Problem 29 - University Physics with Modern Physics

Question

Germanium has a band gap of 0.67 eV. Doping with arsenic adds donor levels in the gap 0.01 eV below the bottom of the conduction band. At a temperature of 300 \(\mathrm{K}\) , the probability is \(4.4 \times 10^{-4}\) that an electron state is occupied at the bottom of the conduction band. Where is the Fermi level relative to the conduction band in this case?

Step-by-Step Solution

Answer

The Fermi level is approximately 0.215 eV below the conduction band.

Step 1: Identify the Given Data

The band gap of Germanium is 0.67 eV. The donor level is 0.01 eV below the conduction band. The probability of occupancy of an electron at the bottom of the conduction band is given as \(4.4 \times 10^{-4}\) at 300 K.

Step 2: Use the Fermi-Dirac Probability Function

The Fermi-Dirac probability function is given by:\[P(E) = \frac{1}{1 + e^{\frac{E - E_F}{kT}}}\]Here, \(P(E) = 4.4 \times 10^{-4}\) is the probability at the conduction band edge \(E_c\), \(E_F\) is the Fermi level, \(k = 8.617 \times 10^{-5} \, \text{eV/K}\) is the Boltzmann constant, and \(T = 300 \, \text{K}\).

Step 3: Calculate the Exponential Term

First, solve for the exponential term:\[e^{\frac{E_c - E_F}{kT}} = \frac{1 - P(E)}{P(E)}\]Substitute \(P(E) = 4.4 \times 10^{-4}\):\[e^{\frac{E_c - E_F}{kT}} = \frac{1 - 4.4 \times 10^{-4}}{4.4 \times 10^{-4}} \approx 2272.73\]

Step 4: Solve for the Fermi Level \(E_F\)

Take the natural logarithm of both sides:\[\frac{E_c - E_F}{kT} = \ln(2272.73)\]Then, solve for \(E_F\):\[E_c - E_F = kT \cdot \ln(2272.73)\]Substitute \(k = 8.617 \times 10^{-5} \) eV/K and \(T = 300 \) K:\[E_c - E_F = (8.617 \times 10^{-5}) \times 300 \times \ln(2272.73)\]Calculate the value:\[E_c - E_F \approx 0.215 \, \text{eV}\]

Step 5: Conclusion

The Fermi level \(E_F\) is approximately 0.215 eV below the bottom of the conduction band \(E_c\) at 300 K.

Key Concepts

Germanium band gap

Germanium is a semiconductor material with a specific property known as the "band gap." The band gap refers to the energy difference between the valence band, where electrons are bound to atoms, and the conduction band, where electrons are free to move and conduct electricity. In the case of Germanium, the band gap is 0.67 eV. This means that to move an electron from the bound state to a free state, it requires an energy input of 0.67 eV.

Understanding the band gap is crucial because it determines how easily a material can conduct electricity. A smaller band gap, like Germanium's, indicates that less energy is needed to push electrons into the conduction band, making it relatively easier for Germanium to conduct electricity compared to materials with larger band gaps, like silicon.

In semiconductor applications, elements like arsenic are often added to materials like Germanium. This process, known as doping, adds extra energy levels (donor levels) within the band gap. For Germanium doped with arsenic, these donor levels are located just 0.01 eV below the conduction band, providing an easier path for electrons to enter the conduction band and improve conductivity.

Fermi-Dirac probability function

The Fermi-Dirac probability function is a fundamental concept to understand electron behavior in materials at any temperature. It gives the probability that a given energy level within a material is occupied by an electron. This is essential for predicting how a material will conduct electricity under different conditions.

The function is expressed as:
\[P(E) = \frac{1}{1 + e^{\frac{E - E_F}{kT}}}\]
Where:

• \(P(E)\) is the probability of an energy level \(E\) being occupied by an electron.
• \(E_F\) is the Fermi level, which is a reference energy level where the probability of occupancy is 50% at absolute zero temperature.
• \(k\) is the Boltzmann constant.
• \(T\) is the absolute temperature in Kelvin.

In our exercise, the bottom of the conduction band has a very low occupation probability of \(4.4 \times 10^{-4}\) at 300 K. This implies that there are very few electrons at that energy level under the given temperature conditions. By inserting this value into the Fermi-Dirac function, we calculated how the Fermi level \(E_F\) is positioned relative to the conduction band.

The Fermi level is crucial because it determines how the electrons populate the energy levels, which in turn affects the conductivity of the material.

Boltzmann constant

The Boltzmann constant \(k\) is a fundamental constant used in physics, providing a bridge between macroscopic and microscopic energy properties. Its value is \(8.617 \times 10^{-5} \text{ eV/K}\). In semiconductor physics, it is crucial in determining how particles behave at different temperatures.

Using the Boltzmann constant allows us to link temperature with energy, which is particularly useful when studying semiconductors, where temperature changes affect electron occupancy of energy levels. In the context of the Fermi-Dirac probability function, the Boltzmann constant helps quantify the effect of temperature on the probability of electron filling energy levels.

The constant is instrumental in calculations involving the distribution of particles over energy states. In this exercise, we utilized the Boltzmann constant to determine how much energy separation there is between the Fermi level and the conduction band, explaining the behavior of electrons at a given thermal energy. Its role ensures precise calculations of energy distributions, critical for predicting semiconductor behavior in various electric applications.