Question

In a diode laser, electrons dropping from the conduction bend, across the gap, and into the valence band produce the photons that add to the coherent light. The ZnTe bser has a band gap of \(2.25 \mathrm{eV}\), About what wavelength Iaser light would you expect it to produce?

Short answer

The wavelength of the produced laser light can be calculated with the procedure mentioned in steps 1 to 3. It is the solution of the equation λ = h*c/E, where λ represents the wavelength, h represents Planck's constant, c represents the speed of light, and E represents the converted energy from eV to Joules.

Step-by-step solution

Convert Energy from eV to Joules

The energy of the band gap is provided in electron volts. To work with the Planck-Einstein relation, we need to convert this energy into Joules. We use the fact that 1 eV equals \(1.602176634×10^{-19}\) Joules. Thus, our energy E in Joules is \(2.25 \mathrm{eV} * 1.602176634×10^{-19}\) Joules/\(\mathrm{eV}\).

Solving for the Wavelength

We now use the Planck-Einstein relation, E = h*c/λ, and solve for λ, the wavelength. Here, E is the energy we just calculated, h is Planck's constant (\(6.62607015×10^{-34}\) J*s), and c is the speed of light (\(3.00×10^{8}\) m/s). Thus, λ = h*c/E.

Calculate the Wavelength

Substitute the values of E, h and c into equation obtained in Step 2 will yield the wavelength of the light that is emitted.

Key concepts

Understanding the Planck-Einstein Relation

The Planck-Einstein relation is one of the fundamental concepts that bridges the gap between the quantum world and the classical world. It describes how the energy (\(E\)) of a photon is directly proportional to its frequency (\(ν\)), with the proportionality constant being Planck's constant (\(h\)). In equation form, it's expressed as \(E = hν\).

However, since the frequency of a photon is related to the speed of light (\(c\) ) and its wavelength (\(λ\) ) by the relation \(c = λν\), we can restate the Planck-Einstein relation in terms of wavelength as \(E = \frac{hc}{λ}\). In this scenario, it becomes clearer why this equation is so crucial when dealing with the physics of diode lasers. The energy gap of the semiconductor material dictates the color of the light, or rather, its wavelength. Thus, understanding this relation enables us to link the known energy of emitted photons to the light's wavelength that we can observe.

Semiconductor Band Gap and Its Role in Lasers

The term 'semiconductor band gap' refers to the energy difference between the top of the valence band and the bottom of the conduction band in semiconductor materials.

For an electron to contribute to electrical conductivity, it must move from the valence band to the conduction band, a process that requires an amount of energy equal to the band gap. In diode lasers, the opposite happens; electrons fall from the conduction band back to the valence band, releasing energy in the form of photons. The energy of these photons corresponds exactly to the band gap energy, and this energy determines the color — or wavelength — of the laser light. Therefore, knowing the semiconductor band gap is pivotal when predicting the wavelength of light a diode laser will emit, as demonstrated in the exercise.

Photon Wavelength Calculation in Laser Diodes

To calculate the photon wavelength emitted by a diode laser, we use the band gap energy and the Planck-Einstein relation. By transforming the electron volts value into joules and applying the relation \(E = \frac{hc}{λ}\), we can solve for the wavelength (\(λ\) ) that corresponds to the known energy.For instance, with the ZnTe laser having a band gap of 2.25 eV, we would first convert this energy into joules using the conversion factor (\(1 \text{eV} = 1.602176634×10^{-19} \text{J}\) ). After that, the inverse of the Planck-Einstein relation helps us find the exact wavelength of the emitted light. Knowing the wavelength is essential for various applications, such as optical communications, where specific wavelengths correspond to certain transmission properties. This direct ability to calculate photon wavelength allows us to tailor diode lasers for a multitude of scientific and technological purposes.