Doping in Semiconductors: Understanding N-Type and P-Type Materials, Exercises of Physics

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Semiconductor devices

BS Physics

Dr. Shahzada Qamar Hussain

Doping in semiconductors

 Doping – Adding impurities to the silicon crystal lattice to increase

the number of carriers

Add a small number of atoms to increase either the number of

electrons or holes

Intrinsic Silicon

Acceptors Make p-Type Material Acceptors

• (^) Add atoms with only 3 valence-band electrons
• (^) ex. Boron (B)
• (^) “Accepts” e–^ and provides extra h+^ to freely travel around
• (^) Leaves behind a negatively charged nucleus (cannot move)
• (^) Overall, the crystal is still electrically neutral
• (^) Called “p-type” silicon (added positive carriers)
• NA = the concentration of acceptor atoms [atoms/cm^3 or cm-3]
• (^) Movement of the hole requires breaking of a bond! (This is hard, so mobility is low, μp ≈ 500cm^2 /V)

The Fermi Function

The Fermi Function The Fermi Function

• (^) Probability distribution function (PDF)
• (^) The probability that an available state at an energy E will be occupied by an e - E  Energy level of interest Ef  Fermi level  Halfway point  Where f(E) = 0. k  Boltzmann constant = 1.38× - J/K = 8.617×10-5^ eV/K T  Absolute temperature (in Kelvins)    E Ef  kT e f E    1 1 f(E) 1

1. 5 Ef E

Effect of Doping on Fermi Level

• (^) High probability of a free e-^ in the conduction band
• Moving Ef closer to EC (higher doping) increases the number of available majority carriers

E

f

is a function of the impurity-doping level

Effect of Doping on Fermi Level

• (^) Low probability of a free e-^ in the conduction band
• (^) High probability of h+^ in the valence band
• Moving Ef closer to EV (higher doping) increases the number of available majority carriers

E

f

is a function of the impurity-doping level

Equilibrium Carrier Concentrations

Non-degenerate Silicon

• (^) Silicon that is not too heavily doped
• Ef not too close to Ev or Ec

Assuming non-degenerate silicon

   E E  kT i E E kT i i f f i p n e n n e     2 i np  n

Multiplying together

Charge Neutrality Relationship

• (^) For uniformly doped semiconductor
• (^) Assuming total ionization of dopant atoms

Common Special Cases in Silicon

1. Intrinsic semiconductor ( NA = 0, ND = 0 )
2. Heavily one-sided doping
3. Symmetric doping Intrinsic Semiconductor ( N

A

=0, N

D

=0 ) i i i n p n p n n n    

Carrier concentrations are given by

Heavily One-Sided Doping A D A i D A D i N N N n N N N n      

This is the typical case for most semiconductor applications

D A D i If N^ ^ N , N  n (Nondegenerate, Total Ionization)

Then

D i D N n p n N 2   A D A i If N^ ^ N , N  n (Nondegenerate, Total Ionization)

Then

A i A N n n p N 2  

Determination of E f in Doped Semiconductor A D A i i A i f D A D i i D f i N N N n n N E E kT N N N n n N E E kT                         ln , ln , for for

Also, for typical semiconductors (heavily one-sided doping)

                    i i f i n p kT n n E E kT ln ln [units eV]