# Extrinsic Semiconductors

Extrinsic semiconductors are semiconductors that contain foreign elements known as dopants in the structure.

### Introduction to Semiconductors

These impurities are carefully added into a highly pure intrinsic semiconductor by a process known as doping. Figure 1.1 shows an example of a typical extrinsic semiconductor structure. There are two types of extrinsic semiconductors. Extrinsic semiconductors with atoms from the V group on the periodic table are predominated by negative charge carries, such as a electrons, and are called n-type semiconductors. The other type is predominated by the positive charge from elements in the III group and is called p-type semiconductors. These imperfections control the electrical conductivity of the material better than a pure, intrinsic semiconductor.

Figure 1.1

N-type semiconductors donate electrons to the conduction band, because the number of valence electrons of the dopant exceeds that of the atom it is replacing. Usually group V is used for this purpose because the atoms have one more valence electrons than group IV, which makes up typical intrinsic semiconductors. Four of the valence electrons are then used for bonding and one electron is left over becoming unstable. So at temperatures above zero this electron will break the small potential barrier and be excited into the conduction band. The opposite is true of p-type semiconductors. Group III is often used as a dopant. These atoms have one less electron than Group IV atoms so they accept electrons from the valence band creating holes in the valence band.

Figure 1.2

### Conductivity and Temperature Dependence

The equation to calculate conductivity of a extrinsic semiconductor is

σ=n*q*μ(e)  n-type

σ=n*q*μ(h) p-type

where N is the density of the charge carries, q is charge and μ is the mobility of each carrier. At room temperature N is equal to the number of impurities in the material, but at temperatures lower than room temperature the equation for N then becomes

N= -4.84x10^15(m*/m0)^(3/2)*T^(3/2)*e-Eg/2kb*T

N-type semiconductors follow the Arrhenius equation

σ=σ(0)exp[-(E(g)-E(d))/kT]

Figure 2.1Figure 2.2

At low T or high 1/T the material will produce extrinsic behaviors (-(Eg-Ed)/k) as electrons are being promoted to the conduction band. Once it reaches a certain temperature, all electrons will have been promoted to the conduction band; this is called the "Exhaustion Range" (See figure 2.1). With all the electrons promoted to the conduction band while temperature still rises, the material will have intrinsic behaviors (-Eg/2k) at high T and low 1/T. Here the Fermi energy levels lie between the donor level and the conduction band.

P-type semiconductors follow the Arrhenius equation of

σ=σ(0)exp[-E(a)/kT]

Figure 2.3Figure 2.4

For p-type semiconductors the graph looks very similiar to the n-type graph, except for where the graph evens out. This is called the "Saturation Range"(see figure 2.3). This is where all the acceptor levels are occupied with electrons. The Fermi energy levels here lie between the Acceptor level (Ea) and the the valence band (O).

*Note these effects take place only in low doped semiconductors. The effects are much less pronouced in highly doped semiconductors because of the number of charge carriers.

### Hall Effect

The Hall Effect is used to measure the number and type of charge carrier in an extrinsic semiconductor. With these measurements scientists are able to measure down to lower than 1010 particles/m3. These measurements are made by having an electric field running through a material  in the positive x-direction while a magnetic field is applied in the z-direction forcing the electrons path to bend, by a force known as the Lorentz Force. This leads to a build- up of electrons on one side of the material, which creates an electric field in the negative y-direction. This is called the Hall Field. The Hall Force given by the equation

Eq.1)  FH= -e*E,

balances the Lorentz Forces, given by the equation,

Eq. 2)  FL=v*B*e,

where v is the velocity of electrons, e is the electron charge and B is the magnetic induction. Setting them equal to each other yields

Eq. 3)  E=v*B,

Then combining

Eq.4)  j= -N*v*e,

With Eq. 3 and solving for N yields

Eq.5)  N=(j*B)/(e*E)=(I*B)/(t*e*Vh),

where t is the thickness of the semiconductor, I is the current and Vh is the Hall Voltage. The Hall voltage is represented by

Vh =(I*B)/(t*e*N)

With this information, the number of charge carrier density(per unit volume) can be solved.

Rh=-1/N*e

RH represents the Hall Constant. The sign of the Hall constant determines whether electrons or holes will predominate. A negative number means electrons predominate and a positive number means holes predominate.

### Problems

1) You are given a n-type semiconductor with Eg=2.3 eV while Ed=1.2 eV, the conductivity of this material is 150 Ω-1 ∙m-1 at room temperature (25°C). Calculate the conductivity at 45°C. Assume that the extrinsic behavior ends at 45°C.

2) In a Arsenic-doped germanium semiconductor, the Fermi-energy level increased by 0.5 eV. What is the likelihood that an electron being thermally advanced up into the conduction band (Eg=0.66 eV) at 22 °C?

3) Calculate the photon wavelength, in nanometers, that is necessary to advance an electron into the conduction band of Aluminum doped Germanium semiconductor. E(g) for Germanium is 0.66 eV.

4) In a Arsenic-doped Germanium semiconductor, where the it has a thickness of 0.5mm, a current of 75 Amps, a magnetic flux density of .40 T. Find a) the Hall voltage and b) the Hall constant. Arsenic-doped Germanium has charge carrier density of 80x1028 m-1 and the charge of an electron is equal to 1.602x10-19 C.

5) For a n type semiconductor, calculate the charge carrier density (N) at 14°C. The effective mass is 0.067 and Eg is 1.2 eV.

### References

1. Bube, Richard H. Electrons in Solids: An Introduction Survey. 3rd Edition. California: Academic Press Inc. 1992
2. Hummel, Rolf. E. Electronic Properties of Materials. 3rd Edition. Florida, Springer, LLC. 2005
3. Shackleford, James F. Material Science for Engineers. 5th Edition. New Jersey, Prentice-Hall. 2000

14:37, 31 Jul 2015

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This material is based upon work supported by the National Science Foundation under Grant Number 1246120