The document discusses carrier concentration calculations in semiconductors. It defines density of states and distribution functions, which are used to calculate the number of electrons and holes. The Fermi-Dirac distribution gives the probability that an energy state is occupied. For non-degenerate semiconductors, the intrinsic carrier concentration is proportional to the exponential of the bandgap divided by temperature. For degenerate semiconductors with high doping, the Fermi level moves into the bands and the effective bandgap is reduced.

1. Homogeneous Semiconductors
• Dopants
• Use Density of States and Distribution
Function to:
• Find the Number of Holes and Electrons.

2. Energy Levels in Hydrogen Atom

3. Energy Levels for Electrons in a Doped Semiconductor

4. Assumptions for Calculation

8. Density of States
(Appendix D)
Energy Distribution Functions
(Section 2.9)
Carrier Concentrations
(Sections 2.10-12)

9. GOAL:
• The density of electrons (no) can be found
precisely if we know
1. the number of allowed energy states in a small energy
range, dE: S(E)dE
“the density of states”
2. the probability that a given energy state will be
occupied by an electron: f(E)
“the distribution function”
no = bandS(E)f(E)dE

10. For quasi-free electrons in the conduction band:
1. We must use the effective mass (averaged over all directions)
2. the potential energy Ep is the edge of the conduction band (EC)
For holes in the valence band:
1. We still use the effective mass (averaged over all directions)
2. the potential energy Ep is the edge of the valence band (EV)
 2
1
2
*
2
2
2
1
)
( C
dse
E
E
m
E
S 











 2
1
2
*
2
2
2
1
)
( E
E
m
E
S V
dsh













11. Energy Band Diagram
conduction band
valence band
EC
EV
x
E(x)
S(E)
Eelectron
Ehole
note: increasing electron energy is ‘up’, but increasing hole energy is ‘down’.
 2
1
2
*
2
2
2
1
)
( C
dse
E
E
m
E
S 











 2
1
2
*
2
2
2
1
)
( E
E
m
E
S V
dsh













12. Reminder of our GOAL:
• The density of electrons (no) can be found
precisely if we know
1. the number of allowed energy states in a small energy
range, dE: S(E)dE
“the density of states”
2. the probability that a given energy state will be
occupied by an electron: f(E)
“the distribution function”
no = bandS(E)f(E)dE

13. Fermi-Dirac Distribution
The probability that an electron occupies an
energy level, E, is
f(E) = 1/{1+exp[(E-EF)/kT]}
– where T is the temperature (Kelvin)
– k is the Boltzmann constant (k=8.62x10-5 eV/K)
– EF is the Fermi Energy (in eV)
– (Can derive this – statistical mechanics.)

14. f(E) = 1/{1+exp[(E-EF)/kT]}
All energy levels are filled with e-’s below the Fermi Energy at 0 oK
f(E)
1
0
EF
E
T=0 oK
T1>0
T2>T1
0.5

15. Fermi-Dirac Distribution for holes
Remember, a hole is an energy state that is NOT occupied by
an electron.
Therefore, the probability that a state is occupied by a hole
is the probability that a state is NOT occupied by an electron:
fp(E) = 1 – f(E) = 1 - 1/{1+exp[(E-EF)/kT]}
={1+exp[(E-EF)/kT]}/{1+exp[(E-EF)/kT]} -
1/{1+exp[(E-EF)/kT]}
= {exp[(E-EF)/kT]}/{1+exp[(E-EF)/kT]}
=1/{1+ exp[(EF - E)/kT]}

16. The Boltzmann Approximation
If (E-EF)>kT such that exp[(E-EF)/kT] >> 1 then,
f(E) = {1+exp[(E-EF)/kT]}-1  {exp[(E-EF)/kT]}-1
 exp[-(E-EF)/kT] …the Boltzmann approx.
similarly, fp(E) is small when exp[(EF - E)/kT]>>1:
fp(E) = {1+exp[(EF - E)/kT]}-1  {exp[(EF - E)/kT]}-1
 exp[-(EF - E)/kT]
If the Boltz. approx. is valid, we say the semiconductor is non-degenerate.

17. Putting the pieces together:
for electrons, n(E)
f(E)
1
0
EF
E
T=0 oK
T1>0
T2>T1
0.5
EV EC
S(E)
E
n(E)=S(E)f(E)

18. Putting the pieces together:
for holes, p(E)
fp(E)
1
0
EF
E
T=0 oK
T1>0
T2>T1
0.5
EV EC
S(E)
p(E)=S(E)f(E)
hole energy

19. Finding no and po
 
 
2
/
3
2
*
/
2
/
3
2
*
2
(min)
0
2
2
...
]
/
)
(
exp[
2
2
1
)
(
)
(






















 









kT
m
N
where
kT
E
E
N
dE
e
E
E
m
dE
E
f
E
S
p
dsh
V
V
F
V
kT
E
E
Ev
V
dsh
Ev
Ev
p
F
 
