**2.6 Intrinsic Semiconductors**

Table of Contents -
Glossary -
Study Aids -
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In this Section

- Introduction
- Calculation of the intrinsic carrier density
- The mass action law
- Calculation of the intrinsic Fermi energy
- The carrier densities as expressed as a function of the intrinsic parameters
- Temperature dependence of the intrinsic
carrier density

**2.6.1 Introduction**

Intrinsic semiconductors are semiconductors which
do not contain impurities. They do contain electrons
as well as holes. The electron density equals the hole
density since the thermal activation of an electron
from the valence band to the conduction band yields a
free electron in the conduction band as well as a free
hole in the valence band. We will identify the intrinsic hole
and electron density using the symbol *n*_{i}, and
refer to it as the intrinsic carrier density.

**2.6.2 Calculation of the intrinsic carrier density**

Intrinsic
semiconductors are almost always non-degenerate, so that
the expressions for the electron and hole densities in
non-degenerate semiconductors apply.
Labeling the Fermi energy of intrinsic material *E*_{i},
we can then write two relations between the intrinsic
carrier density and the intrinsic Fermi energy, namely:

(f7)

It is possible to eliminate the intrinsic Fermi energy from both
equations, simply by multiplying both equations and taking the square root.
This provides an expression
for the intrinsic carrier density as a function of the effective density of
states in the conduction and valence band and the bandgap energy
*E*_{g} = E_{c} - E_{v}.

(f16)

A numeric calculation of the intrinsic carrier density for Ge, Si and GaAs
as well as its temperature dependence can be found in section
2.6.6.

**2.6.3 The mass action law**

It turns out that the product of the electron and hole density, in
a non-degenerate semiconductor is always equal to the square of the
intrinsic carrier density, and not only for intrinsic semiconductors.
Multiplying the expressions for the electron and hole densities in a
non-degenerate semiconductor yields:
(f17)

This property is refered to as the *mass action law*
^{1}. It is
a powerful relation which enables to quickly find the hole density
if the electron density is known or vice versa.

**2.6.4 Calculation of the intrinsic Fermi energy**

The above equations for the intrinsic electron and hole density
can be solved for the intrinsic Fermi
energy, yielding:
(f8)

The intrinsic Fermi energy is typically close to the *midgap* energy,
half way between the conduction and valence band edge.
The intrinsic Fermi energy can also
be expressed as a function of the
effective masses of the electrons and holes in the
semiconductor. For this we use the above expressions for the
effective density of states in the conduction and valence band,
yielding:

(f9)

**2.6.5 The carrier densities as expressed
as a function of the intrinsic parameters**

Dividing the expression for the carrier densities, by
the one for the intrinsic density allows to write the
carrier densities as a function of the intrinsic density
and the Fermi energy, or:
(f2)

and

(f4)

We will use primarily these two equations to find the electron and
hole density in a semiconductor. The same relations can also be
rewritten
to obtain the Fermi energy from either carrier density, namely:

(f20)

and

(f21)

**2.6.6 Temperature dependence of
the intrinsic carrier density**

The temperature dependence of the intrinsic carrier density is
dominated by the exponential dependence on the energy bandgap,
as derived in section
2.6.2. In addition one has to consider the
temperature dependence of the effective densities of states and
that of the energy bandgap. A plot of the intrinsic carrier density
versus temperature is shown below. The temperature dependence of the
effective masses was not included since it is small compared to
the others.

bandgap.xls - intrinsi.gif

**Fig.2.6.1** *Intrinsic carrier density versus temperature
in GaAs (top/black curve), Silicon (blue curve) and Germanium
(bottom/red curve). The markers correspond to T = 300 K*

2.5
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2.7

- This terminology is
also used when describing
chemical reactions.

© Bart J. Van Zeghbroeck, 1996, 1997