Question

An ideal quantum dot for use in TVs does not contain any cadmium due to concerns about disposal. One potential material for this purpose is InP, which adopts the zinc blende (ZnS) structure (face-centered cubic). The unit cell edge length is \(586.9 \mathrm{pm} .(\mathbf{a})\) If the quantum dot is shaped like a cube, how many of each type of atom are there in a cubic crystal with an edge length of \(3.00 \mathrm{nm} ? 5.00 \mathrm{nm} ?(\mathbf{b})\) If one of the nanoparticles in part (a) emits blue light and the other emits orange light, which color is emitted by the crystal with the \(3.00-\mathrm{nm}\) edge length? With the \(5.00-\mathrm{nm}\) edge length?

Short answer

A crystal of 3.00 nm edge length contains 432 In atoms and 432 P atoms and emits blue light. A crystal of 5.00 nm edge length contains 1458 In atoms and 1458 P atoms and emits orange light.

Step-by-step solution

Calculate the number of unit cells in the cubic crystal

We begin by calculating the number of unit cells along each edge (n) of the crystal: \(n=\frac{\text{Edge length of crystal}}{\text{Edge length of unit cell}}\) For the 3.00 nm and 5.00 nm edge length crystals: \(n_{3.00\mathrm{nm}}=\frac{3.00\mathrm{nm}}{586.9\mathrm{pm}}=\frac{3000\mathrm{pm}}{586.9\mathrm{pm}}\approx 5.11\) \(n_{5.00\mathrm{nm}}=\frac{5.00\mathrm{nm}}{586.9\mathrm{pm}}=\frac{5000\mathrm{pm}}{586.9\mathrm{pm}}\approx 8.52\) Since we can't have a partial unit cell, we round up the number of unit cells to the nearest integer for both cases: \(n_{3.00\mathrm{nm}}=6\) \(n_{5.00\mathrm{nm}}=9\) Now we have 6 unit cells along each edge for the 3.00 nm crystal and 9 unit cells along each edge for the 5.00 nm crystal.

Calculate the number of atoms in the cubic crystal

As we know the zinc blende unit cell has 4 atoms (2 In and 2 P) and we have the number of unit cells along each edge of the cubic crystal, we can calculate the total number of unit cells and then the number of atoms of each type in the crystal. Total number of unit cells in the crystal = \(n^3\) For the 3.00 nm and 5.00 nm edge length crystals: Total number of unit cells in the 3.00 nm crystal = \(6^3=216\) Total number of unit cells in the 5.00 nm crystal = \(9^3=729\) Now we know that each unit cell contains 2 In atoms and 2 P atoms. We can multiply the total number of unit cells by the number of In and P atoms to find the total number of atoms of each type in the crystal: For the 3.00 nm crystal: In atoms: \(216 \times 2 = 432\) P atoms: \(216 \times 2 = 432\) For the 5.00 nm crystal: In atoms: \(729 \times 2 = 1458\) P atoms: \(729 \times 2 = 1458\)

Determine the color emitted by the crystals

Considering that one of the nanoparticles in part (a) emits blue light and the other emits orange light, the crystal's emitted color will be determined by the size of the nanoparticles inside the crystal. It is known that as the size of the quantum dot increases, its emitted wavelength of light decreases (shifts to a longer wavelength or lower frequency) from blue to orange. Therefore, we can conclude that: In the case of the 3.00 nm crystal, with fewer and smaller nanoparticles, it emits blue light. In the case of the 5.00 nm crystal, with more and larger nanoparticles, it emits orange light. To summarize: A crystal of 3.00 nm edge length contains 432 In atoms and 432 P atoms and emits blue light. A crystal of 5.00 nm edge length contains 1458 In atoms and 1458 P atoms and emits orange light.

Key concepts

Quantum Dots

Quantum dots are tiny semiconductor particles, often only a few nanometers in size. These miniature marvels have unique optical and electronic properties. One of their most intriguing features is the ability to emit specific light colors based on their size.
When light excites these particles, they can re-emit light at different wavelengths depending on their size. The smaller the quantum dot, the shorter the wavelength of emitted light, which translates into blue light. Conversely, larger quantum dots emit light with longer wavelengths, resulting in emissions ranging from yellow to red.
Their ability to emit different colors makes them essential in display technologies, especially in quantum dot television screens. Quantum dots in screens enhance color vividness by providing more accurate and vibrant hues. This color versatility is due to their control over the specific frequencies of light they emit when stimulated. They also find uses in biological imaging and solar cells owing to their efficiency and customizability.

Nanoparticles

Nanoparticles, like quantum dots, are exceptionally small particles measured in nanometers. To put this in perspective, one nanometer is one-billionth of a meter! Due to their tiny size, nanoparticles possess unique properties that differ significantly from their bulk material counterparts.
The effects experienced at the nanoscale make nanoparticles ideal for various technological and medical applications. Nanoparticles can prioritize certain properties, such as optical, thermal, or electrical, which can be fine-tuned simply by modifying their size and shape.
These characteristics are employed in diverse industries ranging from electronics and cosmetics to pharmaceuticals and environmental remediation. In the context of our study, nanoparticles within the quantum dot crystals dictate the color of light emitted. For example, in displays or LEDs, nanoparticles greatly enhance image quality or light efficiency by allowing precise control over tiny reaction spaces.
The manipulation of nanoparticles is unlocking solutions and innovations across scientific fields, underscoring their massive potential in shaping future technology.

Unit Cell Calculation

Understanding unit cell calculation is crucial when studying crystal structures like those found in zinc blende forms. The unit cell is the smallest structural entity that repeats in all directions in a crystal lattice. Calculating the number of unit cells can help us determine the atoms within the crystal structure.
To find the number of unit cells in a larger structure, divide the overall edge length of the crystal by the edge length of the unit cell. For example, if a cubic quantum dot is 3 nm in length, and the unit cell edge length is 586.9 pm, we first convert units (since 1 nm = 1000 pm) to find out how many unit cells fit along each edge. This allows us to figure out the spatial configuration of atoms.
Once the number of unit cells on one edge is known, we can calculate the entire cube's unit cell count by raising this number to the power of three ( ^3). This gives a three-dimensional perspective of how the fundamental structures repeat throughout the space, ensuring accurate counting of atoms present, necessary in calculating light emission properties of quantum dots.

Light Emission Colors

The light emission colors of quantum dots are based heavily on their size due to a phenomenon known as quantum confinement. As we highlighted, smaller dots emit blue light, while larger ones emit colors like orange. This shift in emitted light wavelength as a function of size is critical in applications such as LEDs and display technology.
Color emission is strongly controlled by the energy gap between the quantum dot's conduction band and its valence band—the smaller the quantum dot, the larger this energy gap. This results in the emission of shorter wavelengths when the quantum dot relaxes from an excited state back to normal, essentially producing blue light.
As quantum dots grow in size, the energy gap shrinks, allowing them to emit light at longer wavelengths, spanning across the visible spectrum into the reds and oranges. This ability to fine-tune light emission by altering dot size is harnessed to enhance display technologies, paving the way for screens that can present more vibrant, lifelike images. This adaptability is why quantum dots are so pivotal for advancing display technology and other optical applications.