 
2
/
3
2
*
/
2
/
3
2
*
2
(max)
0
2
2
...
]
/
)
(
exp[
2
2
1
)
(
)
(






















 








kT
m
N
where
kT
E
E
N
dE
e
E
E
m
dE
E
f
E
S
n
dse
C
F
C
C
kT
E
E
Ec
C
dse
Ec
Ec
F
the effective density of states
at EC
the effective density of states
at EV

20. Energy Band Diagram
intrinisic semiconductor: no=po=ni
conduction band
valence band
EC
EV
x
E(x)
n(E)
p(E)
EF=Ei
where Ei is the intrinsic Fermi level
nopo=ni
2

21. Energy Band Diagram
n-type semiconductor: no>po
conduction band
valence band
EC
EV
x
E(x)
n(E)
p(E)
EF
]
/
)
(
exp[
0 kT
E
E
N
n F
C
C 


nopo=ni
2

22. Energy Band Diagram
p-type semiconductor: po>no
conduction band
valence band
EC
EV
x
E(x)
n(E)
p(E) EF
]
/
)
(
exp[
0 kT
E
E
N
p V
F
V 


nopo=ni
2

23. A very useful relationship
kT
E
V
C
kT
Ev
Ec
V
C
V
F
V
F
C
C
g
e
N
N
e
N
N
kT
E
E
N
kT
E
E
N
p
n
/
/
)
(
0
0 ]
/
)
(
exp[
]
/
)
(
exp[











…which is independent of the Fermi Energy
Recall that ni = no= po for an intrinsic semiconductor, so
nopo = ni
2
for all non-degenerate semiconductors.
(that is as long as EF is not within a few kT of the band edge)
kT
E
V
C
i
i
kT
E
V
C
g
g
e
N
N
n
n
e
N
N
p
n
2
/
2
/
0
0






24. The intrinsic carrier density
kT
E
V
C
i
i
kT
E
V
C
g
g
e
N
N
n
n
e
N
N
p
n
2
/
2
/
0
0





is sensitive to the energy bandgap, temperature, and
(somewhat less) to m*
2
/
3
2
*
2
2 










kT
m
N dse
C

25. The intrinsic Fermi Energy (Ei)
]
/
)
(
exp[
]
/
)
(
exp[ kT
E
E
N
kT
E
E
N V
i
V
i
C
C 




For an intrinsic semiconductor, no=po and EF=Ei
which gives
Ei = (EC + EV)/2 + (kT/2)ln(NV/NC)
so the intrinsic Fermi level is approximately
in the middle of the bandgap.

38. Higher Temperatures
Consider a semiconductor doped with NA ionized
acceptors (-q) and ND ionized donors (+q), do not assume
that ni is small – high temperature expression.
positive charges = negative charges
po + ND = no + NA
using ni
2 = nopo
ni
2/no + ND = no+ NA
ni
2 + no(ND-NA) - no
2 = 0
no = 0.5(ND-NA)  0.5[(ND-NA)2 + 4ni
2]1/2
we use the ‘+’solution since no should be increased by ni
no = ND - NA in the limit that ni<<ND-NA

39. Simpler
Expression
Temperature variation of some important “constants.”

45. Degenerate Semiconductors
1. The doping concentration is so
high that EF moves within a few kT
of the band edge (EC or EV).
Boltzman approximation not valid.
2. High donor concentrations cause
the allowed donor wavefunctions to
overlap, creating a band at Edn.
First only the high states overlap, but
eventually even the lowest state
overlaps.
This effectively decreases the
bandgap by
DEg = Eg0 – Eg(ND).
EC
EV
+ + + +
ED1
Eg0
Eg(ND)
for ND > 1018 cm-3 in Si
impurity band

46. Degenerate Semiconductors
As the doping conc. increases more, EF rises above EC
EV
EC (intrinsic)
available impurity band states
EF
DEg
EC (degenerate) ~ ED
filled impurity band states
apparent band gap narrowing:
DEg
* (is optically measured)
-
Eg
* is the apparent band gap:
an electron must gain energy Eg
* = EF-EV

47. Electron Concentration
in degenerately doped n-type semiconductors
The donors are fully ionized: no = ND
The holes still follow the Boltz. approx. since EF-EV>>>kT
po = NV exp[-(EF-EV)/kT] = NV exp[-(Eg
*)/kT]
= NV exp[-(Ego- DEg
*)/kT]
= NV exp[-Ego/kT]exp[DEg
*)/kT]
nopo = NDNVexp[-Ego/kT] exp[DEg
*)/kT]
= (ND/NC) NCNVexp[-Ego/kT] exp[DEg
*)/kT]
= (ND/NC)ni
2 exp[DEg
*)/kT]

48. Summary
non-degenerate:
nopo= ni
2
degenerate n-type:
nopo= ni
2 (ND/NC) exp[DEg
*)/kT]
degenerate p-type:
nopo= ni
2 (NA/NV) exp[DEg
*)/kT]