Question

GaP has a band gap of \(2.26 \mathrm{eV}\). If GaP is illuminated with ultraviolet light, it emits light equal to the band gap energy. (a) What color is the emitted light? (b) Would appropriately sized GaP quantum dots be able to emit blue light? (c) What about red light?

Short answer

(a) The emitted light is green-yellow, as the wavelength is around 551 nm. (b) No, appropriately sized GaP quantum dots cannot emit blue light, as the wavelength of GaP is larger than the maximum wavelength for blue light. (c) No, appropriately sized GaP quantum dots cannot emit red light, as the wavelength of GaP is smaller than the minimum wavelength for red light.

Step-by-step solution

(Step 1: Convert energy to wavelength)

First, we need to convert the given band gap energy of GaP to the corresponding wavelength. To do this, we use Planck's constant and the relation between energy and wavelength: \[ E = \frac{hc}{\lambda} \] Where, \(E\) = energy (in eV) \(h\) = Planck's constant (\(4.14 × 10^{-15}\) eVs) \(c\) = speed of light (\(3 × 10^{8} \,\text{m/s}\)) \(\lambda\) = wavelength (in meters) We need to find the wavelength (\(\lambda\)), so rearrange the formula to solve for \(\lambda\): \[ \lambda = \frac{hc}{E} \] Now plug in the given energy and constants to calculate the wavelength: \[ \lambda = \frac{(4.14 × 10^{-15}\,\text{eVs})(3 × 10^{8}\,\text{m/s})} {2.26\,\mathrm{eV}} \]

(Step 2: Calculate the wavelength)

After plugging in the values and performing the calculation, we obtain the wavelength of emitted light: \[ \lambda = 5.51 × 10^{-7}\,\text{m} \] We will now convert this wavelength to nanometers (nm) for easier comparison to visible light wavelengths: \[ \lambda = 5.51 × 10^{-7}\,\text{m} \times \frac{10^9\,\text{nm}}{1\,\text{m}} = 551\,\text{nm} \]

(Step 3: Determine the emitted light color)

Now we can determine the color of light corresponding to the calculated wavelength. The wavelengths of visible light range from approximately 380 nm (violet) to 750 nm (red). Based on this range and the calculated wavelength, we see that GaP emits light in the green-yellow region of the spectrum (around 551 nm). Answer: (a) The emitted light is green-yellow.

(Step 4: Check GaP quantum dots for blue light emission)

Blue light has a wavelength around 450 nm to 495 nm. To check if GaP quantum dots can emit blue light, we compare this range with the wavelength of light emitted by GaP. Since the emitted wavelength of GaP is around 551 nm, which is larger than the maximum wavelength for blue light, GaP quantum dots cannot emit blue light. Answer: (b) No, appropriately sized GaP quantum dots cannot emit blue light.

(Step 5: Check GaP quantum dots for red light emission)

Red light has a wavelength between 620 nm to 750 nm. To check if GaP quantum dots can emit red light, we compare this range with the wavelength of light emitted by GaP. Since the emitted wavelength of GaP is around 551 nm, which is smaller than the minimum wavelength for red light, GaP quantum dots cannot emit red light. Answer: (c) No, appropriately sized GaP quantum dots cannot emit red light.

Key concepts

Quantum Dots

Quantum dots are tiny semiconductor particles that are only a few nanometers in size. Their small size gives them unique properties, particularly regarding their electronic and optical characteristics. One of the principal features of quantum dots is their ability to emit light of specific energies or colors when they are illuminated with light. This occurs because of quantum mechanical effects, which dominate at this small scale.

In the case of GaP (Gallium Phosphide) quantum dots, their band gap—the energy difference between the valence band and the conduction band—determines the color of light they can emit. Since the band gap is fixed, quantum dots will typically emit light equal in energy to this band gap. By changing the size of the quantum dots, it's possible to adjust the energy levels and hence the wavelength of emitted light. This means a carefully crafted size of GaP quantum dots could shine light across a narrow spectrum of colors, although not all colors are achievable with a single material like GaP.

The distinctive tunability of quantum dots makes them incredibly useful in applications like displays, where precise color control is essential, and in biomedical imaging, where specific emission wavelengths can be selected to avoid background interference.

Wavelength Conversion

Wavelength conversion is a process where the wavelength of light is transformed. This often involves converting light to a longer or shorter wavelength through physical or chemical processes. In quantum dots, this conversion takes place as they absorb photons (light particles) of one energy and emit photons of another energy, typically corresponding to their band gap energy.

For GaP quantum dots, wavelength conversion happens naturally due to the unique interaction between electrons and the band gap energy. When ultraviolet or other high-energy light illuminates GaP, the electrons get excited above the band gap and, upon returning to their original state, they release energy in the form of light that corresponds to the band gap.

In the exercise, this conversion results in an emission at the wavelength of approximately 551 nm, which is in the green-yellow part of the visible spectrum. It's crucial to note that the ability to convert wavelengths is confined by the inherent band gap of the material in the quantum dot. Thus, GaP cannot be tuned to emit blue or red light naturally given its intrinsic band gap properties.

Visible Light Spectrum

The visible light spectrum is the segment of the electromagnetic spectrum that is visible to the human eye. It ranges from about 380 nm to 750 nm and includes all the colors we perceive, from violet to red. Different wavelengths within this range correlate to different colors: shorter wavelengths correspond to violet and blue, while longer wavelengths appear red or orange.

In the context of GaP quantum dots, the emitted light's wavelength can help determine its color within the visible spectrum. Our step-by-step exercise showed that GaP emits light at approximately 551 nm. Within the visible spectrum, this corresponds to a green-yellow hue, falling between green (495-570 nm) and yellow (570-590 nm).

Understanding the position of a specific wavelength within the visible spectrum is crucial for designing optical devices and lighting solutions that require specific color emissions. This knowledge is leveraged in creating displays, sensors, and even lighting that matches human circadian rhythms.