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Wide band gap semiconductors have a band gap between 2 and 7 electron volts \((\mathrm{eV})\), where \(1 \mathrm{eV}=96.485 \mathrm{~kJ} / \mathrm{mol}\). The wide band-gap semiconductor GaN, used to construct the laser in Bluray DVD players, has a band gap of \(3.44 \mathrm{eV}\). The material in the laser, \(\mathrm{Ga}_{x} \operatorname{In}_{1-x} \mathrm{~N}\), has some indium substituted for gallium. (a) What wavelength of light (in \(\mathrm{nm}\) ) would GaN emit, based on its band gap? (b) If the light from the device is blue, does partial substitution of indium for gallium increase or decrease the band gap of \(\mathrm{Ga}_{x} \mathrm{In}_{1-x} \mathrm{~N}\) compared to \(\mathrm{GaN}\) ?

Step by step solution

01

Understanding the Energy-Wavelength Relationship

The energy of a photon can be related to its wavelength using the equation \( E = \frac{hc}{\lambda} \), where \( E \) is the energy in joules, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J} \cdot \text{s}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength in meters. We will use this to find the wavelength of light emitted by GaN.

02

Convert Energy from Electron Volts to Joules

First, we need to convert the band gap energy from electron volts to joules. Given that 1 eV = \( 1.602 \times 10^{-19} \) J, the energy of GaN is \( 3.44 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 5.51 \times 10^{-19} \text{ J} \).

03

Solve for Wavelength

With the photon energy in joules, we use \( E = \frac{hc}{\lambda} \) to solve for \( \lambda \):\[ \lambda = \frac{hc}{E} = \frac{(6.626 \times 10^{-34} \text{ J} \cdot \text{s})(3.00 \times 10^8 \text{ m/s})}{5.51 \times 10^{-19} \text{ J}} \approx 3.60 \times 10^{-7} \text{ m} \].Convert this to nanometers: \( 3.60 \times 10^{-7} \text{ m} = 360 \text{ nm} \).

04

Analyzing Indium Substitution Effect

The light emitted by the GaN with indium substitution is blue, which has a longer wavelength than UV light produced by GaN. Longer wavelength corresponds to lower energy. Since energy is inversely proportional to the band gap, the band gap decreases when indium is substituted for gallium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Photon Energy-Wavelength Relationship

The relationship between the energy of a photon and its wavelength is a fundamental concept in understanding how semiconductors like Gallium Nitride (GaN) emit light. According to the formula \( E = \frac{hc}{\lambda} \), where:

• \( E \) represents the energy of the photon in joules (J).
• \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).
• \( c \) is the speed of light, about \( 3.00 \times 10^8 \text{ m/s} \).
• \( \lambda \) is the wavelength of the photon in meters.

This formula shows that photon energy is inversely proportional to its wavelength. This means that higher energy photons have shorter wavelengths, which is especially relevant when considering the emission properties of materials like GaN used in Blu-ray technology.

Energy Conversion (eV to J)

Converting energy from electron volts (eV) to joules (J) is pivotal in quantifying and understanding the energy properties of semiconductors. One electron volt is defined as \( 1.602 \times 10^{-19} \) joules. Therefore, to convert an energy given in electron volts to joules, you multiply the energy value by this conversion factor.For instance, GaN has a band gap of 3.44 eV. Converting this value:\[ 3.44 \text{ eV} \times 1.602 \times 10^{-19} \text{ J/eV} = 5.51 \times 10^{-19} \text{ J}\]This calculation is crucial for determining the corresponding wavelength of light emitted, as it allows us to use the energy-wavelength relationship in the correct units.

Indium Substitution Effects

The addition of indium in a GaN substrate results in an alloy known as \( \text{Ga}_x \text{In}_{1-x} \text{N} \), affecting the band gap and consequently the emissions of the material. When indium partly substitutes gallium in GaN:

• It introduces new electron states, slightly altering the energy levels.
• This results in a decrease in the band gap energy because the resulting alloy's structure is different.
• The energy difference reflects in a longer emission wavelength and lower photon energy.

The change makes the emitted light appear blue instead of the ultraviolet produced by pure GaN, showing how alloying allows for the tuning of optical properties in semiconductors.

Gallium Nitride (GaN) Band Gap

The band gap of a semiconductor like Gallium Nitride (GaN) refers to the energy difference between the valence band and the conduction band. It is crucial in determining the electronic and optical properties of the material. GaN has a wide band gap of 3.44 eV, which:

• Makes it suitable for high-power applications, like laser diodes used in Blu-ray players.
• Allows it to emit light in the ultraviolet to visible spectrum.
• Contributes to its utility in lighting and optoelectronics by ensuring efficiency and thermal stability.

Understanding the band gap is essential for designing devices that use these materials to their full potential and allows customization of the material for specific applications through processes like alloying.