Chapter 12 Doping To Change The Carrier Concentrations

12.1 Introduction

In silicon, n_i=1.45× 10¹⁰cm^-3 so the atomic density is 5× 10²²cm^-3 .

For example small fractions of phospherous will change their carrier concentrations dramatically.

We need to start with very pure materials for this to work in the way we what it to behave.

12.2 n-doping: Hydrogenic Impurity Theory

For example substituting Silicon for Phospherous you get a spare electron. The spare electron is very loosely bound to the negative charge on the Phospherous site

Figure 12.1 - Loosely Bound Electrons

Like a hydrogen atom, we can model this where the binding energy is

e⁴m₀

2(4pe₀ )²

=13.6eV

this was calculated by Rydberg. The solid gives the electron's mass m₀ to become m^* and also e₀ to become e₀e_r . The impurity binding energy is

R^*=13.6eV×

m^*

m₀

e_r²

~ few eV

For small values of R^* the electron is thermally excited off the impurity site and so is free to conduct. If the donor concentration is N_D , and if N_D>>n_i then n~ N_D . These are called Extrinsic Carriers.

12.3 p-doping: Holes

For example replacing silicon atoms with aluminium (group III), Aluminium accepts electrons from the valence band leaving a hole.

Figure 12.2 - p-doping Leads to Hole

If we apply an electric field all the electrons get integer K vector increases towaards the right hand side ( K=eE ). We label the states as carrying the current however it is easier to say the holes are current carriers and act as if they had charge |e| and mass |m^*| . The hole concentration is p~ N_A where N_A is the acceptence concentration.

12.4 Law of Mass Action

In an intrinsic semiconductor

E_F~

E_g

º E_i

this is the Intrinsic Fermi Energy. When you dope the semiconductor E_F moves and

n=N_ce

E_c-E_F

p=N_ve

E_F-E_v

where N_c and N_v are the effective band edge densities of states.

Figure 12.3 - ???

The product np is independent of E_F and is also independent of the purity of the material

np=N_cN_ve

E_v-E_c

=N_cN_ve

E_g

In pure materials n=p=n_i , where n_i is known as the intrinsic carrier concentration, so

np=n_i²

this is the Law of Mass Action.