Question

CdS has a band gap of \(2.4 \mathrm{eV}\). If large crystals of \(\mathrm{CdS}\) are illuminated with ultraviolet light they emit light equal to the band gap energy. (a) What color is the emitted light? (b) Would appropriately sized CdS quantum dots be able to emit blue light? (c) What about red light?

Short answer

The emitted light for CdS with a band gap of 2.4 eV is green with a wavelength of 517 nm. Appropriately sized CdS quantum dots may be able to emit blue light due to the band gap shifting with size reduction. However, it is unlikely for CdS quantum dots to emit red light, as it would require an increase in size, making them closer to the bulk material.

Step-by-step solution

Convert the band gap energy into wavelength

First, we need to calculate the wavelength of the emitted light by converting the given band gap energy (2.4 eV). Use the formula: \( E = \dfrac{hc}{\lambda} \) Here: - E (energy) = 2.4 eV, but we should convert it to Joules to be consistent with the units of h and c: \( 2.4 \mathrm{eV} \times \dfrac{1.6 \times 10^{-19} \mathrm{J}}{1 \mathrm{eV}} = 3.84 \times 10^{-19} \mathrm{J} \) - h (Planck's constant) = \( 6.63 \times 10^{-34} \mathrm{Js} \) - c (speed of light) = \( 3.0 \times 10^8 \mathrm{m/s} \) Now, rearrange the formula to find λ: \( \lambda = \dfrac{hc}{E} \) Plug in the values and calculate λ: \( \lambda = \dfrac{(6.63 \times 10^{-34} \mathrm{Js}) (3.0 \times 10^8 \mathrm{m/s})}{3.84 \times 10^{-19} \mathrm{J}} \) \( \lambda = 5.17 \times 10^{-7} \mathrm{m} \) or \( 517 \mathrm{nm} \) So, the wavelength of the emitted light is 517 nm.

Identify the color of the emitted light

Now that we have the wavelength of the emitted light, we can identify its color. The visible light spectrum ranges from 400 nm (violet) to 700 nm (red). Here, 517 nm falls within the green region of the visible spectrum. Therefore, the color of the emitted light is green.

Determine if CdS quantum dots can emit blue and red light

Quantum dots exhibit band gap shifting due to their size. As the size of the quantum dots decreases, the bandgap energy increases, and the emitted light shifts towards shorter wavelengths (blue color). Conversely, as the size of the quantum dots increases, the bandgap energy decreases, and the emitted light shifts towards longer wavelengths (red color). (a) Blue light has wavelengths between 450 nm and 495 nm. Given that CdS crystals emit green light; appropriately sized CdS quantum dots may be able to emit blue light if their size is reduced, resulting in increased bandgap energy. (b) Red light has wavelengths between 620 nm and 750 nm. Since CdS crystals emit green light (517 nm), it would require an increase in size to shift the wavelength towards red. However, CdS quantum dots are already small compared to bulk CdS, and increasing their size would make them closer to the bulk material. Hence it is unlikely that CdS quantum dots will emit red light.

Key concepts

Quantum Dots

Quantum dots (QDs) are nanoscale semiconductor particles that possess unique optical and electronic properties due to their extremely small size, often referred to as the quantum confinement effect. The energy levels in quantum dots are discrete, like those of atoms, which is why they're sometimes called 'artificial atoms.'

What makes quantum dots particularly intriguing is their tunable band gap energy, which depends on the size of the crystal. In quantum dot materials, as the size of the particle decreases, the band gap increases. This leads to a phenomenon known as a 'blue shift' because the emitted light moves towards the blue, shorter-wavelength end of the spectrum when the quantum dot size is reduced.

On the other hand, when quantum dots are made larger, the band gap decreases, resulting in a 'red shift'—where the emitted light moves to the longer-wavelength, red end of the spectrum. Due to this tunability, QDs are significant for various applications, including bio-imaging, displays, and photovoltaic devices.

Photoluminescence

Photoluminescence is an optical phenomenon and a type of luminescence where a material absorbs photons (light energy) and then re-emits photons. This process is inherent in quantum dots: when they are illuminated, electrons absorb energy and jump to a higher energy level. Upon returning to their original state, electrons release energy in the form of light—photoluminescence.

The energy and, thus, the color of the emitted light directly correspond to the band gap energy of the QDs. If quantum dots are illuminated with ultraviolet light, as with the CdS quantum dots in our exercise, they will emit light that corresponds with their specific band gap energy determined by their size. Thus, tailoring the size of the quantum dots can directly dictate which color of light is emitted when they are excited, making use of the photoluminescent properties of these tiny structures.

Visible Light Spectrum

The visible light spectrum encompasses the range of electromagnetic wavelengths that human eyes can perceive, ranging approximately from 400 nm (violet) to 700 nm (red). Each wavelength corresponds to a different color, creating the rainbow of colors we see. Blue light falls between 450 nm and 495 nm, while green light, such as the 517 nm light emitted by CdS, is between 495 nm and 570 nm. Red light spans from 620 nm to 750 nm.

Understanding the visible light spectrum is crucial when discussing photoluminescence because it is the range within which photoluminescent emissions can be seen by the human eye. As evidenced in our exercise, knowing the energy of a band gap allows us to calculate the wavelength of emitted light using Planck's equation, affirming which part of the visible spectrum will correspond to this energy and, consequently, predicting the color of emitted light.