}\text{ lm}=1.32\text{ lm}\\]

The external quantum efficiency of an LED is the probability for each charged carrier that is injected into the LED to give rise to one emitted photon. It is determined by three factors, each measured by a characteristic efficiency of its own.

First, the current injected into an LED consists of the diffusion current, which inject carriers into the active region, and other components such as surface recombination current and space-charge recombination current, which do not contribute to the carriers injected into the active region. The fraction of the total injection current that actually contributes to the injected carriers in the active region of an LED is the **_injection efficiency_** , \\(\eta_\text{inj}\\).

Of the carriers injected into the active region, only those that recombine radiatively are responsible for the generation of photons. The probability of radiative recombination is quantified by the radiative efficiency, or the **_internal quantum efficiency_** , \\(\eta_\text{i}\\), which is defined in (13-4) [refer to the radiative recombination in semiconductors tutorial].

Not all of the photons generated by those carriers that recombine radiatively can be extracted out of the LED, however. A large portion of them is trapped inside the LED and is eventually reabsorbed by the LED material. The **_extraction efficiency_** , \\(\eta_\text{t}\\), quantifies the probability that a photon generated in the active region of the LED can successfully escape to the outside and contribute to the optical output of the LED.

The external quantum efficiency can thus be expressed as a product of these three characteristic efficiencies:

\\[\tag{13-67}\eta_\text{e}=\eta_\text{inj}\eta_\text{t}\eta_\text{i}\\]

Clearly, the characteristic efficiencies of all three factors have to be maximized in order for an LED to have a high external quantum efficiency.

To have a high injection efficiency, surface recombination and carrier leakage have to be avoided by proper choice of the doping concentration and the thickness of each layer in the LED and by careful design of the electrical contacts.

The injection efficiency is generally not a limiting factor for the external quantum efficiency of an LED, however, because it can easily be made higher than \\(80\%\\) for a well-designed LED.

The internal quantum efficiency is normally very high for direct-gap semiconductors but it low for indirect-gap semiconductors. Because the radiative recombination rate increases with carrier concentration, according to (13-5) [refer to the radiative recombination in semiconductors tutorial], the use of a DH (double heterostructure) geometry as discussed in the semiconductor junction structures tutorial can significantly improve the internal quantum efficiency of an LED by providing effective carrier confinement in the active layer.

For a direct-gap semiconductor LED operating at a properly chosen injection current level, the radiative efficiency can be close to unity and, like the injection efficiency, is not a limiting factor for the external quantum efficiency of the LED, either.

For an indirect-gap semiconductor LED, however, the radiative efficiency is a limiting factor though it can be improved with the doping of isoelectronic centers. For example, the radiative efficiency for GaP : Zn,O is about \\(30\%\\), but that for GaP : N is only about \\(3\%\\).

The most significant limiting factor is normally the extraction efficiency, which depends on the details of the LED structure discussed below. A typical LED can have an extraction efficiency between \\(3\\) and \\(30\%\\), depending on the geometry and the material of the LED.

Combining all three factors, the external quantum efficiency of an LED ranges from lower than \\(1\%\\) to higher than \\(30\%\\).

## **LED Construction**

The construction of an LED is determined largely by the consideration of maximizing the extraction efficiency of the LED.

For LEDs used in fiber-optic applications, including the infrared LEDs used in optical communications, the ultimate efficiency that counts includes not only the external quantum efficiency but also the coupling efficiency of the emission into the fiber.

Therefore, the coupling efficiency to an optical fiber is also a factor to be considered in the construction of an LED that is used in a fiber-optic application.

Because spontaneous emission radiates in all directions, a properly designed surface-emitting LED, which allows emission output in many different directions, generally has a much higher extraction efficiency than a comparable edge-emitting LED, which limits its emission output to a narrow angular spread in only one direction.

However, because of the optical waveguiding effect in an edge-emitting device discussed in the lateral structures tutorial, the emission of an edge-emitting LED is much more collimated, thus allowing for much more efficient direct coupling to an optical fiber, than that of a surface-emitting LED.

For these reasons, conventional LEDs are surface-emitting devices, but edge-emitting LEDs are often used in fiber-optic applications.

The limitation on the extraction efficiency of a surface-emitting LED is caused by the absorption of the LED material and the Fresnel reflection between the high-index semiconductor and the low-index air.

If nothing is done to optimize the structure of the LED, the extraction efficiency is less than \\(2\%\\). Clearly, this is an important factor to be considered, and there is plenty of room for it to be improved through careful design of the LED structure.

