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Chapter 9: Problem 18

Why are the sharp transitions at the Fermi energy "smoothed out" by increasing the temperature?

Short Answer

Expert verified
The sharp transitions at the Fermi energy are "smoothed out" by increasing the temperature because the Fermi-Dirac distribution function, which indicates the probability of electron occupation at specific energy levels, becomes smoother at higher temperatures. This allows electrons from lower energy levels to occupy states slightly above the Fermi energy and some states slightly below it to be less occupied. As a result, the abrupt change in electron occupation around the Fermi energy at zero temperature is no longer present at higher temperatures.

Step by step solution

01

Understanding Fermi energy and the density of energy states

The Fermi energy is the energy level at zero temperature at which an electron can occupy a state in a system. In solid-state physics, it is connected with the electron distribution in the energy levels of the system. The density of states (DOS) is the number of energy levels available in a given energy range. At zero temperature, all states below the Fermi energy are fully occupied and states above are empty.
02

Introduce the Fermi-Dirac distribution

The Fermi-Dirac distribution function gives the probability of the occupation of a state with energy E by an electron at a specific temperature. It is defined as: \(f(E) = \dfrac{1}{e^{\frac{E - E_F}{k_B T}} + 1}\) Where \(f(E)\) is the occupation probability, \(E\) is the energy level, \(E_F\) is the Fermi energy, \(k_B\) is the Boltzmann constant, and \(T\) is the temperature. At \(T = 0\), the distribution becomes a step function, with all states occupied below the Fermi energy and none occupied above it.
03

Analyze the effect of temperature on the Fermi-Dirac distribution

As the temperature increases, the Fermi-Dirac function becomes smoother. This means that the electrons from lower energy levels can now occupy a state slightly above the Fermi energy, and some states slightly below the Fermi energy are not fully occupied. Consequently, the sharp transition at the Fermi energy gets "smoothed out" at higher temperatures. At high temperatures, the Fermi-Dirac distribution starts to resemble the classical Maxwell-Boltzmann distribution.
04

Conclusion

The sharp transitions at the Fermi energy get smoothed out by increasing the temperature because the Fermi-Dirac distribution becomes smoother at higher temperatures. The electrons from lower energy levels now have a probability to occupy states slightly above the Fermi energy, and some states slightly below it are not fully occupied. Thus, the abrupt change in electron occupation around the Fermi energy at zero temperature is no longer present at higher temperatures.

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