Problem 79 - Chemistry The Central Science In Si Units

Question

GaAs and GaP make solid solutions that have the same crystal structure as the parent materials, with As and P randomly distributed throughout the crystal. \(\mathrm{GaP}_{x} \mathrm{As}_{1-x}\) exists for any value of \(x\). If we assume that the band gap varies linearly with composition between \(x=0\) and \(x=1\), estimate the band gap for \(\mathrm{GaP}_{0.8} \mathrm{As}_{0.2}\). (GaAs and GaP band gaps are \(1.43 \mathrm{eV}\) and \(2.26 \mathrm{eV}\), respectively.) What wavelength of light does this correspond to?

Step-by-Step Solution

Answer

The band gap for GaP_{0.8}As_{0.2} is 2.094 eV, which corresponds to a wavelength of approximately 593 nm (red light).

Step 1: Identify Given Variables

We're told that GaAs has a band gap of \(1.43 \text{ eV}\) and GaP has a band gap of \(2.26 \text{ eV}\). The composition we need to examine is \(\text{GaP}_{0.8}\text{As}_{0.2}\), which means \(x = 0.8\).

Step 2: Apply Linear Variation Formula

Since the band gap varies linearly with the composition \(x\), use the formula for linear interpolation between the two band gaps. The band gap \(E_g\) for \(\text{GaP}_x\text{As}_{1-x}\) is given by:\[E_g = x \cdot E_{g,\text{GaP}} + (1-x) \cdot E_{g,\text{GaAs}}\]where \(E_{g,\text{GaP}} = 2.26 \text{ eV}\) and \(E_{g,\text{GaAs}} = 1.43 \text{ eV}\).

Step 3: Calculate Band Gap

Substitute the values into the formula:\[E_g = 0.8 \cdot 2.26 + 0.2 \cdot 1.43\]Calculate each term:\[0.8 \cdot 2.26 = 1.808 \quad \text{and} \quad 0.2 \cdot 1.43 = 0.286\]Add the results:\[E_g = 1.808 + 0.286 = 2.094 \text{ eV}\]

Step 4: Convert Band Gap to Wavelength

To convert the band gap energy to wavelength, use the formula:\[\lambda = \frac{hc}{E_g}\]where \(h = 4.135667696 \times 10^{-15} \text{ eV} \cdot \text{s}\) (Planck's constant) and \(c = 3.00 \times 10^8 \text{ m/s}\) (speed of light).Calculate the wavelength:\[\lambda = \frac{4.135667696 \times 10^{-15} \times 3.00 \times 10^8}{2.094}\]

Step 5: Solve for Wavelength

Perform the calculation for the wavelength:\[\lambda \approx 5.93 \times 10^{-7} \text{ m} \approx 593 \text{ nm}\]This falls within the visible spectrum of light, appearing as a wavelength corresponding to red light.

Key Concepts

GaAs

Gallium Arsenide, commonly referred to as GaAs, is a compound semiconductor composed of gallium (Ga) and arsenic (As). GaAs is known for its direct band gap, which means that electrons can directly emit light when transitioning between energy levels. This property makes GaAs extremely useful in optoelectronic applications, such as light-emitting diodes (LEDs) and laser diodes.
GaAs also plays a significant role in high-frequency applications due to its high electron mobility compared to silicon. This means that electrons can move quickly through the material, making GaAs a preferred choice for microwave and millimeter-wave devices. For instance:

• It is used in satellite and radar communications.
• It benefits high-speed integrated circuits.

Understanding GaAs and its properties, like its band gap of 1.43 eV, is crucial for estimating the properties of more complex materials such as GaPAs alloys.

GaP

Gallium Phosphide, or GaP, is another important semiconductor material composed of gallium (Ga) and phosphorus (P). Unlike GaAs, GaP exhibits an indirect band gap, meaning that the transition of electrons between the conduction and valence bands requires a change in momentum with the assistance of phonons.
This indirect band gap is larger, measuring 2.26 eV. Although GaP is less efficient in light-emitting applications compared to GaAs, it is still valuable in certain contexts. GaP is often used in:

• Green and red LEDs due to its semiconducting properties.
• Optoelectronic devices where long-wavelength infrared signals are necessary.

When combined with GaAs to form GaPAs alloys, GaP offers unique properties by influencing the band gap and the electronic characteristics of the material.

Linear Interpolation

Linear interpolation is a method used to estimate values between two known points. In the context of semiconductors like GaP and GaAs, it can be applied to predict changes in band gap as the composition of elements within the alloy changes.
For GaAs and GaP, the band gap shifts linearly between their individual band gaps when forming alloys. The formula used is:
\[ E_g = x \, \cdot \, E_{g,\text{GaP}} + (1-x) \, \cdot \, E_{g,\text{GaAs}} \]
where:

• \(E_g\) is the desired band gap.
• \(x\) is the proportion of GaP in the alloy \(\text{GaP}_x\text{As}_{1-x}\).
• \(E_{g,\text{GaP}}\) and \(E_{g,\text{GaAs}}\) are the band gaps of GaP and GaAs, respectively.

This formula is essential for determining the energy properties of semiconductor materials tailored for specific applications.

Wavelength Conversion

Wavelength conversion involves translating the energy of a semiconductor's band gap into light wavelength, enabling the determination of what color or type of light these materials can emit or absorb.
The band gap energy \(E_g\), measured in electron volts (eV), can be converted to wavelength \(\lambda\) using the formula:
\[ \lambda = \frac{hc}{E_g} \]
where:

• \(h\) is Planck’s constant \(4.135667696 \times 10^{-15} \text{ eV} \cdot \text{s}\).
• \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\).

This conversion is critical for understanding which parts of the electromagnetic spectrum a particular material can interact with, and hence what potential applications it might have in photonics and optoelectronics.

Visible Spectrum

The visible spectrum is the portion of the electromagnetic spectrum that can be detected by the human eye, typically ranging from approximately 380 nm to 750 nm.
Light in this range includes red, orange, yellow, green, blue, indigo, and violet. The precise wavelength corresponding to each color depends on the material and its energy band gap.

• Red light, for example, is typically around 620-750 nm.
• Materials like GaAs and GaP, when alloyed, can be engineered to emit light at specific points within this spectrum.

Calculating the emission wavelength based on band gap energy helps guide the development of:

Lighting applications like LEDs.

Display technologies.

The visible spectrum is a core concept in designing and utilizing semiconductor materials for consumer electronics and communication systems.