ECE 875: Electronic Devices

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ECE 875Electronic Devices

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

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Lecture 08, 27 Jan 14
Chp. 01 Concentrations Degenerate Nondegenerate
Effect of temperature Contributed by traps

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ExampleConcentration of conduction band
electrons for a nondegenerate semiconductor n
hot approximation of Eqn (16)
3D Eqn (14)
Three different variables (NEVER ignore this)
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AnswerConcentration of conduction band
electrons for a nondegenerate semiconductor n
MC
NC The effective density of states at the
conduction band edge.
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AnswerConcentration 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
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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 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
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HW03 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
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HW03 Pr 1.10
Define Average Kinetic Energy
Single band assumption
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HW03 Pr 1.10
hot approximation of Eqn (16)
3D Eqn (14)
Average Kinetic Energy
Single band assumption
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HW03 Pr 1.10
hot approximation of Eqn (16)
3D Eqn (14)
Average Kinetic Energy
Equation 14
Single band definition
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Considerations
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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 Eqn (16)
3D Eqn (14)
Start Average Kinetic Energy
Finish Average Kinetic Energy
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Reference http//en.wikipedia.org/wiki/Gamma_func
tionIntegration_problems Some commonly used
gamma functions
n is always a positive whole number
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Because nondegenerate used the Hot limit
E
C
E
F
E
i
E
V
-
F(E)
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Consider as the Hot limit approaches the Cold
limitwithin the degenerate limit
E
C
E
F
E
i
E
V
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 Eqn (14)
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
hot approximation of Eqn (16)
3D Eqn (14)
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
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(No Transcript)
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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 gt hF 0 Read off the
F1/2(hF) integral value at hF 0 0.6
Appendix G
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ExampleWhat 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 eqns (21) and (23) for
nondegenerate
Compare with degenerate
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Lecture 08, 27 Jan 14
Chp. 01 Concentrations Degenerate Nondegenerate
Effect of temperature Contributed by traps

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Nondegenerate will show this is the Temperature
dependence of intrinsic concentrations ni pi
ECE 474
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Intrinsic n pIntrinsic EF Ei Egap/2
Correct definition of intrinsic
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
pi
ni
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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
pi
ni
Units of 4.9 x 1015 ? cm-3 K-3/2
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Plot ni versus T
ni
Note temperature is not very low
1018
106
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Dotted 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
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