The spontaneous photons generated in the active region of an LED are emitted isotropically in all directions, but only those photons that reach a surface of the LED at angles of incidence smaller than the critical angle, \\(\theta_\text{c}\\), can be transmitted through that surface.

This critical angle defines an escape cone of a solid angle \\(\Omega_\text{esc}\\) with respect to each surface of the LED. Because spontaneous emission is distributed isotropically over the \\(4\pi\\) solid angle, the probability for emitted photons to escape through a given surface is

\\[\tag{13-68}\eta_\text{esc}=\frac{\Omega_\text{esc}}{4\pi}T\\]

where \\(T\\) is the transmittance of the surface.

For a flat interface between the LED material of a refractive index \\(n_1\\) and the ambient medium of a refractive index \\(n_2\\), with \\(n_1\gt{n}_2\\), \\(\Omega_\text{esc}=2\pi(1-\cos\theta_\text{c})\\) and

\\[\tag{13-69}\eta_\text{esc}\approx\frac{n_2^3}{n_1(n_1+n_2)^2}\\]

where we have approximated the transmittance \\(T\\) with that of normal incidence.

_**Example 13-11**_

In this example, we calculate the escape efficiency of an AlGaInP/GaP LED like the one considered in Example 13-10. The refractive index of AlGaInP is \\(n=3.4\\). The AlGaInP LED surface is exposed directly to the air without any treatment.

(a) Find the critical angle \\(\theta_\text{c}\\) and the escape solid angle \\(\Omega_\text{esc}\\) for the interface between AlGaInP and air. What is the transmittance of this surface?

(b) Find the escape efficiency \\(\eta_\text{esc}\\) using the relation in (13-68).

(c) Find \\(\eta_\text{esc}\\) usign the approximation given in (13-69). Compare the result with that obtained in (b).

(d) How does this escape efficiency compare with the external quantum efficiency of the AlGaInP/GaP LED described in Example 13-10?

**(a)**

With \\(n_1=3.4\\) and \\(n_2=1\\), we find that

\\[\theta_\text{c}=\sin^{-1}\frac{1}{3.4}=17.1^\circ\\]

We then find that

\\[\Omega_\text{esc}=2\pi(1-\cos17.1^\circ)=0.0884\pi\\]

The transmittance

\\[T=\frac{4n_1n_2}{(n_1+n_2)^2}=\frac{4\times3.4\times1}{(3.4+1)^2}=0.70\\]

**(b)**

Using the parameters obtained in (a) for (13-68), we find that

\\[\eta_\text{esc}=\frac{\Omega_\text{esc}}{4\pi}T=\frac{0.0884\pi}{4\pi}\times0.70=1.55\%\\]

**(c)**

Using \\(n_1=3.4\\) and \\(n_2=1\\) for (13-69), we find that

\\[\eta_\text{esc}\approx\frac{n_2^3}{n_1(n_1+n_2)^2}=\frac{1^3}{3.4\times(3.4+1)^2}=1.52\%\\]

This approximate result of \\(\eta_\text{esc}=1.52\%\\) is \\(98\%\\) of the result of \\(\eta_\text{esc}=1.55\%\\) obtained in (b). Therefore, the convenient relation given in (13-69) is a very accurate approximation.

**(d)**

The transparent substrate (TS) AlGaInP/GaP LED described in Example 13-10 has an external quantum efficiency of \\(\eta_\text{e}=23.7\%\\), which is more than 15 times the escape efficiency of \\(\eta_\text{esc}=1.55\%\\) found for the AlGaInP surface to the air.

In the face of the small value of \\(\eta_\text{esc}\\), such a high external quantum efficiency looks quite impossible, but it is real. Many techniques can be applied to realize such a high external quantum efficiency. The basic concepts of such techniques are described in the following text.

In a surface-emitting LED, a transparent window layer grown on top of the active layer allows light emission from the top surface. The device can have either an absorbing substrate (AS) or a transparent substrate (TS).

For an AS LED with a thin window layer of a thickness less than approximately 10 μm, as shown in Figure 13-22(a), we find that \\(\eta_\text{t}\le\eta_\text{esc}\\) because only those photons that are emitted directly toward the top surface do not get totally absorbed by the substrate before reaching a surface.

If an AS LED has a thick window layer, as shown in Figure 13-22(b), half of the photons emitted toward each side surface of the LED chip can reach the side surface without being absorbed by the substrate, thus increasing the extraction efficiency to \\(\eta_\text{t}\le3\eta_\text{esc}\\).

