Question

Silicon for computer chips is grown in large cylinders called aboules" 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 -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.)

Short answer

(a) Approximately 2661 wafers can be cut from a single boule. (b) The mass of a single silicon wafer is approximately 123.7 grams.

Step-by-step solution

Finding the Volume of the Boule

Use the given formula for the volume of a cylinder: \(V = \pi r^{2}h\), where \(r\) is the radius and \(h\) is the height. The diameter of the boule is 300 mm, so the radius is half of that, 150 mm. The height is given as 2 m, which needs to be converted to millimeters (2000 mm). Now, plug in the values into the formula: \(V_{boule} = \pi (150 \,\mathrm{mm})^{2}(2000 \,\mathrm{mm}) \)

Calculate the Volume of the Boule

Calculate the volume of the boule: \(V_{boule} = \pi (150 \,\mathrm{mm})^{2}(2000 \,\mathrm{mm}) = \pi (22500 \,\mathrm{mm}^{2})(2000 \,\mathrm{mm})\) \(V_{boule} \approx 141372000 \, \mathrm{mm}^{3}\)

Finding the Volume of a Single Wafer

Use the same formula for the volume of a single wafer: \(V_{wafer} = \pi r^{2}h\) The diameter of the wafer is 300 mm, so the radius is 150 mm. The height (thickness) of the wafer is 0.75 mm. Plug in the values into the formula: \(V_{wafer} = \pi (150 \,\mathrm{mm})^{2}(0.75 \,\mathrm{mm}) \)

Calculating Volume of the Wafer

Calculate the volume of a single wafer: \(V_{wafer} = \pi (150 \,\mathrm{mm})^{2}(0.75 \,\mathrm{mm}) = \pi (22500 \,\mathrm{mm}^{2})(0.75 \,\mathrm{mm})\) \(V_{wafer} \approx 53093.374 \, \mathrm{mm}^{3}\)

Finding the Number of Wafers per Boule

Divide the volume of the boule by the volume of a single wafer to find the number of wafers that can be cut from the boule: (number of wafers) = \(\frac{V_{boule}}{V_{wafer}}\) (number of wafers) = \(\frac{141372000 \, \mathrm{mm}^{3}}{53093.374 \, \mathrm{mm}^{3}}\)

Calculate the Number of Wafers

Calculate the number of wafers: (number of wafers) ≈ 2661 So, approximately 2661 wafers can be cut from a single boule.

Calculate the Mass of a Single Wafer

To find the mass of a single wafer, use the density formula (mass = density × volume). The density of silicon is given as 2.33 g/cm³. First, we need to convert the volume of the wafer from mm³ to cm³: \(V_{wafer} \approx 53093.374 \, \mathrm{mm}^{3} = 53.093\, \mathrm{cm}^{3}\) Now, use the formula: mass of a wafer = (density of silicon) × (volume of a wafer) mass of a wafer = (2.33 g/cm³) × (53.093 cm³)

Finding the Mass of a Single Wafer

Calculate the mass of a single wafer: mass of a wafer = (2.33 g/cm³) × (53.093 cm³) mass of a wafer ≈ 123.707 g The mass of a single silicon wafer is approximately 123.7 grams.

Key concepts

Volume of Cylinder

The concept of volume is fundamental when dealing with three-dimensional objects like cylinders. A cylinder is a shape with two parallel circular bases joined by a curved surface at a fixed distance from the center. To find the volume of such an object, we use the formula:\[ V = \pi r^{2} h \]Here, \( V \) represents the volume, \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.
In the given problem, silicon is grown in cylindrical boules, so understanding this formula is essential. By plugging in the values for the radius and height, where the radius is half of the diameter (which is given as 300mm, hence the radius is 150mm), and the height is 2000mm (since 2 meters is equivalent to 2000 millimeters), we calculate:- The volume of the boule as: \[ V_{boule} = \pi \times (150 \, \mathrm{mm})^{2} \times 2000 \, \mathrm{mm} \approx 141,372,000 \, \mathrm{mm}^{3} \]- Volume of a single wafer using its thickness as the height: \[ V_{wafer} = \pi \times (150 \, \mathrm{mm})^{2} \times 0.75 \, \mathrm{mm} \approx 53,093.374 \, \mathrm{mm}^{3} \]

Density of Silicon

Density is another important concept in physics and engineering, defined as mass per unit volume. The formula for density is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]In our problem, the density of silicon is known, and it is given as 2.33 g/cm³. When converting the volume of an object from cubic millimeters (\(\mathrm{mm}^{3}\)) to cubic centimeters (\(\mathrm{cm}^{3}\)), remember that:- 1 \(\mathrm{cm}^{3}\) = 1000 \(\mathrm{mm}^{3}\)
Therefore, the volume of the silicon wafer discussed in the exercise is first converted, and we use the relation:- If the wafer's volume is approximately 53,093.374 \(\mathrm{mm}^{3}\), in cm³ it is: \[ 53.093 \, \mathrm{cm}^{3} \]With this conversion, to find the mass of the wafer, multiply the volume by the density:- Mass of wafer = Density \(\times\) Volume = \(2.33 \, \mathrm{g/cm}^{3} \times 53.093 \, \mathrm{cm}^{3} \approx 123.7 \, \mathrm{g}\)

Integrated Circuits

Integrated circuits, or ICs, are crucial in modern electronics. They are composed of many tiny components such as transistors, diodes, resistors, and capacitors on a semiconductor material, usually silicon. These components work together to perform electrical functions. Integrated circuits help power our digital world by being the backbone of electronic devices. Silicon is the base material for these ICs, due to its properties as a semiconductor. This characteristic allows it to control electrical current efficiently, which is essential for the operation of electronic circuits.
This exercise involves slicing silicon wafers from a boule (which is a large cylinder of silicon) and then using those wafers as a base for building integrated circuits. Each wafer must be precisely thin, around 0.75mm, to ensure it functions well in its applications. Once sliced, the wafer undergoes a series of processes, including doping, etching, and metal deposition, to form the complex pathways and components of an integrated circuit.
In summary, understanding concepts like the volume of a cylinder and density is crucial when preparing silicon wafers for the creation of integrated circuits, as these initial steps underpin the entire manufacturing process.