Question

Silicon has the diamond structure with a unit cell edge length of \(5.43 \AA\) and eight atoms per unit cell. (a) How many silicon atoms are there in \(1 \mathrm{~cm}^{3}\) of material? (b) Suppose you dope that \(1 \mathrm{~cm}^{3}\) sample of silicon with 1 ppm of phosphorus that will increase the conductivity by a factor of a million. How many milligrams of phosphorus are required?

Short answer

In 1 cm³ of silicon material, there are \(5.0 \times 10^{23}\) silicon atoms. To dope the silicon sample with 1 ppm concentration of phosphorus, 0.02329 mg of phosphorus is required.

Step-by-step solution

Find the volume of a silicon unit cell.

The unit cell of silicon has a cubic shape with an edge length of 5.43 Å. First, we need to convert this to cm: Edge length (in cm) = \(5.43 \times 10^{-8}\) cm Now, we can find the volume of the unit cell: Volume of unit cell = edge length³ = \((5.43 \times 10^{-8})^3\) = \(1.6021 \times 10^{-22} \mathrm{~cm}^{3}\)

Determine the number of unit cells in 1 cm³

To find the number of unit cells in 1 cm³ of silicon, divide the volume of 1 cm³ by the volume of a unit cell: Number of unit cells = \(\frac{1 \mathrm{~cm}^{3}}{1.6021 \times 10^{-22} \mathrm{~cm}^{3}}\) = \(6.25 \times 10^{22}\)

Calculate the total number of silicon atoms in 1 cm³

Since there are 8 silicon atoms per unit cell, we can find the total number of silicon atoms in 1 cm³ by multiplying the number of unit cells by 8: Total number of silicon atoms = \(8 \times 6.25 \times 10^{22}\) = \(5.0 \times 10^{23}\)

Calculate the mass of 1 cm³ of silicon

Now we can find the mass of 1 cm³ of silicon sample. The molar mass of silicon is approximately 28 g/mol. We know that 1 mole of silicon consists of \(6.022 \times 10^{23}\) atoms. So, we can find the mass of the total number of silicon atoms that we calculated in step 3: Mass of 1 cm³ of silicon = \(\frac{5.0 \times 10^{23} \ \text{atoms} \times 28 \ \text{g/mol}}{6.022 \times 10^{23} \ \text{atoms/mol}}\) = 23.29 g

Find the mass of phosphorus required for 1 ppm concentration

We are asked to dope the silicon sample with 1 ppm (parts per million) of phosphorus, which means that we need to add 1 mg of phosphorus per 1g of silicon. Mass of phosphorus required = \(\frac{23.29\text{g}}{10^{6} \text{g/mg}}\) = 0.00002329 g So, 0.00002329 g or 0.02329 mg of phosphorus is required to dope the 1 cm³ sample of silicon with a 1 ppm concentration.

Key concepts

Crystal Lattice

The concept of a crystal lattice is fundamental in understanding the structure of crystalline solids like silicon. A crystal lattice is an arrangement of atoms, molecules, or ions in a highly ordered microscopic structure, forming a lattice that extends in all directions. For silicon, which has a diamond cubic crystal structure, each unit cell—a repeating unit that forms the crystal—contains eight atoms arranged in a way that resembles two interpenetrating face-centered cubic lattices.

The edges of a silicon unit cell measure 5.43 angstroms, and these precise measurements are crucial when determining how many atoms are in a given volume. For any calculations involving unit cells, remember that understanding their three-dimensional arrangement assists in visualizing how many unit cells fill a certain space, such as one cubic centimeter, and subsequently how many atoms are present.

Doping Semiconductors

Doping semiconductors is a process where impurities are intentionally introduced to a semiconductor to enhance its conductivity. The introduction of a dopant, such as phosphorus, into a silicon crystal lattice creates additional free charge carriers that greatly increase the material's electrical conductivity.

The proprietary level of doping is measured in parts per million (ppm), indicating how many dopant atoms are added relative to the number of semiconductor atoms. Even with a minuscule amount of phosphorus, the conductivity can increase by factors of millions. This principle is at the heart of creating various electronic components, from microchips to solar cells, where precise control over the semiconductor properties is paramount.

Molar Mass

The molar mass, often expressed in grams per mole (g/mol), represents the mass of one mole of any substance. For elements, the molar mass in grams per mole is numerically equal to the atomic mass of the element as listed on the periodic table.

Silicon's molar mass is approximately 28 g/mol, essential for converting between the number of atoms and the mass of the substance. During calculations involving molar mass, it serves as a conversion factor that links the macroscopic mass of a sample to its microscopic atomic count. For instance, knowing the molar mass allows the calculation of how many atoms are in a 1 cm³ silicon sample and how much mass of a dopant, such as phosphorus, is needed for a specific doping concentration.