In a TS LED with a thick window, as shown in Figure 13-22(c), it is possible for photons emitted in any direction to reach a surface, and the theoretical limit of the extraction efficiency is further increased to \\(\eta_\text{t}\le6\eta_\text{esc}\\).

A highly reflecting mirror surface is normally applied to the bottom surface of a TS LED to reflect light back to the top window surface.

By examining Table 13-1, we see that among the AlGaInP LEDs, those that have a transparent GaP substrate have higher efficiencies than those that have an absorbing GaAs substrate. The InGaN/sapphire LEDs also have transparent substrates, which is part of the reason for their high efficiencies.

**Figure 13-22**. (a) Surface-emitting LED with a thin window and an absorbing substrate; only one top cone. (b) Surface-emitting LED with a thick window and an absorbing substrate; top cone plus four half-side cones. (c) Surface-emitting LED with a thick window and a transparent substrate; top and bottom cones plus four side cones.

From (13-69), we find that \\(\eta_\text{esc}\\) for a flat semiconductor/air interface has a very small value. In Example 13-11 with \\(n_1=3.4\\) for AlGaInP and \\(n_2=1\\) for air, we get \\(\eta_\text{esc}\approx1.55\%\\). This value is exceedingly small.

Even for a TS LED with a thick window, this value only permits a maximum extraction efficiency of less than \\(10\%\\). Something has to be done to increase the value of \\(\eta_\text{esc}\\) if the extraction efficiency is to be increased further.

Clearly from (13-68), the value of \\(\eta_\text{esc}\\) can be increased by increasing either the value of \\(\Omega_\text{esc}\\), thus allowing photons reaching the surface at large angles of incidence to be transmitted through the surface, or the value of \\(T\\), thus allowing a higher probability of transmittance for a photon striking the surface at a given angle within the cone of \\(\Omega_\text{esc}\\), or both. A few solutions have been developed for this purpose.

One method to increase the value of \\(\Omega_\text{esc}\\) significantly, but not that of \\(T\\), is to shape the surface of the top window layer of an LED into a hemisphere with a radius much larger than the thickness of the active layer so that all spontaneous photons radiating toward this spherical surface come close to normal incidence, thus completely avoiding total reflection.

This approach, however, requires polishing the semiconductor into a spherical surface and is therefore expensive and not practically useful.

A second technique to increase the effective value of \\(\Omega_\text{esc}\\) is to roughen the surfaces of the LED, thus randomizing the angles of incidence to reduce the probability of total reflection. The needed microscopic surface textures can be easily created by chemical etching or by coating with small polymer spheres. A factor of 2 increase in \\(\eta_\text{esc}\\) can be accomplished by proper surface texturing; therefore, this technique is practical.

Another practical solution is to encapsulate the LED chip in a transparent plastic epoxy, which normally has a refractive index close to 1.5. In this approach, both the value of \\(\Omega_\text{esc}\\) and that of \\(T\\) are increased because \\(n_2\\) is increased. For \\(n_1=3.4\\) and \\(n_2=1.5\\), we find that \\(\eta_\text{esc}\approx4.2\%\\), an improvement of nearly three-fold over that of an LED without encapsulation.

The last two techniques can be combined by encapsulating an LED that has textured surfaces to improve \\(\eta_\text{esc}\\) further. Applying these solutions to a TS LED properly, a high extraction efficiency of \\(\eta_\text{t}\gt30\%\\) is possible.

Figure 13-23 shows the construction of a surface-emitting LED with plastic encapsulation that is shaped into a spherical dome lens. When a TS LED, such as an AlGaInP/GaP LED, is assembled in this package, the LED can be placed in a miniature dish-shaped reflector that is coined into the top of the cathode post and is used to direct the emission from the side surfaces of the LED toward the dome lens.

The shape and size of the plastic dome control the radiation pattern of the LED. Various radiation patterns for different applications can be obtained by tailoring the shape and size of the plastic encapsulation. In addition to the design of the spherical dome lens, aspherical dome lenses and rectangular packages are also used.

**Figure 13-23**. Construction of an LED encapsulated in plastic epoxy with a spherical dome lens. If a TS LED is used, it can be placed in a dish-shaped reflector to send side emission to the dome lens.

LEDs do not couple efficiently into single-mode fibers because the incoherent emission of an LED has a large divergence. Therefore, only multimode fibers are used in fiber-optic applications of LEDs.

