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1
Chapter 2: Semiconductor Fundamentals
2.1 Introduction2.2 Crystals and Crystal Structures2.3 Energy Bands2.4 Density of States2.5 Carrier Distribution Functions2.6 Carrier Densities2.7 Carrier Transport2.8 Carrier Recombination and Generation2.9 Continuity Equation 2.10 The Drift-Diffusion Model2.11 Semiconductor Thermodynamics
Chapter 2: Semiconductor Fundamentals
2.1 Introduction
Carrier densityCrystal structuresEnergy bandsDensity of statesDistribution function
Carrier transportDriftDiffusionRecombination/Generation
Chapter 2: Semiconductor Fundamentals
2.2 Crystals and Crystal Structures
Bravais lattice2D3DMiller indices
Cubic crystalsDiamond latticePacking density
Types of solid materials
Amorphous materialNo order“Frozen” liquid state
Poly-crystalline materialShort range order onlySmall crystalline regions with random
orientationCrystalline material
Both short and long range order
2
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Bravais lattice construction
32rr
2ar
1ar2132 23 aar rrr
+=
α
Figure 2.2.1 The construction of lattice points using unit vectors
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2D Bravais Lattices
2ar
1ar
o90 2ar
1aro90 2ar
1aro90
2ar
1ar
o120 2ar
1arα
(a)
(d) (e)
(b) (c)
Figure 2.2.2 The five Bravais lattices of two-dimensional crystals: (a) square, (b) rectangular, (c) centered rectangular, (d) hexagonal and (e) oblique
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2D Bravais Lattices
Name Number of Bravais lattices Conditions
Square 1 a1 = a2 , α = 90°
Rectangular 2 a1 ≠ a2 , α = 90°
Hexagonal 1 a1 = a2 , α = 120°
Oblique 1 a1 ≠ a2 , α ≠ 120°, α ≠ 90°
Table 2.2.1 Bravais lattices of two-dimensional crystals
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3D Bravais Lattices
Table 2.2.2 Bravais lattices of three-dimensional crystals (14 total)
Name Number of
Bravais lattices
Conditions Primitive Base-centered
Body-centered
Face-centered
Triclinic 1 a1 ≠ a2 ≠ a3 , α ≠ β ≠ γ
Monoclinic 2 a1 ≠ a2 ≠ a3 , α = β = 90° ≠ γ
Orthorhombic 4 a1 ≠ a2 ≠ a3 , α = β = γ = 90°
Tetragonal 2 a1 = a2 ≠ a3 α = β = γ = 90°
Cubic 3 a1 = a2 = a3 , α = β = γ = 90°
Trigonal 1 a1 = a2 = a3 , α = β = γ < 120° ≠ 90°
Hexagonal 1 a1 = a2 ≠ a3 , α = β = 90° , γ = 120°
3
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3D Bravais LatticesTriclinic
Monoclinic
Base-centered
a1 ≠ a2 ≠ a3
α ≠ β ≠ γ
a1 ≠ a2 ≠ a3
α = β = 90ο ≠ γ
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3D Bravais Lattices
Orthorhombic
Base-centered Face-centeredBody-centered
a1 ≠ a2 ≠ a3 α = β = γ = 90ο
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3D Bravais Lattices
Tetragonal
Body-centered
a1 = a2 ≠ a3 α = β = γ = 90ο
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Cubic Crystals
(a) (b) (c)
a a a
Figure 2.2.3 The simple cubic (a), the body-centered cubic (b)and the face centered cubic (c) lattice
SimpleCubic
Body-centeredCubic
Face-centeredCubic
a1 = a2 = a3 α = β = γ = 90ο
4
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3D Bravais Lattices
Trigonal
a1 = a2 = a3
α = β = γ < 120ο ≠ 90ο
Hexagonal
a1 = a2 ≠ a3
α = β = 90o γ = 120ο
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Miller Indices
x
z
y
p
r
q
x
z
y
p
r
q
Figure 2.2.4 Intersections of a plane and the x, y and z axes, as used to determine the Miller indices of the plane.
