Problem 4 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

Silicon has a density of \(2.40 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the concentration of the silicon atoms per cubic centimeter? (b) Phosphorus is added to the silicon to make it an \(n\)-type semiconductor with a conductivity of \(1 \mathrm{mho} / \mathrm{cm}\) and an electron mobility of \(1700 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}\). What is the concentration of the conduction electrons per cubic centimeter?

Step-by-Step Solution

Answer

(a) \(5.14 \times 10^{22} \text{ atoms/cm}^3\); (b) \(3.68 \times 10^{15} \text{ electrons/cm}^3\).

Step 1: Determine the molar mass of silicon

The molar mass of silicon is approximately \(28.09 \text{ g/mol}\). This will be used to find the number of silicon atoms in a given mass.

Step 2: Calculate the number of moles of silicon per cm³

Using the density \(\rho = 2.40 \text{ g/cm}^3\) and the molar mass \( M = 28.09 \text{ g/mol}\), calculate the number of moles in one cubic centimeter: \[\text{Number of moles} = \frac{\rho}{M} = \frac{2.40}{28.09} \approx 0.0854 \text{ mol/cm}^3\].

Step 3: Convert the moles to atoms per cm³

Using Avogadro's number, \(6.022 \times 10^{23} \text{ atoms/mol}\), convert the moles into atoms: \[\text{Number of atoms/cm}^3 = 0.0854 \text{ mol/cm}^3 \times 6.022 \times 10^{23} \text{ atoms/mol} \]This results in approximately \(5.14 \times 10^{22} \text{ atoms/cm}^3\).

Step 4: Use conductivity formula to find electron concentration

The conductivity formula is given by \(\sigma = ne\mu\), where \(\sigma = 1 \text{ mho/cm}\), \(e = 1.6 \times 10^{-19} \text{ C}\), and \(\mu = 1700 \text{ cm}^2/\text{V-s}\).Rearranging gives \(n = \frac{\sigma}{e\mu}\).

Step 5: Calculate conduction electron concentration

Substitute the known values into the formula to find the concentration of conduction electrons:\[n = \frac{1}{(1.6 \times 10^{-19} \text{ C}) \times 1700 \text{ cm}^2/\text{V-s}}\] This yields approximately \(3.68 \times 10^{15} \text{ electrons/cm}^3\).

Key Concepts

Density of Silicon

Silicon is an element that forms the basis of modern electronics. Understanding its properties is crucial for applications in semiconductors.
One of these properties is its density, measured at 2.40 grams per cubic centimeter (g/cm³).
Density is a measure of how much mass is contained in a given volume. It’s an important factor when calculating the number of atoms present in a specific volume of material.

• Knowing the density allows for computation of other key properties, like the concentration of atoms in a cubic centimeter.
• It helps predict how silicon behaves in different physical situations.

With a density of 2.40 g/cm³, silicon's atomic structure is tightly packed, which has implications on its conductivity and other electronic properties.

Molar Mass of Silicon

The molar mass of an element like silicon is essentially the weight of one mole of its atoms. For silicon, this is approximately 28.09 grams per mole (g/mol).
Molar mass is essential because it helps in converting between mass and the number of atoms, which is critical in calculations relevant to silicon-based devices.

• This property allows us to find the number of moles of silicon in a specified mass.
• It is calculated using the atomic weight of silicon and Avogadro's number.

By understanding silicon's molar mass, scientists and engineers can determine how many atoms are used in the manufacturing of an electronic component.

Electron Concentration

Electron concentration refers to the number of free electrons available for conduction in a semiconductor material like silicon.
In pure silicon, these free electrons come from thermal generation, but doping silicon with elements like phosphorus can significantly increase electron concentration.

• This is because phosphorus atoms introduce additional electrons that become free to move and contribute to electrical current.
• High electron concentration generally equates to better conductivity.

Understanding electron concentration is crucial for designing semiconductors with desired electrical properties.
In the given problem, calculating the electron concentration involves knowing the material's conductivity and electron mobility.

n-type Semiconductor

An n-type semiconductor is made by adding certain donor impurities to pure silicon, which introduces more free electrons, the carriers of electric charge.
Phosphorus is a common dopant used for this purpose.
The additional electrons introduced make the semiconductor rich in negative charge carriers, hence the "n-type" tag.

• They typically have better conductivity than their intrinsic counterparts due to the increased electron availability.
• n-type semiconductors are integral in electronic devices, controlling current flow and logic operations in circuits.

By understanding how n-type semiconductors work, one can appreciate how different impurities and doping levels affect a material's electrical characteristics.

Conductivity and Mobility

Conductivity is the measure of a material's ability to conduct electric current. In semiconductors, conductivity is largely influenced by their electron mobility and concentration.
The mobility refers to how quickly an electron can move through a material when an electric field is applied.
In this context, the conductivity equation is essential: \[ \sigma = ne\mu \] where \( \sigma \) is the conductivity, \( n \) is the electron concentration, \( e \) is the elementary charge, and \( \mu \) is the electron mobility.

• The higher the mobility and concentration, the greater the conductivity.
• In the exercise, using known values, you can predict how well silicon, doped with phosphorus, will conduct electricity.

Understanding these properties enables engineers to tailor the electronic characteristics of silicon for specific applications, enhancing the performance of electronic devices.