When a surface-emitting LED is used, special arrangement has to be made to bring the fiber tip into close proximity to the emitting active region of the LED for coupling of its emission into the fiber. A famous design is the Burrus type shown in Figure 13-24. Other designs feature a microlens on the fiber tip or on the emitting surface of the LED to facilitate efficient coupling.

**Figure 13-24**. Surface-emitting Burrus-type LED for fiber-optic applications.

An edge-emitting LED has a structure similar to that of an edge-emitting semiconductor laser. The LED is prevented from laser oscillation by adding antireflection coating on the emitting facet or by leaving an unpumped absorbing section in the structure. The nonemitting facet is normally coated to be highly reflective to enhance the output from the emitting facet.

The emission of an edge-emitting LED is much more collimated than that of a surface-emitting LED. As a result, its radiance is typically about 10 times higher than that of a surface emitting LED.

Figure 13-25 shows the structure of a stripe-geometry edge-emitting LED. Its emission has a vertical spread of about \\(30^\circ\\) and a horizontal spread of about \\(120^\circ\\).

With its emission radiating from a very small area on the emitting facet, an edge-emitting LED allows a fiber easy access to its emission output for a good coupling efficiency to the fiber.

**Figure 13-25**. Stripe-geometry edge-emitting LED.

## **Light-Current Characteristics**

An LED is basically a p-n, P-n, or p-N junction diode though it may have a sophisticated DH structure for improved performance, as discussed in the preceding section. Therefore, the general electrical properties of an LED are those of a semiconductor junction diode described in the semiconductor junctions tutorial with the current-voltage characteristics shown in Figure 12-12.

The excess carriers in an LED that has an active layer of a thickness \\(d\\) much smaller than the diffusion length of the injected minority carriers can be considered uniformly distributed in the active region with a uniform density \\(N\\). This is normally true for a DH device with a thin active layer.

In this situation, the temporal variation of the carrier density in response to the variation in the injection current can be expressed as

\\[\tag{13-70}\frac{\text{d}N}{\text{d}t}=\frac{J}{ed}-\frac{N}{\tau_\text{s}}\\]

where \\(e\\) is the electronic charge, \\(\tau_\text{s}\\) is the spontaneous carrier lifetime defined in (13-2) [refer to the radiative recombination in semiconductors tutorial], and \\(J\\) is the injection current density in the active region.

Taking into consideration the carrier injection efficiency, the current density \\(J\\) that actually contributes to carrier injection is related to the total current supplied to the device as follows:

\\[\tag{13-71}J=\eta_\text{inj}\frac{I}{\mathcal{A}}\\]

where \\(\eta_\text{inj}\\) is the carrier injection efficiency defined earlier and \\(\mathcal{A}\\) is the area of the junction.

As a light-emitting device that is pumped by current injection, a very important property of an LED is its output optical power as a function of the injection current, known as the **_light-current characteristics_** , or simply as the **_\\(L-I\\) characteristics_** , or the **_power-current characteristics_** , or simply as the **_\\(P-I\\) characteristics_**.

The steady-state solution with \\(\text{d}N/\text{d}t=0\\) for (13-70) results in the following ideal power-current relation for an LED:

\\[\tag{13-72}P_\text{out}=\eta_\text{e}\frac{h\nu}{e}I\\]

which indicates that the output power of an LED increases linearly with the injection current.

The \\(L-I\\) characteristics of a typical LED, shown in Figure 13-26, are not exactly linear throughout the entire range of operation, however.

These characteristics have several important features that distinguish an LED from a laser.

First, there is no threshold in the \\(L-I\\) characteristics of an LED, indicating that an LED is turned on and starts emitting light once it is forward biased with any amount of injection current.

The \\(L-I\\) curve of an LED is indeed quite linear, particularly at moderate current levels, as indicated by (13-72). This linearity is useful for analog modulation of an LED.

Nonlinearities in the \\(L-I\\) relationship are usually found at very low and very high current levels. Because of these nonlinearities, the efficiency of an LED changes as the injection current is varied. The efficiency starts low at low injection levels, increases sharply with an increasing injection current, but saturates or even decreases at high injection levels.

**Figure 13-26**. Typical light-current characteristics of an LED. The solid curve shows the output power, \\(P_\text{out}\\), and the dashed curve shows the power conversion efficiency, \\(\eta_\text{c}\\).

_**Example 13-12**_

The optical output power and luminous flux of an LED at a given injection current level can be found without knowing the bias voltage if the external quantum efficiency is known. Find them for the AlGaInP/GaP LED described in Example 13-10 without using the knowledge of its bias voltage. Compare the results with those found in Example 13-10.