)rA
qA
pA(
Calculate inverses
Miller indices Example
)346(_
p = 2, q = -3, r = 4
Multiply with smallest common denominator, A = 12
1/2, -1/3, 1/4
Minus signs are replaced with a bar above the number
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Diamond Lattice
Figure 2.2.5 The diamond lattice of silicon and germanium
a
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The diamond lattice
a
)346(_
Model detail Front view
a /2
109.5o
5
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Zincblende Lattice
aFigure 2.2.6 The zincblende crystal structure of GaAs and InP
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Packing Density of Cubic Crystals
Radius Atoms/ unit cell
Packing density
# Neighbors
Simple cubic 2
a 1 % 52 6
=π 6
Body centered cubic 4
3 a 2 % 68 8
3=
π 8
Face centered cubic 4
2 a 4 % 74 6
2=
π 12
Diamond 83 a 8 % 34
163
=π 4
3
3
34
cellunit theofVolumeatoms of VolumeDensity Packing
a
rN
π==
1631)
83(
3481
34
33
33 πππ ==
aa
arN
621)
42(
3441
34
33
33 πππ ==
aa
arN
831)
43(
3421
34
33
33 πππ ==
aa
arN
61)
2(
3411
34
33
33 πππ ==
aa
arN
Chapter 2: Semiconductor Fundamentals
2.3 Energy Bands
Free electron modelKronig-Penney modelReal energy band diagramsSimplified energy band diagramTemperature and doping dependenceMetals, insulators and semiconductorsEffective mass
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Free electron model of a metal
MqΦ
VACUUME
E
0 L
x
States filledwith electrons
Figure 2.3.1 The free electron model of a metal.
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Energy levels versus lattice constant
E
Ec
a
a0
2p orbital
2s orbitalEv
closely-packed states
Figure 2.3.2 Energy levels in a carbon crystal versus lattice constant, a.
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Band diagrams of common semiconductors
Figure 2.3.7 Energy band diagram of (a) germanium, (b) silicon and (c) gallium arsenide
a) b) c)
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Simplified band diagram
E Evacuum
Ec
Ev
Conduction
qχ
Eg
band
{{{
Energybandgap
Valenceband
Figure 2.3.8 A simplified energy band diagram used to describe semiconductors.
Vacuum level
Conduction band edge
Valence band edge
ElectronAffinity
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Temperature dependence of the energy bandgap
00.20.40.60.8
11.21.41.6
0 200 400 600 800 1000 1200
Temperature (K)
Ene
rgy
Ban
dgap
(eV
)
GaAsSi
Ge
βα
+−=
TTETE gg
2
)0()(
Germanium Silicon GaAs
E g(0) (eV) 0.7437 1.166 1.519α (meV/K) 0.477 0.473 0.541β (K) 235 636 204
Figure 2.3.9 Temperature dependence of the energy bandgap of germanium (Ge), silicon (Si) and gallium arsenide (GaAs)
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Doping dependence of the energy bandgap
-250
-200
-150
-100
-50
0
1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20
Doping Density (cm-3)
Ban
dgap
cha
nge
(meV
)
1016 1017 1018 1019 1020
Ge
GaAs
SikTNqqNE
ssg εεπ
22
163)( −=Δ
Figure 2.3.10 Doping dependence of the energy bandgap of silicon (Si), gallium arsenide (GaAS) and germanium (Ge)
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Band DiagramsE
Metal Semiconductor Insulatora) b) c) d)
Eg
Conductionband
Valenceband
Figure 2.3.11 Possible energy band diagrams of a crystal.
Half filledband
Overlappingbands
Filled bandSmall energy gap
Filled bandLarge energy gap
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Electron and Hole transport
E
Allmostempty band
{{{
Energybandgap
Allmostfilled band
E
Figure 2.3.12 Energy band diagram in the presence of a uniform electric field.
