In the figure below, Intrinsic carrier concentration is given as a function of temperature for Ge and Si. By using the data in this figure calculate the band gap of these materials. For both materials find the energy difference between the Fermi Energy Level and the conductivity band.

Materials Science and Engineering An Introduction by William D. Callister Figure 18.16

My edition shows in that figure ln charge carrier vs temp. You can relate the slope of those lines to energy. Choose the most vertical slopes at low temps as energy valence band, and the upper temps as slope of conduction. the halfway point is Fermi energy.

I haven't looked at this since EE in the 1960s, since then I became a systems engineer, and done my thing with that. The text I looked at is very dated (11 years), so the figures may be different.

To calculate the band gap of Ge and Si using the data in Figure 18.16, you need to use the relationship between intrinsic carrier concentration (ni) and temperature (T), as well as the relationship between band gap (Eg) and ni.

1. Find the values of ni for both Ge and Si at the given temperatures in Figure 18.16.
- Locate the data points for Ge and Si on the graph.
- Determine the ni values corresponding to the given temperatures.

2. Plot the natural logarithm of ni as a function of temperature.
- Take the natural logarithm (ln) of the ni values obtained in step 1.
- Plot ln(ni) on the y-axis and temperature on the x-axis.

3. Draw a best-fit line through the data points.
- Based on the trend of the data points, draw a line that approximates the relationship between ln(ni) and temperature.

4. Determine the slope of the best-fit line.
- Calculate the slope of the best-fit line using the equation: slope = (ln(ni2) - ln(ni1)) / (T2 - T1), where ni2 and ni1 are the ln(ni) values corresponding to two temperatures T2 and T1.

5. Calculate the band gap (Eg) using the relationship between ni and Eg.
- Use the equation: ln(ni) = (-Eg / (2 * k * T)) + (C) where k is the Boltzmann constant and C is a constant.
- Rearrange the equation to solve for Eg: Eg = (-(2 * k * T) * slope) / 2, where slope is the slope calculated in step 4.

6. Calculate the energy difference between the Fermi energy level and the conduction band.
- The Fermi energy level (EF) is the energy level where the probability of an electron being occupied is 0.5.
- The energy difference between EF and the conduction band (EC) is given by the equation: EC - EF = (k * T * ln(ni) / 2).

By following these steps, you can use the data in Figure 18.16 to calculate the band gap of Ge and Si as well as the energy difference between the Fermi energy level and the conduction band for both materials.