Problem 40 - Chemistry The Central Science In Si Units

Question

Silicon for computer chips is grown in large cylinders called "boules" that are \(300 \mathrm{~mm}\) in diameter and \(2 \mathrm{~m}\) in length, as shown. The density of silicon is \(2.33 \mathrm{~g} / \mathrm{cm}^{3}\). Silicon wafers for making integrated circuits are sliced from a \(2.0-\mathrm{m}\) boule and are typically \(0.75 \mathrm{~mm}\) thick and \(300 \mathrm{~mm}\) in diameter. (a) How many wafers can be cut from a single boule? (b) What is the mass of a silicon wafer? (The volume of a cylinder is given by \(\pi r^{2} h,\) where \(r\) is the radius and \(h\) is its height.)

Step-by-Step Solution

Answer

266 wafers can be cut; a wafer weighs approximately 1236 g.

Step 1: Calculate the Volume of the Silicon Boule

To find the volume of the silicon boule, we need to use the formula for the volume of a cylinder: \( V = \pi r^2 h \). The diameter of the boule is given as 300 mm, which means the radius \( r \) is 150 mm or 15 cm (since 1 cm = 10 mm). The height \( h \) is 2 m or 200 cm. Plug these values into the formula: \[ V = \pi (15)^2 (200) = 45000\pi \text{ cm}^3. \]

Step 2: Calculate the Volume of One Wafer

Each wafer is a cylinder with the same diameter as the boule, 300 mm, thus the radius is 15 cm. The thickness of a wafer is 0.75 mm, which is 0.075 cm. Use the volume formula for a cylinder: \( V_{\text{wafer}} = \pi r^2 h \), where \( h \) is the thickness of the wafer: \[ V_{\text{wafer}} = \pi (15)^2 (0.075) = 168.75\pi \text{ cm}^3. \]

Step 3: Calculate Number of Wafers from the Boule

To find out how many wafers can be cut, divide the volume of the boule by the volume of one wafer: \[ \text{Number of wafers} = \frac{45000\pi}{168.75\pi} = \frac{45000}{168.75} \approx 266.67. \] Since you cannot have a fraction of a wafer, 266 whole wafers can be cut.

Step 4: Calculate the Mass of a Silicon Wafer

The density of silicon is given as 2.33 g/cm³. Use the volume of one wafer, 168.75π cm³, to calculate the mass: \[ \text{Mass} = \text{Density} \times \text{Volume} = 2.33 \times 168.75\pi \approx 1235.51 \text{ g}. \]

Key Concepts

Volume Calculation

When working with silicon boules, volume calculation is a fundamental step. To find the volume of a silicon boule, which is cylindrical in shape, we use the formula for the volume of a cylinder: \( V = \pi r^2 h \). The diameter of our silicon boule is 300 mm, leading us to a radius \( r \) of 150 mm, or 15 cm when converted (as 1 cm equals 10 mm). The height \( h \) of the boule is given as 2 meters, equal to 200 cm. By substituting these values into the cylinder volume formula, we calculate:

• \( V = \pi (15)^2 (200) \ = 45000\pi \text{ cm}^3 \).

This result gives us the total amount of space the silicon boule occupies, crucial for determining how many wafers we can produce.

Cylinder Geometry

Understanding cylinder geometry is crucial when making calculations for silicon boules and wafers. Cylinders are 3D shapes with circular bases attached by a curved surface, characterized by their radius and height. For the silicon wafers, which are also cylindrical as they share the same diameter as the boule, the radius remains 15 cm. These slices have a thickness (or height in the context of the formula) of 0.75 mm, converted to 0.075 cm. Using the same volume formula \( V = \pi r^2 h \), we find:

• \( V_{\text{wafer}} = \pi (15)^2 (0.075) = 168.75\pi \text{ cm}^3 \).

This allows us to understand the individual volume of each wafer, necessary to further explore how many such wafers can be extracted from a single boule.

Density and Mass Calculations

To determine the mass of a silicon wafer, density and mass calculations play a key role. The density of silicon, provided as 2.33 g/cm\(^3\), allows us to connect volume with mass. For each silicon wafer with a volume of \(168.75\pi \text{ cm}^3\), we use the density equation:

• \( \text{Mass} = \text{Density} \times \text{Volume} = 2.33 \times 168.75\pi \approx 1235.51 \text{ g} \).

This calculation provides the mass of a single silicon wafer, which is vital for both manufacturing settings and economic considerations, as it affects cost and production planning.

Integrated Circuit Manufacturing

In the realm of integrated circuit (IC) manufacturing, silicon wafers are the canvas upon which circuits are etched. A large boule is sliced into thin wafers, each functioning as an individual substrate for circuitry. To compute how many wafers a single boule can yield, we divide the total boule volume by the volume of one wafer:

• \( \text{Number of wafers} = \frac{45000\pi}{168.75\pi} = \frac{45000}{168.75} \approx 266.67 \).

Since partial wafers aren't viable, we conclude that 266 wafers can be produced from each boule. This efficiency in material use is crucial in IC production, where maximizing yield from raw silicon directly impacts technology manufacturing worldwide.