EqdxdE
=
∑=
statesfilled
1ivb (-q)v
VJ
))()((1
statesempty
states all
∑ −−∑ −= iivb vqvqV
J
∑ +=
statesempty
)(1ivb vq
VJ
Current in an almost filled band
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Effective Mass Germanium Silicon GaAs
Smallest energy bandgap at 300 K Eg (eV) 0.66 1.12 1.424
Electron effective mass for density of states calculations
0
*,
mm dose
0.55 1.08 0.067
Hole effective mass for density of states calculations
0
*,
mm dosh
0.37 0.811 0.45
Electron effective mass for conductivity calculations
0
*,m
m conde0.12 0.26 0.067
Hole effective mass for conductivity calculations
0
*,m
m condh0.21 0.386 0.34
Table 2.3.2 Effective mass of carriers in germanium, silicon and gallium arsenide (GaAs)
8
Chapter 2: Semiconductor Fundamentals
2.4 Density of States
Derivation of 3D density of states
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kx
kz
ky
Number of states in k-spaceL
nkxπ
= , n = 1, 2, 3, …
Figure 2.4.1 Calculation of the number of states with wavenumber less than k.
3334)(
812 kLN ×××××= π
π
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Density of states derivation33
34)(
812 kLN ×××××= π
π
dEdkk
dEdk
dkdN
dEdN )(== 23L ππ
*
22
2=)(
mkkE h , providing
km
dEdk
2
*
=h
and h
Emk*2
=
0for ,281)(2/3*
33≥== EEm
hdEdN
LEg π
ccc EEEEmh
Eg ≥−= for ,28)( 2/3*3
π
cc EEEg <= for ,0)(
Conduction Band
Derivation (3 Dimensions)
Chapter 2: Semiconductor Fundamentals
2.5 Carrier Distribution Functions
Fermi-Dirac distributionExample
Impurity distribution functionBose-Einstein distribution functionMaxwell-Boltzmann distribution function
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Statistical Thermodynamics• Describes a large collection of particles
– Parameters• Temperature, volume, number/type of particles
– Statistical thermodynamics• All configurations are equally likely
• Needed concepts– Thermal equilibrium– Basic laws– Thermodynamic identity (energy conservation)– Fermi energy, EF– Thermal voltage, Vt
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10/1010/10
1/102/104/106/108/109/10
0/100/100/10
ExampleN = 10E = 23T > 0K
N = 10E = 20T = 0K
1234567
0
89
10
1234567
0
89
10
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1/102/104/106/108/109/10
10/101234567
10/100
0/108
0 %
100 %80 %60 %40 %40 %20 %
1 2 3 4 5 6 70 8 9 10
0/1090/1010
Example (cont.)
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The Fermi-Dirac distribution function
0%
20%
40%
60%
80%
100%
120%
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
E-E F (eV)Pr
obab
ility
600 K
150 K300 K
kTEEE
Ff
/)()(
e11−+
=
f(EF) = 1/2
Figure 2.5.1 The Fermi function at three different temperatures
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Example (cont.)
Energy(eV)
0
5
10
15
#1 #24#2 #3 #4 #23#22#21
Figure 2.5.2 Eight of the 24 possible configurations in which 20 electrons can be placed having a total energy of 106 eV
N = 20E = 106 eV
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Example (cont.)All 24 possible configurations for N = 20 and E = 106 eV
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Example (cont.)
