Question

A silicon sample is doped with phosphorus at 1 part per \(1.00 \cdot 10^{6} .\) Phosphorus acts as an electron donor, providing one free electron per atom. The density of silicon is \(2.33 \mathrm{~g} / \mathrm{cm}^{3},\) and its atomic mass is \(28.09 \mathrm{~g} / \mathrm{mol}\) a) Calculate the number of free (conduction) electrons per unit volume of the doped silicon. b) Compare the result from part (a) with the number of conduction electrons per unit volume of copper wire (assume each copper atom produces one free (conduction) electron). The density of copper is \(8.96 \mathrm{~g} / \mathrm{cm}^{3},\) and its atomic mass is \(63.54 \mathrm{~g} / \mathrm{mol}\)

Short answer

Based on our calculations, the number of free (conduction) electrons per unit volume of doped silicon is approximately \(4.99 \cdot 10^{16} \ \text{electrons/cm}^3\), whereas the number of conduction electrons per unit volume for copper wire is approximately \(8.49 \cdot 10^{22} \ \text{electrons/cm}^3\). Comparing these values, we find that the doped silicon has a relative concentration of about \(5.88 \cdot 10^{-7}\) times the number of conduction electrons per unit volume of copper wire.

Step-by-step solution

Finding the number of silicon atoms per unit volume

First, we need to find the number of silicon atoms per unit volume. We can do this by using the density of silicon and its atomic mass. The formula to find the number of atoms per unit volume is: Number of atoms per unit volume = (Density \(\times\) Avogadro's number) / Atomic mass The Avogadro's number is \(6.022 \cdot 10^{23} \ \text{atoms/mol}\), the density of silicon is \(2.33 \ \text{g/cm}^3\), and its atomic mass is \(28.09 \ \text{g/mol}\).

Calculate the number of silicon atoms per unit volume

Using the formula and the given values, we can now find the number of silicon atoms per unit volume. Number of silicon atoms per unit volume = \(\frac{2.33 \ \text{g/cm}^3 \times 6.022 \cdot 10^{23} \ \text{atoms/mol}}{28.09 \ \text{g/mol}}\) Number of silicon atoms per unit volume ≈ \(4.99 \cdot 10^{22} \ \text{atoms/cm}^3\)

Calculate the number of free electrons per unit volume

Now, we can find the number of free electrons per unit volume as the silicon sample is doped with phosphorus, providing one free electron per phosphorus atom. Since the doping concentration is 1 part per \(1.00 \cdot 10^6\), we can multiply the number of silicon atoms per unit volume by the doping concentration. Number of free electrons per unit volume = \(4.99 \cdot 10^{22} \ \text{atoms/cm}^3 \times \frac{1}{1.00 \cdot 10^6}\) Number of free electrons per unit volume ≈ \(4.99 \cdot 10^{16} \ \text{electrons/cm}^3\) #b) Compare the result from part (a) with the number of conduction electrons per unit volume of copper wire.#

Finding the number of copper atoms per unit volume

First, we need to find the number of copper atoms per unit volume. We can use the formula similarly to step 1. The density of copper is \(8.96 \ \text{g/cm}^3\), and its atomic mass is \(63.54 \ \text{g/mol}\). Number of copper atoms per unit volume = \(\frac{8.96 \ \text{g/cm}^3 \times 6.022 \cdot 10^{23} \ \text{atoms/mol}}{63.54 \ \text{g/mol}}\) Number of copper atoms per unit volume ≈ \(8.49 \cdot 10^{22} \ \text{atoms/cm}^3\)

Calculate the number of conduction electrons per unit volume of copper wire

As mentioned, each copper atom produces one free conduction electron. Therefore, the number of conduction electrons per unit volume of copper wire is the same as the number of copper atoms per unit volume. Number of conduction electrons per unit volume of copper wire ≈ \(8.49 \cdot 10^{22} \ \text{electrons/cm}^3\) Now, we have the number of conduction electrons per unit volume for both doped silicon and copper wire. We can compare these values to find the relative concentrations.

