1. States and state filling
So far, we saw how to calculate bands for solids
Kronig-Penny was a simple example
Real bandstructures more complex
Often look like free electrons with effective mass m*
Given E-k, we can calculate ‘density of states’
High density of conducting states would imply metallicity
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2. Carrier Statistics Carrier populations depend on
number of available energy states (density of states)
statistical distribution of energies (Fermi-Dirac function)
Assume electrons
act ‘free’ with a
parametrized
effective mass m*
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3. Labeling states allows us to count them!Si
GaAs
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4. Where are the states? E dE dE dk k x k
For 1D parabolic bands, DOS peaks at edges
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5. Where are the states? E dE dE dk k x k (# states) = 2(dk/Dk) = Ldk/p
Dk = 2p/L
(# states) = 2(dk/Dk) = Ldk/p
DOS = g = # states/dE = (L/p)(dk/dE)
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6. Analytical results for simple bandsdE dE dk k x k
Dk = 2p/L
E = h2k2/2m* + Ec  dk/dE = m/2ħ2(E-Ec)
DOS = Lm*/2p2ħ2(E-Ec) ~ 1/(E-Ec)
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7. Increasing Dimensions
# k points increases
due to angular integral
along circumference,
as (E-Ec)
dNs = 2 x 2pkdk/(2p/L)2
g ~ Sq(E-Ec), step fn E dE dE dE dk
2pkdk . k k
In higher dimensions, DOS has complex shapes
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8. From E-k to Density of StatesΣ1
dNs = g(E)dE =  2 (dk/[2p/L]) for each dimension k
Use E = Ec + ħ2k2/2mc to convert
kddk into dE
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9. From E-k to DOS for free elsEc
DOS
3-D
2-D
1-D
(Wmc/2p2ħ3)[2mc(E-Ec)]1/ (Smc/2pħ2)q(E-Ec) (mcL/pħ)/√2mc[E-Ec]
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10. Real DOS needs computation
VB CB
E (eV)
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11. Keep in mind 3-D Density of States
In the interest of simplicity, we’ll try to reduce all
g‘s to this form…
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12. (Ev-E)
(Ev-E) lh
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13. What if ellipsoids? Such that number of states is preserved
E – EC = ħ2k12/2ml*
+ ħ2k22/2mt*
+ ħ2k32/2mt* b a
1 = k12/a2
+ k22/b2
+ k32/b2
a = 2ml*(E-EC)/ħ2
b = 2mt*(E-EC)/ħ2
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14. What if ellipsoids? mn* = (ml*mt*2)1/3(Nel)2/3
a = 2ml*(E-EC)/ħ2
b = 2mt*(E-EC)/ħ2
Total k-space volume of Nel ellipsoids
= (4pab2/3)Nel
where
k-space volume of equivalent sphere
= (4pk3/3)
k = 2mn*(E-EC)/ħ2
where
mn* = (ml*mt*2)1/3(Nel)2/3
Equating
K-space
Volumes
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15. Valence bands more complex
where
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16. Valence bands more complexhh lh
But can try to fit two paraboloids for heavy and light holes and sum
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17. Valence bands more complexhh lh
4pk3/3
= 4pk13/3 +
4pk23/3
k = 2mp*(EV-E)/ħ2
k1 = 2mhh(EV-E)/ħ2
k2 = 2mlh(EV-E)/ħ2
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18. So map onto 3D isotropic free-electron DOS
gC(E) = mn*[2mn*(E-EC)]/p2ħ3
gV(E) = mp*[2mp*(EV-E)]/p2ħ3
with the right masses
mn* = (ml*mt*2)1/3(Nel)2/3
mp* = (mhh3/2 + mlh3/2)2/3
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19. Density of states effective mass for various solids
mn* = (ml*mt*2)1/3(Nel)2/3
mp* = (mhh3/2 + mlh3/2)2/3
ml*
mt*
Nel
mhh
mlh
mn*
mp*
GaAs Si Ge 1 6 8
0.98
0.19
0.49
0.16
1.64
0.082
0.067
0.45
0.28
0.042
0.067
0.473
1.084
0.5492
0.89
0.29
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20. But how do we fill these states?

21. Fermi-Dirac Function Find number of carriers in CB/VB - need to know
Number of available energy states (g(E))
Probability that a given state is occupied (f(E))
Fermi-Dirac function derived from statistical mechanics of “free” particles with three assumptions:
Pauli Exclusion Principle – each allowed state can accommodate only one electron
The total number of electrons is fixed N=Ni
The total energy is fixed ETOT =  EiNi
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22. Fermi-Dirac Function
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23. Carrier concentrations
Can figure out # of electrons in conduction band
And # of holes in valence band
gc(E)
State
Density
Occupancy
per state
El. Density
f(E)
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24. Full F-D statistics
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26. Carrier Concentrations
If EC-EF >> kT the integral simplifies – nondegenerate semiconductors Fermi level more than ~3kT away from bottom/top of band
We then have h << -1, so drop +1 in denominator of F1/2 function
For electrons in conduction band:
Or equivalently
CB lumped density of states
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27. Carrier Concentrations
Can do same thing for holes (nondegenerate approximation)
VB lumped density of states
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28. Intrinsic Semiconductors
For every electron in the CB there is a hole in the VB
Fermi level is in middle of bandgap if effective masses not too different for e and h
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29. Intrinsic Semiconductors
Can plug in Fermi energy to find intrinsic carrier concentrations
Electrical conductivity proportional to n so intrinsic semiconductors have resistance change with temperature (thermistor) but not useful for much else.
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30. Doped Semiconductors Boron has 1 less el than P has 1 more el than
tetrahedral
Extra el loosely tied (why?)
It n-dopes Si
(1016/cm3 means 1 in 5 million)
Boron has 1 less el than
tetrahedral
It steals an el from a nb. Si
to form a tetrahedron. The
deficit ‘hole’ p-dopes Si
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31. Carrier Concentrations - nondegenerate
Independent of Doping
(This is at Equilibrium)
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32. A more useful form for n and p
For intrinsic semiconductors
n = nie(EF-Ei)/kT
p = nie(Ei-EF)/kT EF Ei EF Ei
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33. Charge Neutrality Poisson’s Equation In equilibrium, E=0 and =0 - -
Relationship
Number of ionized donors:
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
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34. Charge Neutrality ED “1” PN  e-(EN-EFN)/kT “0”ED
“1”
PN  e-(EN-EFN)/kT
“0”
<N> = 0.P0 + 1.P1 = f(ED-EF) = 1/[1 + e-(EF-ED)/kT]
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
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35. Charge Neutrality   2ED + U0 PN  e-(EN-EFN)/kT   ED
<N> = 0.P0 + 1.(P + P) + 2.P
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
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36. Charge Neutrality
Can equivalently alter ED to account for degeneracy n
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38. Nondegenerate Fully Ionized Extrinsic Semiconductors
n-type (donor) ND >> NA, ND >> ni
n  ND
p = ni2/ND
p-type (acceptor) NA >> ND, NA >> ni
p  NA
n = ni2/NA
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39. Altering Fermi Level with doping (… and later with fields)
Recall:
Take ln
n-type
p-type
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41. In summary
Labeling states with ‘k’ index allows us to count and get a DOS
In simple limits, we can get this analytically
The Fermi-Dirac distribution helps us fill these states
For non-degenerate semiconductors, we get simple
formulae for n and p at equilibrium in terms of Ei and EF, with EF determined by doping
Let’s now go away from equilibrium and see what happens