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25
Energy (eV)
Prob
abili
ty (%
)
Figure 2.5.3 Probability versus energy averaged over the 24 possible configurations (circles) fitted with a Fermi-Dirac function (solid line) using kT = 1.447 eV and EF = 9.998 eV
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Other distribution functionsImpurity distributions Bose-Einstein distribution
For indistinguishable integer spin particles
Maxwell-Boltzmann distributionfor distinguishable particles
Donors
Acceptors with degenerate valence band
kTEEEMB
Ff
/)()(
e1
−=
1e1
/)()(
−=
− kTEEEBE
Ff
kTEEEacceptor
Faaf
/)()(
e411
−+=
kTEEEdonorFd
df /)(21
)(e1
1−+
=
11
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Other distribution functionsMaxwellBolzmann
FermiDirac
BoseEinstein
MaxwellBolzmann
FermiDirac
BoseEinstein
0%
25%
50%
75%
100%
125%
150%
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Energy (eV)
Occ
upan
cy p
roba
bilit
y
0.01
0.1
1
10
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Energy (eV)O
ccup
ancy
pro
babi
lity
Figure 2.5.4 Probability of occupancy versus energy of the Fermi-Dirac, the Bose-Einstein and the Maxwell-Boltzmann distribution
kTE /e−
Chapter 2: Semiconductor Fundamentals
2.6 Carrier Densities
Carrier density integralNon-degenerate semiconductorsDegenerate semiconductors
Joyce-Dixon approximationIntrinsic semiconductors
Mass action lawIntrinsic material as a reference
Doped semiconductorsDonors and acceptorsGeneral solutionTemperature dependence
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Electron and hole densities per unit energy
)()()( EfEgEn c=
)](1[)()( EfEgEp v −=
ccec EEEEmh
Eg ≥−= for ,28)( 2/3*3
π
vvhv EEEEmh
Eg ≤−= for ,28)( 2/3*3
π
electrons
holes
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Carrier density integral
E F
0.0E+00
1.0E+21
2.0E+21
3.0E+21
4.0E+21
5.0E+21
6.0E+21
-0.2 -0.1 0 0.1 0.2 0.3 0.4
Energy (eV)D
ensi
ty (c
m-3
eV-1
)
0%
20%
40%
60%
80%
100%
120%
Prob
abili
tyf(E)n(E)
g c (E)
6 x 1021
5 x 1021
4 x 1021
3 x 1021
2 x 1021
1 x 1021
0 x 1021
Figure 2.6.1 The carrier density integral. Shown are the density of states, gc(E), the density per unit energy, n(E), and the probability of occupancy, f(E).
∫=∫=band conduction theof topband conduction theof top
)()()(cc E
cE
dEEfEgdEEnn
12
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-2-1.5
-1-0.5
00.5
11.5
22.5
3
0.0E+00 5.0E+25 1.0E+26 1.5E+26 2.0E+26 2.5E+26
Density (cm-3eV-1)
Ene
rgy
(eV
)
0 0.2 0.4 0.6 0.8 1Probability of occupancy
f(E)E F
n (E)
p (E)
g c (E)
g v (E)
5 x 10250 10 x 1025 15 x 1025 20 x 1025 25 x 1025
Calculation of the electron and hole density
Figure 2.6.2 The density of states and carrier densities in the conduction and valence band
∫=∞
cco
EdEEfEgn )()(
∫ −=∞−
vE
vo dEEfEgp )](1)[(
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Solution to the Fermi integral
Non-degenerate semiconductors: Ev +3kT < EF < Ec + 3kT
kTEE
co
cF
eNn−
= 2/32
*]
2[2
h
kTmN e
cπ
=
kTEE
vo
Fv
eNp−
= 2/32
*]
2[2
h
kTmN h
vπ
=
electrons
holes
with
with
∫+
−=∞
−c
FEkT
EEceo dE
e
EEmh
n
1
128 2/3*3
π
∫
+
−=∞−
−
v
F
E
kTEEvho dE
e
EEmh
p
1
128 2/3*3
π
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Other solution to the Fermi integral
Joyce-Dixon approximation (Degenerate semiconductors)
... ))(93
163(
81ln 2 +−−+≅
−
c
o
c
o
c
ocF
Nn
Nn
Nn
kTEE
... ))(93
163(
81ln 2 +−−+≅
−
v
o
v
o
v
oFv
Np
Np
Np
kTEE
electrons
holes
Zero Kelvin solution
electrons
holes
cFcFe
o EEEEmn ≥−= for ,)(232 2/3
3
2/3*
2 hπ
FvFvh
o EEEEmp ≥−= ,)(232 2/3
2
2/3*
2 forhπ
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Fermi integral comparison
1.E+19
1.E+20
1.E+21
-0.1 0 0.1 0.2
Fermi Energy (eV)
Car
rier
Den
sity
(cm
-3)
1021
1020
1019
Joyce-Dixonapproximation
Non-degeneratesemiconductor
Zero Kelvin solution
Numericsolution
Figure 2.6.3 Carrier (electron) density versus Fermi energy, for Εc = 0 eV.