From Example 13-10, we have \\(\lambda=636\text{ nm}\\), \\(\eta_\text{e}=23.7\%\\), and \\(I=20\text{ mA}\\). First, we note that

\\[\frac{h\nu}{e}=\frac{1239.8\text{ eV}}{636}\times\frac{1}{e}=\frac{1239.8}{636}\text{ V}\\]

Then, when we use (13-72) to find the optical output power, we simply have

\\[P_\text{out}=\eta_\text{e}\frac{h\nu}{e}I=23.7\%\times\frac{1239.8}{636}\times20\text{ mW}=9.25\text{ mW}\\]

With \\(K=683\text{ lm W}^{-1}\\) and \\(V(\lambda_0)=0.20816\\) as found in Example 13-10, we find from (13-66) that

\\[\Phi_\text{l}=KV(\lambda_0)P_\text{out}=683\times0.20816\times9.25\times10^{-3}\text{ lm}=1.32\text{ lm}\\]

Compared with the results obtained in Example 13-10, we find exactly the same values for both \\(P_\text{out}\\) and \\(\Phi_\text{l}\\), as expected.

## **Spectral Characteristics**

The spectral characteristics of an LED include the emission wavelength, the spectral width, and the spectral shape.

The emission wavelength of a direct-gap LED is determined by the bandgap of the active layer. Because of the **_band-filling effect_** of the injected electrons and holes taking up the states near the edges of the conduction and valence bands, respectively, the peak emission wavelength tends to be somewhat shorter than \\(\lambda_\text{g}=hc/E_\text{g}\\), corresponding to a photon energy somewhat larger than the bandgap energy.

However, if the active layer is heavily doped, the formation of bandtail states can lead to a long emission wavelength corresponding to a photon energy smaller than the bandgap energy, as discussed in the introduction to semiconductors tutorial.

For an indirect-gap LED doped with isoelectronic impurities, the emission wavelength is longer than \\(\lambda_\text{g}\\) with a photon energy smaller than the bandgap energy, as is also discussed in the introduction to semiconductors tutorial and illustrated in Figure 12-1.

The peak emission wavelength of an LED varies with injection current and temperature. Because the bandgap of a III-V semiconductor normally decreases with increasing temperature, the peak emission wavelength of an LED becomes longer as the operating temperature increases. The rate of change depends on the specific semiconductor material of the LED.

When the injection current increases, the band-filling effect caused by the corresponding increase in the concentration of the injected carriers lead to an increase in the emitted photon energy and a corresponding reduction in the peak emission wavelength. This effect is often abated by the shrinkage of the bandgap due to heating of the junction that accompanies the increase in injection current.

The spectral width and shape of the emission are intrinsically defined by the spontaneous emission spectrum in (13-42) [refer to the spontaneous emission in semiconductors tutorial]. The emission spectra of an LED, however, are often further complicated by frequency-dependent absorption and scattering by impurities and other materials, which have different bandgaps, in the layered structure of an LED.

The spectral width in terms of the photon energy is approximately \\(h\Delta\nu=3k_\text{B}T\\), but it can range between \\(2k_\text{B}T\\) and \\(4k_\text{B}T\\).

At room temperature, the spectral width of an LED is approximately 80 meV, but it can be as narrow as 50 meV or as broad as 100 meV in some devices.

In terms of optical wavelength, the spectral width \\(\Delta\lambda\\) ranges from approximately 20 nm for InGaN LEDs emitting short-wavelength ultraviolet or blue light to the order of 100 nm for InGaAsP LEDs with long-wavelength infrared emission.

The spectral width of an LED normally increases with both temperature and injection current.

Because an LED emits spontaneous radiation without an optical cavity, the longitudinal and transverse mode structures that are characteristic of a laser spectrum do not exist in the emission spectrum of an LED. Figure 13-27 shows a representative emission spectrum of an LED.

**Figure 13-27**. Representative emission spectrum of an LED.

## **Modulation Characteristics**

An LED can be directly modulated by applying the modulation signal to the injection current, an approach known as direct current-modulation.

For high-speed applications, a large modulation bandwidth is desired. There are two factors that limit the modulation bandwidth of an LED: the junction capacitance, \\(C_\text{j}\\), and the diffusion capacitance, \\(C_\text{d}\\), both discussed in the semiconductor junctions tutorial.

Because an LED is operated under a forward bias, the diffusion capacitance is the dominating factor for its frequency response. The diffusion capacitance is a function of the carrier lifetime \\(\tau_\text{s}\\) because it is associated with the injection or removal of carriers in the diffusion region in response to the modulation on the injection current.

