# 2.6 Intrinsic Semiconductors

In this Section

1. Introduction
2. Calculation of the intrinsic carrier density
3. The mass action law
4. Calculation of the intrinsic Fermi energy
5. The carrier densities as expressed as a function of the intrinsic parameters
6. 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 ni, 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 Ei, 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 Eg = Ec - Ev.

(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 ¬ ­ ® 2.7

1. This terminology is also used when describing chemical reactions.

© Bart J. Van Zeghbroeck, 1996, 1997