# 2.4.7 Thermodynamics of semiconductors

Table of Contents - 1 2 3 4 5 6 7 8 9 R S ¬ ­ ®
In this section:
1. Thermal equilibrium
2. Thermodynamic identity
3. Meaning of the Fermi energy
4. Example: an ideal electron gas
5. Quasi-Fermi energies
6. Energy loss in recombination processes
7. Thermo-electric effects
8. Thermo-electric cooler
9. Hot probe experiment

## 2.4.7.1 Thermal equilibrium

A system is in thermal equilibrium if detailed balance is obtained: i.e. every process taking place in the system is exactly balanced by its inverse process so that there is no net effect on the system.

This definition implies that in thermal equilibrium no energy (heat, work or particle energy) is being exchanged between the parts within the system and between the system and the environment. Thermal equilibrium is obtained by isolating a system from its environment, removing any internal sources of energy, and waiting for a long enough time until the system does not change any more.

The concept of thermal equilibrium is of interest since a variety of thermodynamic results assume that the system under consideration is in thermal equilibrium. Few systems of interest rigorously satisfy this condition so that we often apply the thermodynamical results to systems which are "close" to thermal equilibrium. Agreement between theories based on this assumption and experiments justify this approach.

## 2.4.7.2 Thermodynamic identity

The thermodynamic identity simply states that a change in energy can be caused by adding heat, work or particles. Mathematically this is expressed by:
(t8)
where U is the energy, Q is the heat and W is the work. m is the energy added to a system when adding one particle without adding either heat or work. The amount of heat exchanged depends on the temperature and the entropy, while the amount of work delivered to a system depends on the pressure and the volume or:
(t6)
and
(t7)
yielding:
(t1)
Or in words: The change in (total) energy equals the sum of added heat, work and particle energy.

## 2.4.7.3 The Fermi energy

The Fermi energy, EF, is the energy associated with a particle which is in thermal equilibrium with the system of interest. The energy is strictly associated with the particle and does not consist even in part of heat or work. This same quantity is called the electro-chemical potential, m, in most thermodynamics texts.

## 2.4.7.4 Example: an ideal electron gas

As an example to illustrate the difference between the average energy of particles in a system and the Fermi energy we now consider an ideal electron gas. The term ideal refers to the fact that the gas obeys the ideal gas law. To be "ideal" the gas must consist of particles which do not interact with each other.

The total energy of the non-degenerate electron gas containing N particles equals:

(t2)
as each non-relativistic electron has a thermal energy of kT/2 for each degree of freedom in addition to its minimum energy, Ec. The product of the pressure and volume is given by the ideal gas law:
(t4)
While the Fermi energy is given by:
(t5)
The thermodynamic identity can now be used to find the entropy from:
(t11)
yielding:
(t3)
This relation can be visualized on an energy band diagram when one considers the energy, work and entropy per electron and compares it to the electro-chemical potential as shown in figure 1.

idealgas.gif
Fig.1 Energy, work and heat per electron in an ideal electron gas visualized on an energy band diagram.
The distinction between the energy and the electro-chemical potential also leads to the following observations: Adding more electrons to an ideal electron gas with an energy which equals the average energy of the electrons in the gas increases both the particle energy and the entropy as heat is added in addition to particles. On the other hand, when bringing in electrons through an electrical contact whose voltage equals the Fermi energy (in electron volts) one does not add heat and the energy increase equals the Fermi energy times the number of electrons added.

Therefore when analyzing the behavior of electrons and holes on an energy band diagram, one should be aware of the fact that the total energy of an electron is given by its position on the diagram, but that the particle energy is given by the Fermi energy. The difference is the heat minus the work per electron or dQ - dW = T dS + p dV.

Related topics: Thermo-electric effect, thermo-electric cooler and "hot-probe" experiment.

## 2.4.7.5 Quasi-Fermi energies

Quasi-Fermi energies are introduced when the electrons and holes are clearly not in thermal equilibrium with each other. This occurs when an external voltage is applied to the device of interest. The quasi-Fermi energies are introduced based on the notion that even though the electrons and holes are not in thermal equilibrium with each other, they still are in thermal equilibrium with themselves and can still be described by a Fermi energy which is now different for the electrons and the holes. These Fermi energies are refered to as the electron and hole quasi-Fermi energies, EFn and EFp. For non-degenerate densities one can still relate the electron and hole densities to the two quasi-Fermi energies by the following equations:
(t13)
(t14)

## 2.4.7.6 Energy loss in recombination processes

The energy loss in a recombination process equals the difference between the electron and hole quasi-Fermi energies as the energy loss is only due to the energy of the particles which are lost:
(t12)
No heat or work is removed from the system, just the energy associated with the particles. The energy lost in the recombination process can be converted in heat or light depending on the details of the process.
2.4.6 ¬ ­ ® 2.4.7.7

© Bart J. Van Zeghbroeck, 1996, 1997