Title: ECE 875: Electronic Devices
1ECE 875Electronic Devices
- Prof. Virginia Ayres
- Electrical Computer Engineering
- Michigan State University
- ayresv_at_msu.edu
2Lecture 08, 27 Jan 14
Chp. 01 Concentrations Degenerate Nondegenerate
Effect of temperature Contributed by traps
3ExampleConcentration of conduction band
electrons for a nondegenerate semiconductor n
hot approximation of Eqn (16)
3D Eqn (14)
Three different variables (NEVER ignore this)
4AnswerConcentration of conduction band
electrons for a nondegenerate semiconductor n
MC
NC The effective density of states at the
conduction band edge.
5AnswerConcentration of conduction band
electrons for a nondegenerate semiconductor n
Nondegenerate EC is above EF
Sze eqn (21)
Use Appendix G at 300K for NC and n ND when
fully ionised
6Lecture 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
7Similar result for holesConcentration of
valence band holes for a nondegenerate
semiconductor p
Nondegenerate EC is above EF
Sze eqn (23)
Use Appendix G at 300K for NV and p NA when
fully ionised
8HW03 Pr 1.10
Shown kinetic energies of e- in minimum energy
parabolas KE ? E gt EC.
Therefore generic definition of KE as KE E -
EC
9HW03 Pr 1.10
Define Average Kinetic Energy
Single band assumption
10HW03 Pr 1.10
hot approximation of Eqn (16)
3D Eqn (14)
Average Kinetic Energy
Single band assumption
11HW03 Pr 1.10
hot approximation of Eqn (16)
3D Eqn (14)
Average Kinetic Energy
Equation 14
Single band definition
12Considerations
13Therefore Single band assumption means
14Therefore Use a Single band assumption in HW03
Pr 1.10
hot approximation of Eqn (16)
3D Eqn (14)
Start Average Kinetic Energy
Finish Average Kinetic Energy
15Reference http//en.wikipedia.org/wiki/Gamma_func
tionIntegration_problems Some commonly used
gamma functions
n is always a positive whole number
16Because nondegenerate used the Hot limit
E
C
E
F
E
i
E
V
-
F(E)
17Consider as the Hot limit approaches the Cold
limitwithin the degenerate limit
E
C
E
F
E
i
E
V
Use
18Will 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
19Concentration of conduction band electrons for a
semiconductor within the degenerate limit n
3D Eqn (14)
Three different variables (NEVER ignore this)
20Part of strategy pull all semiconductor-specific
info into NC. To get NC
21Next put the integrand into one single variable
22Next put the integrand into one single variable
Therefore have
And have
23Next put the integrand into one single variable
Change dE
Remember to also change the limits to hbottom and
htop
24Now have
Next write Factor in terms of NC
25Write Factor in terms of NC
Compare
26Write Factor in terms of NC
27F1/2(hF)
No closed form solution but correctly set up for
numerical integration
28Note
- 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
29Recall on Slide 5 for a nondegenerate
semiconductor n
hot approximation of Eqn (16)
3D Eqn (14)
F1/2(hF)
30Useful 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
31(No Transcript)
32Why 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
33Why 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?
34Why useful another reason
Answer Degenerate EF EC gt hF 0 Read off the
F1/2(hF) integral value at hF 0 0.6
Appendix G
35ExampleWhat 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
36Answer
37For degenerately doped semiconductors (Sze
degenerate semiconductors) the relative Fermi
level is given by the following approximate
expressions
38Compare Sze eqns (21) and (23) for
nondegenerate
Compare with degenerate
39Lecture 08, 27 Jan 14
Chp. 01 Concentrations Degenerate Nondegenerate
Effect of temperature Contributed by traps
40Nondegenerate will show this is the Temperature
dependence of intrinsic concentrations ni pi
ECE 474
41Intrinsic n pIntrinsic EF Ei Egap/2
Correct definition of intrinsic
Set concentration of e- and holes equal For
nondegenerate
42Solve for EF
EF for n p is given the special name Ei
43Substitute EF Ei into expression for n and p.
n and p when EF Ei are given name intrinsic
ni and pi
ni pi
pi
ni
44Substitute EF Ei into expression for n and p.
n and p when EF Ei are given name intrinsic
ni and pi
ni pi
pi
ni
Units of 4.9 x 1015 ? cm-3 K-3/2
45Plot ni versus T
ni
Note temperature is not very low
1018
106
46Dotted line is same relationship for ni as in the
previous picture.However this is doped Si
lt liquid N2
1017
When temperature T high, most electrons in
concentration ni come from Si bonds not from
dopants
1013