Question

The semiconductor GaP has a band gap of \(2.2 \mathrm{eV}\). Green LEDs are made from pure GaP. What wavelength of light would be emitted from an LED made from GaP?

Short answer

The wavelength of light emitted by a GaP green LED can be calculated using the energy-wavelength formula: \( λ = \frac{h * c}{E} \). Given the band gap of GaP is 2.2 eV, we convert it to joules and plug all the values into the formula. This yields a wavelength of approximately \( 5.65 \times 10^{-7} \mathrm{m} \), which is equivalent to 565 nm.

Step-by-step solution

Convert the band gap energy to joules

We have to convert the given energy value from electron volts (eV) to joules (J) using the relation: 1 eV = 1.602 x 10^-19 J.

Relate energy and wavelength

We can relate the energy and wavelength of light using the formula, E = h * c / λ, where E is the energy, h is the Planck's constant (6.626 x 10^-34 Js), c is the speed of light (3 x 10^8 m/s), and λ is the wavelength.

Calculate the wavelength

Now we will use the energy-wavelength formula to determine the wavelength: \( λ = \frac{h * c}{E} \)

Plug the values and compute

We have all the values we need, let's plug them in and calculate the wavelength: \( λ = \frac{(6.626 \times 10^{-34} Js) \times (3 \times 10^8 m/s)}{(2.2eV) \times (1.602 \times 10^{-19} J/eV)} \)

Simplify and find the final wavelength

Now we will simplify the equation and find the final value of the wavelength: \( λ \approx 5.65 \times 10^{-7} m \)

Convert the wavelength to nanometers

To make the value more convenient to work with, let's convert the wavelength from meters to nanometers (1 meter = 1 x 10^9 nanometers): \( λ \approx 5.65 \times 10^{-7} m \times \frac{1 \times 10^9 nm}{1 m} = 565 nm \) So the wavelength of light emitted by a GaP green LED is approximately 565 nm.

Key concepts

Band Gap Energy

Semiconductors like Gallium Phosphide (GaP) have a specific band gap energy, which is the energy difference between the top of the valence band and the bottom of the conduction band. It is this band gap energy that determines the electrical and optical properties of the semiconductor.
GaP has a band gap of 2.2 electron volts (eV). This might seem abstract, but think of it as the energy required to excite an electron to the point where it can jump from the valence band to the conduction band. When an electron falls back to its original state, it releases energy in the form of light.
In the case of a light-emitting diode (LED) made from GaP, the band gap energy corresponds to the energy of the photons emitted as light, making this concept crucial for designing LEDs that emit specific colors.

Wavelength Calculation

To find out the wavelength of light emitted by a LED, we need to use the relationship between energy and wavelength: \[ E = \frac{hc}{\lambda} \] Where:

• \(E\) is the energy in joules (converted from eV)
• \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \) Js
• \(c\) is the speed of light, roughly \(3 \times 10^{8} \) m/s
• \(\lambda\) is the wavelength in meters

First, convert the band gap energy into joules using the conversion factor: 1 eV = \(1.602 \times 10^{-19}\) J. So, for 2.2 eV:\[ E = 2.2 \times 1.602 \times 10^{-19} = 3.5244 \times 10^{-19} \text{ J} \]With this energy, we can rearrange the formula to solve for wavelength:\[ \lambda = \frac{hc}{E} \]Plug in the known values and calculate the wavelength: \[ \lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{3.5244 \times 10^{-19}} \approx 5.65 \times 10^{-7} \text{ m} = 565 \text{ nm} \]This calculation shows that the wavelength of light emitted is about 565 nanometers, placing it in the green part of the visible spectrum.

LED Emission

Light Emitting Diodes (LEDs) are an application of semiconductors where electrons release energy as they fall to a lower energy state, emitting light in the process. The color of that light is determined by the semiconductor's band gap energy.
In the case of GaP, the energy released by electron transitions corresponds to a wavelength of approximately 565 nm, which is green. This is why GaP semiconductors are used in green LEDs.
LEDs are efficient light sources because they emit light in a narrow wavelength range, minimizing energy loss to heat. Contrast this with traditional bulbs, where much energy is wasted. Understanding the principles of band gap energy and wavelength is key to designing efficient, colorful LED lights.