ECE 663
• 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
States and state filling
Carrier populations depend on
• number of available energy states (density of states)• statistical distribution of energies (Fermi-Dirac function)
Assume electronsact ‘free’ with a parametrized effective mass m*
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Carrier Statistics
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Labeling states allows us to count them!
Si
ECE 663
dE
k
dE
dkx xxxxxxxx
E
k
For 1D parabolic bands, DOS peaks at edges
Where are the states?
ECE 663
dE
k
dE
dkx xxxxxxxx
E
kk = 2/L
(# states) = 2(dk/k) = Ldk/
DOS = g = # states/dE = (L/)(dk/dE)
Where are the states?
ECE 663
dE
k
dE
dkx xxxxxxxx
E
k
Analytical results for simple bands
k = 2/L
E = h2k2/2m* + Ec dk/dE = m/2ħ2(E-Ec)
DOS = Lm*/22ħ2(E-Ec) ~ 1/(E-Ec)
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dE
k
dE
dk
E
k
In higher dimensions, DOS has complex shapes
............
............
dE
2kdk
# k points increasesdue to angular integralalong circumference,as (E-Ec)
dNs = 2 x 2kdk/(2/L)2
g~S(E-Ec), step fn
Increasing Dimensions
ECE 663
From E-k to Density of States
Use E = Ec + ħ2k2/2mc to convert
kddk into dE
dNs = g(E)dE = 2 2 (dk/[2/L]) for each dimensionΣ1 k
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(mc/22ħ3)[2mc(E-Ec)]1/2 (Smc/2ħ2)(E-Ec) (mcL/ħ)/√2mc[E-Ec]
From E-k to DOS for free els
E
Ec
E
DOSDOS
E
DOS
3-D 2-D 1-D
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Real DOS needs computation
VB CB
DOS
E (eV)
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Keep in mind 3-D Density of States
32
)(2)(
cEEmm
Eg
In the interest of simplicity, we’ll try to reduce allg‘s to this form…
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(Ev-E)
(Ev-E)lh
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Such that number of states is preserved
What if ellipsoids?
E – EC = ħ2k12/2ml
* + ħ2k22/2mt
* + ħ2k32/2mt
*
1 = k12/a2 + k2
2/b2 + k32/b2
a = 2ml*(E-EC)/ħ2
b = 2mt*(E-EC)/ħ2
ab
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What if ellipsoids?
a = 2ml*(E-EC)/ħ2
b = 2mt*(E-EC)/ħ2
Total k-space volume of Nel ellipsoids = (4ab2/3)Nel
where
k-space volume of equivalent sphere = (4k3/3)
k = 2mn*(E-EC)/ħ2
where
mn* = (ml
*mt*2)1/3(Nel)2/3
EquatingK-spaceVolumes
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Valence bands more complex
where
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Valence bands more complex
hh
lh
But can try to fit two paraboloids for heavy and light holes and sum
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Valence bands more complex
hh
lh
4k3/3 = 4k13/3 + 4k2
3/3
k = 2mp*(EV-E)/ħ2
k1 = 2mhh(EV-E)/ħ2
k2 = 2mlh(EV-E)/ħ2
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mn* = (ml
*mt*2)1/3(Nel)2/3
mp* = (mhh
3/2 + mlh3/2)2/3
gC(E) = mn*[2mn
*(E-EC)]/2ħ3
gV(E) = mp*[2mp
*(EV-E)]/2ħ3
So map onto 3D isotropic free-electron DOS
with the right masses
ECE 663
Density of states effective mass for various solids
mn* = (ml
*mt*2)1/3(Nel)2/3
mp* = (mhh
3/2 + 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 0.067 0.45 0.082
0.28 0.042
0.067 0.473
1.084 0.5492
0.89 0.29
But how do we fill these states?
ECE 663
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:1. Pauli Exclusion Principle – each allowed state can
accommodate only one electron2. The total number of electrons is fixed N=Ni
3. The total energy is fixed ETOT = EiNi
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Fermi-Dirac Function
kTEE FeEf
1
1)(
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Carrier concentrations
• Can figure out # of electrons in conduction band
• And # of holes in valence band
top
c
E
Ec dEEfEgn )()(
v
bottom
E
Ev dEEfEgp )(1)(
El. Density StateDensity
Occupancyper state
gc(E)
f(E)
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kTEE
ed
F
FNn
CFc
cC
/)(
1)(
)(2
0
21
21
21
Full F-D statistics
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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 << -1, so drop +1 in denominator of F1/2 function
• For electrons in conduction band:
23
2
)(
23
2
*22
*22
hkTm
N
eNn
eh
kTmn
nC
kTEE
c
kTEE
n
FC
CF
Or equivalently
CB lumped density of states
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Carrier Concentrations
• Can do same thing for holes (nondegenerate approximation)
23
2
)(
23
2
*22
*22
h
kTmN
eNp
eh
kTmp
pv
kTEE
v
kTEE
p
vF
FV
VB lumped density of states
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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
**
ln4
3
ln22
)(
)()(
p
nF
C
VVCF
kTEE
VkTEE
C
mmkT
midgapE
NNkTEE
E
eNeN
pnVFFC
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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.
kT
E
VCkT
EE
VCi
GVC
eNNeNNnpn 22
)(
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Doped Semiconductors
Boron has 1 less el than tetrahedral
It steals an el from a nb. Si to form a tetrahedron. Thedeficit ‘hole’ p-dopes Si
P has 1 more el than tetrahedral
Extra el loosely tied (why?)
It n-dopes Si (1016/cm3 means 1 in 5 million)
ECE 663
Carrier Concentrations - nondegenerate
kTEE
v
vF
eNp)(
kTEE
c
FC
eNn)(
2
//)(
i
kTEVC
kTEEVC
nnp
eNNeNNnp GCv
Independent of Doping(This is at Equilibrium)
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A more useful form for n and p
For intrinsic semiconductors
kTEE
ci
iC
eNn)(
kTEE
vi
vi
eNp)(
n = nie(EF-Ei)/kT p = nie
(Ei-EF)/kT
EF
Ei
EF
Ei
ECE 663
Charge Neutrality
0/ k Poisson’s Equation
In equilibrium, E=0 and =0
0
AD
AD
NNnp
NNnpq
Charge NeutralityRelationship
Number of ionized donors:
kTEEDD
D
DFegNN
/)(1
1
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
-
-
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Charge Neutrality
kTEEDD
D
DFegNN
/)(1
1
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
0
ED PN e-(EN-EFN)/kT
<N> = 0.P0 + 1.P1 = f(ED-EF) = 1/[1 + e-(EF-ED)/kT]“0”
“1”
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Charge Neutrality
kTEEDD
D
DFegNN
/)(1
1
(gD = 2 for e’s, 4 for light holes, 1 for deep traps)
0
PN e-(EN-EFN)/kT
<N> = 0.P0 + 1.(P + P) + 2.P
ED
2ED + U0
ECE 663
Can equivalently alter ED to account for degeneracy
01
1
1
1/)(/)(
/)(/)(
kTEEA
kTEED
kTEEC
kTEEV
FADF
CFFV
egegeNeN
Charge Neutrality
n
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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
ECE 663
Altering Fermi Level with doping (… and later with fields)
• Recall:
kTEEi
kTEEi
Fi
iF
enp
enn/)(
/)(
Take ln
i
AFi
i
DiF
iFii
nN
kTEE
nN
kTEE
EEnp
kTnn
kT
ln
ln
lnln
n-type
p-type
ECE 663
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