Problem 2ΒΆ
In your lectures you have derived approximate solutions to the Fermi-Dirac integrals for counting electrons and holes in semiconductors. In the derivation you assumed that \(E_{c0}-E_f >> k_bT\) and \(E_f - E_{v0} >> k_bT\) so you could use the Boltzmann approximation. In this question you will explore the range of scenarios under which this approximation is valid.
Consider the semiconductor AlSb. At 300K it has an energy gap of 1.615 eV and an electron effective mass of \(m_n* = 0.12 m_e\).
Using both the full Fermi-Dirac integral, and the Boltzmann approximation calculate the number of electrons in the conduction band at 300K. What is the difference between these values? Is the Boltzmann approximation valid a room temperature.
Produce a graph of number of electrons in the conduction band as a function of temperature using both the full Fermi-Dirac integral and the Boltzmann approximation. At what temperature does the Boltzmann approximation become inaccurate?