Compare the number of conduction electrons in doped silicon and copper wire

We can compare the values by dividing the number of conduction electrons in doped silicon by the number of conduction electrons in copper wire. Relative concentration = \(\frac{4.99 \cdot 10^{16} \ \text{electrons/cm}^3}{8.49 \cdot 10^{22} \ \text{electrons/cm}^3}\) Relative concentration ≈ \(5.88 \cdot 10^{-7}\) The number of free conduction electrons per unit volume of the doped silicon is about \(5.88 \cdot 10^{-7}\) times the number of conduction electrons per unit volume of copper wire.

Key concepts

Doping in Semiconductors

Doping is a fundamental process in semiconductor physics that introduces impurities into an intrinsic (pure) semiconductor to modify its electrical properties. In essence, doping enhances the semiconductor's ability to conduct electricity. By adding a small, controlled amount of foreign atoms (known as dopants) to the host material, we can significantly alter the number of charge carriers within the semiconductor. In our example problem, silicon is doped with phosphorus, introducing extra electrons into the silicon lattice. This increases the number of free (or conduction) electrons, thereby enhancing the material's conductivity. Doping can be done by adding electron donors or electron acceptors, which either increases or decreases the number of free electrons, respectively. The result of doping is crucial for creating materials with specific electrical characteristics necessary for electronic devices.

Conduction Electrons

Conduction electrons are the charge carriers in a material that are free to move, allowing the material to conduct electricity. In metals, conduction electrons are typically bound very loosely to their respective atoms, which means they can move freely throughout the lattice. In semiconductors, the number of conduction electrons can be manipulated through the doping process. When silicon is doped with phosphorus, each phosphorus atom donates an additional electron to the silicon crystal structure. This increases the number of conduction electrons per unit volume, which in turn affects the material's conductivity. Understanding the behavior of conduction electrons is vital for predicting and manipulating how different materials will respond in electrical circuits.

Silicon

Silicon is the most widely used semiconductor material, thanks to its abundance, cost-effectiveness, and exceptional properties. It has a crystalline structure that can be effectively modified through doping to enhance its electrical conductivity. In its pure form, silicon's electrical conductivity is not particularly high because it has no free charge carriers. However, semiconductor engineering has revolutionized how silicon is used, making it a bastion in the electronics industry. Doping silicon with elements like phosphorus (an n-type dopant) adds additional electrons (negative charge carriers) and increases conductivity. Silicon's importance cannot be overstated - it forms the backbone of various devices, including transistors, photovoltaic cells, and integrated circuits.

Copper

Copper is a metal known for its outstanding electrical and thermal conductivity, making it a popular choice for wiring and electronics. The high number of conduction electrons per unit volume in copper is due to the metal's atomic structure, allowing electrons to move more freely than in semiconductor materials like silicon. In the exercise, copper is compared to doped silicon. Unlike semiconductors, the conductivity of copper is primarily due to its inherent atomic structure rather than the introduction of impurities. Each copper atom contributes one conduction electron, leading to significantly more conduction electrons compared to doped silicon. This difference highlights why copper conducts electricity exceptionally well but lacks the adjustable properties that make semiconductors valuable in electronics.

Electron Donors

Electron donors in semiconductor physics are elements that provide additional electrons when used as dopants. The concept is crucial when engineering semiconductor materials to modify their conductive capabilities. Phosphorus, which acts as an electron donor in silicon, introduces extra free electrons into the structure. These donors are typically elements from group V of the periodic table, like arsenic or antimony. By donating electrons, these elements increase the electron density in the host material, moving the Fermi level up towards the conduction band. This process transforms an intrinsic semiconductor into an n-type semiconductor, where the additional electrons serve as primary charge carriers. Understanding electron donors allows engineers to tailor materials specifically for electronic components by controlling the doping level and type of dopant.