Question

The semiconductor GaP has a band gap of \(2.26 \mathrm{eV}\). What wavelength of light would be emitted from an LED made from GaP? What color is this?

Short answer

The wavelength of light emitted from an LED made from GaP is \(\approx 545 \, nm\), which corresponds to the color green in the visible light spectrum.

Step-by-step solution

Given values and constants

Bandgap energy: \(E = 2.26 \, eV\) Planck's constant: \(h = 4.135667696 \times 10^{-15} \, eV \cdot s\) Speed of light: \(c = 2.99792458 \times 10^8 \, m/s\)

Convert energy to Joules

To work with the Planck's constant in eV, we need to convert the bandgap energy to Joules: 1 eV = 1.602176634 × 10⁻¹⁹ J So, \(E = 2.26 \, eV \times 1.602176634 \times 10^{-19} \, J/eV = 3.62087782 \times 10^{-19} \, J\)

Calculate the wavelength

Use the formula: \(E = h * c / \lambda\) Rearrange for the wavelength: \(\lambda = h * c / E\) Substitute values: \(\lambda = (4.135667696 \times 10^{-15} \, eV \cdot s)(2.99792458 \times 10^8 \, m/s) / 2.26 \, eV\) Calculate the wavelength: \(\lambda = 545.0737 \, nm\)

Determine the color of the emitted light

The wavelength of light emitted from the GaP LED is \(\approx 545 \, nm\), which corresponds to the color green in the visible light spectrum. So, the color of the light emitted from the LED made from GaP is green.

Key concepts

Understanding Band Gap Energy

Band gap energy is a crucial concept in the field of semiconductors and solid-state physics. It refers to the energy difference between the top of the valence band (where the valence electrons reside) and the bottom of the conduction band (where electrons can move freely and contribute to conduction). This gap determines the electrical conductivity of a material: a large band gap results in an insulator, a small band gap indicates a semiconductor, and no band gap points to a conductor.

For semiconductor materials like gallium phosphide (GaP), this band gap energy is a key property. As in the exercise, GaP has a band gap of 2.26 electronvolts (eV). The energy of the emitted photon when an electron transitions from the conduction band to the valence band is approximately equal to the band gap energy. Therefore, the band gap energy not only influences a material's conductivity but also its optical properties, such as the color of light it emits when used in an LED.

LED Wavelength Calculation

The wavelength of light emitted by a Light Emitting Diode (LED) can be calculated if we know the band gap energy of the semiconductor material. By utilizing the energy-wavelength relationship from the Planck-Einstein equation, we can find out the color of the light that will be observed.

Starting with the energy in electronvolts, we first convert it to joules, since Planck's constant is commonly expressed in joule-seconds. After conversion, we apply the formula \( \lambda = \frac{h \cdot c}{E} \) where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( c \) is the speed of light, and \( E \) is the energy of the photon. Using this formula, we can calculate the wavelength that corresponds to the band gap energy of the semiconductor material.

Planck's Constant and its Role in Quantum Mechanics

Planck's constant, denoted by \( h \) and valued at approximately \( 4.135667696 \times 10^{-15} \, eV \cdot s \), is a fundamental quantity in quantum mechanics. It relates the frequency (or wavelength) of a photon to its energy. Essentially, it bridges the worlds of classical and quantum physics by quantifying the idea that energy is not continuous but comes in discrete 'packets' known as quanta.

Its use in the LED wavelength calculation exercise is central. Since energy and wavelength are inversely proportional, Planck's constant helps us determine the energy of a photon from its wavelength, and vice versa. Without this constant, the precise determination of such relationships would be impossible, and understanding the behavior of quantum-scale phenomena would be significantly more challenging.

The Visible Light Spectrum

The visible light spectrum is a small part of the electromagnetic spectrum that can be perceived by the human eye. It ranges from approximately 380 nm (violet) to 750 nm (red). Each color of light within this spectrum corresponds to a specific range of wavelengths. For instance, green light typically falls within the 495–570 nm wavelength range.

As shown in the exercise, by knowing the wavelength of the emitted light from an LED, we can determine its color. The LED made from GaP emits light at approximately 545 nm, which lies in the green region of the visible spectrum. Understanding the visible light spectrum allows us to interpret and predict the colors associated with various wavelengths, which is particularly important in applications such as lighting, display technologies, and optical communications.