NC
13
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Intrinsic semiconductors
kTEE
co
cF
eNn−
= kTEE
vo
Fv
eNp−
=
kTEEcEEoi ci
iFeNnn /)(
)(−
= ==kTEE
vEEoiiv
iFeNpn /)(
)(−
= ==
kTEvci
geNNn 2/−=
)ln(21
2 c
vvci N
NkT
EEE +
+= )ln(
43
2 *
*
e
hvci
m
mkT
EEE +
+=
2/32
*]
2[2
h
kTmN e
cπ
= 2/32
*]
2[2
h
kTmN h
vπ
=
Carrier density for non-degenerate semiconductors
Intrinsic Fermi energy
Intrinsic carrier density, ni
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Mass action law
2/)(i
kTEEvcoo neNNpn cv ==⋅ −
kTEE
co
cF
eNn−
= kTEE
vo
Fv
eNp−
=kTE
vcigeNNn 2/−=
Product of electrons and holes
Mass action law: Product of electrons and holes is constant
2ioo npn =⋅
Electron, hole and intrinsic carrier density
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Intrinsic carrier density versus temperature
1.E-10
1.E-05
1.E+00
1.E+05
1.E+10
1.E+15
1.E+20
0 2 4 6 8 10 12
1000/T (1000/K)
Intr
insic
den
sity
(cm
-3)
1020
1010
10-10
germanium
GaAs
silicon100
100
Temperature (K)
2003001,000 kTEvci
geNNn 2/−=
Figure 2.6.4 Intrinsic carrier density versus temperature in gallium arsenide (GaAs), silicon and germanium.
2/32
*]
2[2
h
kTmN e
cπ
=
2/32
*]
2[2
h
kTmN h
vπ
=
Dotted lined without temperature dependence of the energy bandgap
βα
+−=
TTETE gg
2
)0()(
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Intrinsic material as a reference
kTEEio iFenn /)( −= kTEE
io Fienp /)( −=
i
oiF n
nkTEE ln+=i
oiF n
pkTEE ln−=
kTEE
co
cF
eNn−
= kTEE
vo
Fv
eNp−
=
kTEEcEEoi ci
iFeNnn /)(
)(−
= == kTEEvEEoi
iviF
eNpn /)()(
−= ==
Taking the ratio of n0 and ni, the ratio of po and ni:
Solving for EF:
14
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2D representation of the diamond lattice
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Donors and acceptors in silicon:
-+
Phosphorous and ArsenicGroup V
Boron and AluminumGroup III
Donor Acceptor
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Ionization of donors and acceptors
E
Ec
Ev
Ed
Ei
(a)
Ec
Ev
Ei
Ea
(b)
Figure 2.6.5 Ionization of a) a shallow donor and b) a shallow acceptor
dd NNn =≅ +0
aa NNp =≅ −0
Shallow impurities
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Ionization energy model
+q
Donor ion
electron
r
-q
Figure 2.6.6 Trajectory of an electron bound to a donor ion within a semiconductor crystal. A 2-D square lattice is used for ease of illustration.
