1. Semiconductor Equilibrium
No external forces (voltages, electric fields, temp.gradients)
First
Consider pure crystal
Then
Consider addition of dopants

2. Semiconductor Equilibrium
Charge carriers
Electrons in conductance
n(E) = gc(E)fF(E)
n(E) - prob. dens. of electrons
gc(E) - conductance density
fF(E) - Fermi-Dirac prob. function
Holes in valence
p(E) = gV(E)(1 - fF(E))
p(E) - prob. dens. of holes
gv(E) - valence density

3. Semiconductor Equilibrium
Charge carriers(cont.)

4. Semiconductor Equilibrium
Charge carriers(cont.)
Example
Find the probability that a state in the conduction band is occupied
and calculate the electron concentration in silicon at T = 300K.
Assume Fermi energy is .25 eV below the conductance band
Note low probability per state but large number of states implies reasonable concentration of electrons

5. Semiconductor Equilibrium
Charge carriers(cont.)
For intrinsic semiconductor, concentration of electrons in
conductance band is equal to holes in the valence band. Thus,

6. Semiconductor Equilibrium
Dopant Atoms (n-type semiconductor)
Phosphorous has 5 valence electrons
Energy-band diagram

7. Semiconductor Equilibrium
Dopant Atoms (p-type semiconductor)
Boron has 3 valence electrons
Energy-band diagram

8. Semiconductor Equilibrium
The Extrinsic Semiconductor
n-type p-type

9. Semiconductor Equilibrium
The Extrinsic Semiconductor
Example
Consider doped silicon at 300K. Assume that the
Fermi enery is .25 eV below the conduction band
and .87 eV above the valence band. Calculate the
thermal equilibrium concentration of e’s and holes

10. Semiconductor Equilibrium
The Extrinsic Semiconductor
The n0p0 product
That is, the product of n0 and p0 is a constant for a given semiconductor at a given temperature.

11. Semiconductor Equilibrium
Statistics of donors and acceptors
Ratio of electrons in donor state total electrons
Example
Consider phosporous doped silicon at T = 300K and
at a concentration of Nd = 1016 cm-3. Find the
fraction of electrons in the donor state.

12. Semiconductor Equilibrium
Compensated semiconductors
Formed by adding both donor and acceptor impurities
in the same region
Energy-band diagram

13. Semiconductor Equilibrium
Compensated semiconductors (cont.)
With the assumption of charge neutrality, we can derive
Example
Consider a silicon semiconductor at T = 300K in
which Na = 1016 cm-3 and Nd = cm-3.
Assume ni = cm-3 and find p0 and n0.

14. Semiconductor Equilibrium
Position of Fermi energy level
As a function of doping levels As a function of temperature for a given doping level