Homogeneous_Semiconductors0801c.ppt

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

Energy Levels in Hydrogen Atom

Energy Levels for Electrons in a Doped Semiconductor

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

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

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













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













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

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.)

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

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]}

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.

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)

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

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

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

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

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

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






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

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.

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

Simpler
Expression
Temperature variation of some important “constants.”

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

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

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]

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]

Homogeneous_Semiconductors0801c.ppt

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