eV 6.132
0
*
r
conddc
m
mEE
ε=−
15
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Compensated material
Ec
Ed
Ea
Ev
−+ −=− ad NNpn
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Range of doping densities
intrinsic low high silicondensity density densityni (300K) nSi(cm-3) (cm-3) (cm-3) (cm-3)
1010 1013←⎯⎯⎯→ 1020 1023
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Analysis for non-degenerate semiconductors
0)( =−+−= −+adoo NNnpqρ
−+ −+= adi
o NNn
nn
2
22)2
(2 i
adado n
NNNNn +
−+
−=
−+−+ 22)2
(2 i
dadao n
NNNNp +
−+
−=
+−+−
i
oiF n
nkTEE ln+=i
oiF n
pkTEE ln−=
Charge neutrality
Mass action law
Solution to the quadratic equation
Calculation of the Fermi energy
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n-type material
For Nd+ - Na
- >> ni
p = ni2 /(Nd
+ - Na- )
n = Nd+ - Na
-
ExampleNd = 1017 cm-3
Na = 5 x 1016 cm-3
ni = 1010 cm-3
n = 5 x 1016 cm-3
p = 2 x 103 cm-3
22)2
(2 i
adad nNNNNn +−
+−
=−+−+
16
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.87
General analysis
1.0E+10
1.0E+12
1.0E+14
1.0E+16
1.0E+18
1.0E+20
1.0E+22
-0.5 0 0.5 1 1.5
Fermi energy (eV)
Car
rier
den
sity
(cm
-3)
p n p + N d+n + N a
-
E cE v
E F
1010
1012
1014
1016
1018
1020
1022
Figure 2.6.8 Graphical solution of the Fermi energy based on the general analysis.
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.89
Temperature dependence
1E+14
1E+15
1E+16
1E+17
1E+18
0 5 10 15 20
1000/T (1000/K)
Ele
ctro
n de
nsity
(cm
-3) Intrinsic
Freeze-out
Extrinsic
1014
1015
1016
1017
1018
Figure 2.6.9 Electron density as a function of temperature
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Non-equilibrium carrier densities
)exp(kT
EFnnnn in
io−
=+= δ
)exp(kT
FEnppp pi
io−
=+= δ
Quasi-Fermi energies: Fp and Fp
Chapter 2: Semiconductor Fundamentals
2.7 Carrier Transport
Carrier driftMobility
Doping and temperature dependenceResistivityVelocity saturation
DiffusionDiffusion current derivationEinstein relation
Total currentHall effect
17
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Carrier drift
Ev
V -+
I
A
L
Figure 2.7.1 Drift of a carrier due to an applied electric field.
vLQ
tQIr /
==
vvLA
QAIJ rrr
rρ===
vqnJ rr−=
vqpJ rr=
Electrons: Holes:
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.93
Random carrier motion
x
= 0E = 0 /E
Figure 2.7.2 Random motion of carriers in a semiconductor with and without an applied electric field.
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.94
Mobility vs. doping density
0200400600800
1000120014001600
1.0E+14 1.0E+16 1.0E+18 1.0E+20
Doping density (cm-3)
Mob
ility
(cm
2 /V-s
)
Electrons
Holes
1014 1016 1018 1020
Figure 2.7.3 Electron and hole mobility versus doping density for silicon.
αμμμμ
minmaxmin
)(1rN
N+
−+=
Arsenic Phosphorous Boron
μmin (cm2/V-s) 52.2 68.5 44.9
μmax (cm2/V-s) 1417 1414 470.5
Nr (cm-3) 9.68 x 1016 9.20 x 1016 2.23 x 1017
α 0.68 0.711 0.719
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Doping dependent mobility
−+ += ad NNNwithα
μμμμ)(1minmax
min
rNN
+
−+=
18
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Resistivity of silicon
0.0001
0.001
0.01
0.1
1
10
100
1000
1.E+14 1.E+16 1.E+18 1.E+20
Doping density (cm-3)
Res
istiv
ity ( Ω
−cm
)
1014 1015 1018 1020101910171016
n-typep-type
)(11
pnq pn μμσρ
+==
Figure 2.7.4 Resistivity of n-type and p-type silicon versus doping density.
