ECE 875:
Electronic Devices
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 08, 27 Jan 14
Chp. 01
Concentrations
Degenerate
Nondegenerate
Contributed by traps
}
Effect of temperature
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Example:
Concentration of conduction band electrons for a nondegenerate
semiconductor: n:
3D: Eq’n (14)
n
“hot” approximation
of Eq’n (16)
EC  E
 N E F E dE
EC
Three different variables
(NEVER ignore this)
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Answer:
Concentration of conduction band electrons for a
nondegenerate semiconductor: n:
mde 
3
2
2M C
 2 3
mde
MC
NC
The effective density of states at the
conduction band edge.
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Answer:
Concentration of conduction band electrons for a
nondegenerate semiconductor: n:
E F  EC
n  N C exp
kT
Nondegenerate: EC is above EF:
 EC  E F 
n  N C exp  

kT 

Sze eq’n (21)
 NC 
EC  E F  kT ln 

 n 
Use Appendix G at 300K for NC
and n ≈ ND when fully ionised
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Lecture 07: Would get a similar result for holes:
This part is called NV: the effective density of states at the valence band
edge.
Typically valence bands are symmetric about G: MV = 1
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Similar result for holes:
Concentration of valence band holes for a nondegenerate
semiconductor: p:
EV  EF
p  NV exp
kT
Nondegenerate: EC is above EF:
 EF  EV 
p  NV exp  

kT 

Sze eq’n (23)
 NV
EF  EV  kT ln 
 p
Use Appendix G at 300K for NV
and p ≈ NA when fully ionised



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HW03: Pr 1.10:
Shown: kinetic energies of e- in minimum energy parabolas: KE  E > EC.
Therefore:
generic definition of
KE as:
KE = E - EC
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HW03: Pr 1.10:
EC  E
 E  E dn( E )
C
Define: Average Kinetic Energy

EC
EC  E
 dnE 
EC
Single band assumption
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HW03: Pr 1.10:
“hot” approximation
of Eq’n (16)
3D: Eq’n (14)
EC  E
 E  E N E F E dE
C
Average Kinetic Energy

EC
EC  E
 N E F E dE
EC
Single band assumption
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HW03: Pr 1.10:
“hot” approximation
of Eq’n (16)
3D: Eq’n (14)
EC  E
 E  E N E F E dE
C
Average Kinetic Energy

EC
EC  E
 N E F E dE
EC
Equation 14:
N (E) 
mde 
3
2
2M C
 2 3
E  EC 1/ 2
Single band definition
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Considerations:

mde 
N (E) 
3
2
 
2M C
2 3
E  EC 
1/ 2
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Therefore: Single band assumption: means:
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Therefore: Use a Single band assumption in HW03: Pr 1.10:
“hot” approximation
of Eq’n (16)
3D: Eq’n (14)
EC  E
 E  E N E F E dE
C
Start: Average Kinetic Energy

EC
EC  E
 N E F E dE
EC
Finish: Average Kinetic Energy
3
 kT
2
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Reference:
http://en.wikipedia.org/wiki/Gamma_function#Integration_problems
Some commonly used gamma functions:
n is always a positive
whole number
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Because nondegenerate: used the Hot limit:
EC
EF
Ei
EV
-
= F(E)
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Consider: as the Hot limit approaches the Cold limit:
“within the degenerate limit”
EC
EF
Ei
EV
Use:
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Will find: useful universal graph: from n:
Dotted: nondegenerate
Solid: within the
degenerate limit
y-axis:
Fermi-Dirac
integral: good
for any
semiconductor
x-axis: how much energy do e-s need: (EF – EC)
versus how much energy can they get: kT
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Concentration of conduction band electrons for a
semiconductor within the degenerate limit: n:
3D: Eq’n (14)
n
EC  E
 N E F E dE
EC
Three different variables
(NEVER ignore this)
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Part of strategy: pull all semiconductor-specific info into NC. To get NC:
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Next: put the integrand into one single variable:
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Next: put the integrand into one single variable:
Therefore have:
And have:
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Next: put the integrand into one single variable:
Change dE:
Remember to also change the limits to hbottom and htop:
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Now have:
Next: write “Factor” in terms of NC:
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Write “Factor” in terms of NC:
Compare:
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Write “Factor” in terms of NC:
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F1/2(hF)
No closed form solution
but correctly set up for
numerical integration
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Note:

hF = (EF - EC)/kT is semiconductor-specific

F1/2(hF) is semiconductor-specific

But: a plot of F1/2(hF) versus hF is universal
Could just as easily write this as F1/2(x) versus x
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Recall: on Slide 5 for a nondegenerate semiconductor: n:
3D: Eq’n (14)
n
“hot” approximation
of Eq’n (16)
EC  E
 N E F E dE
EC
 EC  E F 
n  N C exp  

kT 

n  NC

2   
 E F  EC 
 exp h F 
exp 
  NC

 2
  2 
 kT 

2

F1/2(hF)
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Useful universal graph:
Dotted: nondegenerate
Solid: within the
degenerate limit
y-axis:
Fermi-Dirac
integral: good
for any
semiconductor
x-axis: how much energy do e-s need: (EF – EC)
versus how much energy can they get: kT
VM Ayres, ECE875, S14
VM Ayres, ECE875, S14
Why useful: one reason:
Around -1.0
Starts to diverge
-0.35: ECE 874 definition of
“within the degenerate limit”
Shows where hot
limit becomes the
“within the
degenerate limit”
EC
EF
Ei
EV
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Why useful: another reason:
F(hF)1/2 integral is
universal: can read
numerical solution
value off this graph for
any semiconductor
Example: p.18 Sze:
What is the
concentration n for any
semiconductor when
EF coincides with EC?
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Why useful: another reason:
Answer:
Degenerate
EF = EC => hF = 0
Read off the F1/2(hF)
integral value at hF = 0
≈ 0.6
Appendix G
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Example:
What is the concentration of conduction band electrons for degenerately
doped GaAs at room temperature 300K when EF – EC = +0.9 kT?
EF
0.9 kT
EC
Ei
EV
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Answer:
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For degenerately doped semiconductors (Sze: “degenerate semiconductors”):
the relative Fermi level is given by the following approximate expressions:
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Compare: Sze eq’ns (21) and (23): for nondegenerate:
Compare with degenerate:
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Lecture 08, 27 Jan 14
Chp. 01
Concentrations
Degenerate
Nondegenerate
Contributed by traps
}
Effect of temperature
VM Ayres, ECE875, S14
Nondegenerate: will show: this is the Temperature dependence of intrinsic
concentrations ni = pi
ECE 474
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Correct definition of intrinsic:
Intrinsic: n = p
Intrinsic: EF =Ei = Egap/2
Set concentration of e- and holes equal: For nondegenerate:
=
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Solve for EF:
EF for n = p is given the special name Ei
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Substitute EF = Ei into expression for n and p.
n and p when EF = Ei are given
name: intrinsic: ni and pi
ni =
pi =
n i = p i:
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Substitute EF = Ei into expression for n and p.
n and p when EF = Ei are given
name: intrinsic: ni and pi
ni =
pi =
Units of 4.9 x 1015 = ? = cm-3 K-3/2
n i = p i:
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Plot: ni versus T:
ni
1018
Note: temperature is
not very low
106
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Dotted line is same relationship for ni as in the previous picture.
However: this is doped Si:
1017
1013
< liquid N2
When temperature T =
high, most electrons in
concentration ni come
from Si bonds not from
dopants
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