Therefore, the intrinsic speed of an LED is primarily determined by the lifetime of the injected carriers in the active region.

For an LED that is biased at a DC injection current level \\(I_0\\) and is modulated at a frequency \\(\Omega=2\pi{f}\\) with a **_modulation index \\(m\\)_** , we can write the total time-dependent current that is injected to the LED as

\\[\tag{13-73}I(t)=I_0+I_1(t)=I_0(1+m\cos\Omega{t})\\]

In the linear response regime under the condition that \\(m\ll1\\), the output optical power of the LED in response to this modulation can be expressed as

\\[\tag{13-74}P(t)=P_0+P_1(t)=P_0[1+|r|\cos(\Omega{t}-\varphi)]\\]

where \\(P_0\\) is the constant optical output power at the bias current level of \\(I_0\\), \\(|r|\\) is the magnitude of the response to the modulation, and \\(\varphi\\) is the phase delay of the response to the modulation signal.

For an LED modulated in the linear response regime, the complex response as a function of modulation frequency \\(\Omega\\) is

\\[\tag{13-75}r(\Omega)=|r|\text{e}^{\text{i}\varphi}=\frac{m}{1-\text{i}\Omega\tau_\text{s}}\\]

The frequency response and modulation bandwidth of an LED are usually measured in terms of the electrical power spectrum of a broadband, high-speed photodetector that converts the optical output of the LED into an electric current.

In the linear operating regime of the detector, the detector current is linearly proportional to the optical power of the LED. Therefore, the electrical power spectrum of the detector output is proportional to \\(|r|^2\\):

\\[\tag{13-76}R(f)=|r(f)|^2=\frac{m^2}{1+4\pi^2f^2\tau_\text{s}^2}\\]

which has a 3-dB modulation bandwidth of

\\[\tag{13-77}f_\text{3dB}=\frac{1}{2\pi\tau_\text{s}}\\]

as shown in Figure 13-28.

**Figure 13-28**. Normalized current-modulation frequency response of an LED measured in terms of the electrical power spectrum of a photodetector. The spontaneous carrier lifetime is taken to be \\(\tau_\text{s}=10\text{ ns}\\), as in Example 13-13, for this plot.

The spontaneous carrier lifetime \\(\tau_\text{s}\\) is normally on the order of a few hundred to 1 ns for an LED. Therefore, the modulation bandwidth of an LED is typically in the range of a few megahertz to a few hundred megahertz.

A modulation bandwidth up to 1 GHz can be obtained with a reduction in the internal quantum efficiency of the LED by reducing the carrier lifetime to the sub-nanosecond range.

Aside from this intrinsic response speed determined by the carrier lifetime, the modulation bandwidth of an LED can be further limited by parasitic effects from its electrical contacts and packaging, as well as from its driving circuitry.

At an injection current \\(I\\), the output optical power and the small-signal modulation bandwidth of an LED have the following **_power-bandwidth product_** :

\\[\tag{13-78}P_\text{out}f_\text{3dB}=\eta_\text{e}\frac{h\nu}{e}\frac{I}{2\pi\tau_\text{s}}=\eta_\text{inj}\eta_\text{t}\eta_\text{i}\frac{h\nu}{e}\frac{I}{2\pi\tau_\text{rad}}\\]

where \\(h\nu\\) is the photon energy of the LED emission.

Therefore, at a given injection level, the modulation bandwidth of an LED is inversely proportional to its output power. A high-power LED tends to have a low speed, and vice versa.

_**Example 13-13**_

The AlGaInP/GaP LED described in Example 13-10 has a spontaneous carrier lifetime of \\(\tau_\text{s}=10\text{ ns}\\). Find its 3-dB modulation bandwidth and its power-bandwidth product under the operating conditions described in Example 13-10.

Using (13-77) for \\(\tau_\text{s}=10\text{ ns}\\), the 3-dB modulation bandwidth of the LED is easily found:

\\[f_\text{3dB}=\frac{1}{2\pi\tau_\text{s}}=\frac{1}{2\pi\times10\times10^{-9}}\text{ Hz}=15.9\text{ MHz}\\]

Because the output power is \\(P_\text{out}=9.25\text{ mW}\\) according to Example 13-10, the power-bandwidth product is

\\[P_\text{out}f_\text{3dB}=9.25\text{ mW}\times15.9\text{ MHz}=147\text{ KW Hz}^{-1}\\]