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.97
Sheet resistance: Uniform doping
WtLR ρ
=
Resistance of a uniformly doped semiconductor
Units: Ohm/square
Sheet resistance Lt
W
WLRR s=
Material withresistivity ρ
tRs
ρ=
L
W= Rs x number of squares
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Sheet resistance
0.01
0.1
1
10
100
1000
10000
1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20
Doping density (cm-3)
Shee
t res
istan
ce (
Ω/sq
)
1014 1015 1018 1020101910171016
n-typep-type
tRs
ρ=
WLRR s=
Figure 2.7.5 Sheet resistance of a 14 mil thick n-type and p-type silicon wafer versus doping density.
t = 14 mil
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.99
Non-uniform doping
Nd
Na(x)
xj x
Nd,Na
x+dx
x
Figure 2.7.6 Sheet resistance calculation of a non-uniform doping distribution.
∫∫≅=
jj x
ap
x
p
s
dxxNqdxxpxq
R
00
)(
1
)()(
1
μμ
dxxpxqdxdR
ps )()(
1μ
ρ==
19
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E
v
(a)
vsat
E
v
(b)
vsat
v =μ Evpeak
Epeak
Velocity saturation
Figure 2.7.7 Velocity-field relation for a) materials without accessible higher bands and b) materials with an accessible higher band.
satv
vE
EE
μμ
+=
1)(
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Diffusion current derivation
n
x-l l0
n(-l )n(l )
Figure 2.7.8 Carrier density profile used to derive the diffusion current expression
)]()([21
,, lxnlxnvthleftrightnrightleftnn =−−==Φ−Φ=Φ →→
dxdnvl
llxnlxnvl ththn −=
−=−=−=Φ
2)()(
dxdnvlqqJ thnn =Φ−=
dxdnDqJ nn =
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Diffusion current derivation
dxdnvlqqJ thnn =Φ−= dx
dnDqJ nn =
dxdpDqJ pp −=
cth
lvτ
=22
2*thvmkT
=
μτ
qkT
mq
qvm
vl cthth == *
2*
tnnn VqkTD μμ ==
tppp VqkTD μμ ==
Einstein relation Derivation
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Total current
dxdnDqnqJ nnn += Eμ
dxdpDqpqJ ppp −= Eμ
)( pntotal JJAI +=
Total carrier current = Drift current + Diffusion current
Total current = electron current + hole current
20
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.104
L
W
t
Vx
zy
x
vxEx
Bz
Ey
Fyvx
Ey
Fy
-
++
-
VH
L
W
t
VxEx
Bz
-
+
VH+ -
IxIx
L
W
t
Vx
zy
x
vx
Ex
Bz
Ey
vx
Ey
-
+
VH
L
W
t
VxEx
Bz
-
+
VH+ -
IxIx
+ -
+
-
The Hall effect
Figure 2.7.9 Hall setup and carrier motion for a) holes and b) electrons.
a) b)
0
1
pzx
yH qpBJ
==Δ E
R
Chapter 2: Semiconductor Fundamentals
2.8 Carrier Recombination and Generation
Recombination/generation mechanismsOther generation mechanisms
Simple R/G modelBand-to-band recombinationSHR recombinationSurface recombinationAuger recombination
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Carrier recombination and generation
Figure 2.8.1 Carrier recombination mechanisms in semiconductors
E
Et
Ec
Ev
Band-to-bandrecombination
Trap-assistedrecombination
Augerrecombination
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Light absorption and ionization
Figure 2.8.2 Carrier generation due to: a) light absorption and b) ionization due to high-energy particle beams
E
Ec
Ev
generation due tolight absorption
ionization due to chargedhigh-energy particles
Eph > EgEg QEph
21
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Avalanche multiplicationE
Ec
Ev
E
Figure 2.8.3 Impact ionization and avalanche multiplication of electrons and holes in the presence of a large electric field.
