# 2.4.7 Thermodynamics of semiconductors

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In this section:
- Thermal equilibrium
- Thermodynamic identity
- Meaning of the Fermi energy
- Example: an ideal electron gas
- Quasi-Fermi energies
- Energy loss in recombination processes
- Thermo-electric effects
- Thermo-electric cooler
- 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, *E*_{F}, 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, *E*_{c}.
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,
*E*_{Fn} and *E*_{Fp}. 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
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2.4.7.7

© Bart J. Van Zeghbroeck, 1996, 1997