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.110
Carrier recombination and generation
E
Ec
band-to-bandrecombination
trap-assistedrecombination
Augerrecombination
Ev
Et
Figure 2.8.1 Carrier recombination mechanisms in semiconductors
σtht
tii
iSHR vN
kTEEnnp
npnU)cosh(2
2
−++
−=
Trap-assisted recombinationShockley-Hall-Reed recombination
Band-to-band recombination )( 2
ibb nnpbU −=−
)()( 22ipinAuger nnppnnpnU −Γ+−Γ=
Auger recombination
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Simple recombination model
n
ppnnn
nnRGU
τ0−
=−=
p
nnppp
ppRGUτ
0−=−=
p-type
n-type
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Light absorption
E
Eph > Eg
generation due to
light absorption
Eg
Figure 2.8.2 Carrier generation due to: a) light absorption
AExP
GGph
optlightnlightp
)(,, α==
Generation rate due to light
)(
)(xP
dxxPd
optopt α−=
Light absorption
22
Chapter 2: Semiconductor Fundamentals
2.9 Continuity Equation
Derivation of the continuity equationDiffusion equation
Solution for quasi-neutral semiconductors
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.118
Derivation of the continuity equation
Gn
Ec)(xJn )( dxxJn +
Evx
Rn
x x+dxFigure 2.9.1 Electron currents and possible recombination and generation processes
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.119
Derivation of the continuity equation
Gn
Ec)(xJn )( dxxJn +
Evx
Rn
x x+dx
Figure 2.9.1 Electron currents and possible recombination and generation processes
dxAtxRtxGAdxAt
txnnnq
dxxJqxJ nn )),(),(()(),( )()( −+−=
−+
−∂∂
dxxJdxxJ dxxJd
nnn )()()( +=+
),(),(),(1),( txRtxG
xtxJ
qttxn
nnn −+∂
∂=
∂∂
),(),(),(1),( txRtxG
xtxJ
qttxp
ppp −+∂
∂−=
∂∂
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.120
Alternate continuity equations
),(),(),(),(),( ),(2
2
txRtxGx
txnDx
txnx
txnt
txnnnnnn −+
∂∂
+∂
∂+
∂∂
= EE μμ
∂∂
),(),(),(),(),( ),(2
2
txRtxGx
txpDx
txpx
txpt
txpppppp −+
∂∂
+∂
∂−
∂∂
−= EE μμ
∂∂
),,,(),,,(),,,(),,,( 1 tzyxRtzyxGtzyxJt
tzyxnnnnq −+∇=
r
∂∂
),,,(),,,(),,,(),,,( 1 tzyxRtzyxGtzyxJt
tzyxppppq −+∇−=
r
∂∂
3-Dimensional continuity equation
Substitution of the current density
23
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Diffusion equations
n
pppn
p ntxn
x
txnD
ttxn
τ∂
∂∂
∂ 02
2 ),(),(),( −−=
p
nnnp
n ptxpx
txpD
ttxp
τ∂∂
∂∂ 0
2
2 ),(),(),( −−=
Continuity equations applied to a quasi-neutral region
Assumptions:Zero electric fieldSimple recombination/generation modelValid for minority carriers only
n-type material
p-type material
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Steady state solution to the diffusion equations
n
pppn
nxn
xd
xndD
τ0
2
2 )()(0
−−=
p
nnnp
pxpxd
xpdDτ
02
2 )()(0 −−=
nn LxLxpp eDeCnxn //
0)( ++= −pp LxLxnn eBeApxp //
0)( ++= −
nnn DL τ=ppp DL τ=
Steady state equations
General solutions
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.123
Formulation as a function of excess carrier densities
Excess minority carrier densities
Diffusion equations
nnn δ+= 0 ppp δ+= 0
22
2 )(0
n
pp
L
n
xd
nd δδ−= 22
2 )(0
p
nn
Lp
xdpd δδ
−=
Chapter 2: Semiconductor Fundamentals
2.10 The Drift-Diffusion Model
Assumptions10 equations, 10 unknowns
24
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Drift-diffusion model assumptions
Full ionization:all dopants are assumed to be ionized (shallow dopants)
Non-degenerate: the Fermi energy is assumed to be at least 3 kTbelow/above the conduction/valence band edge.
Steady state:All variables are independent of time
Constant temperature:The temperature is constant throughout the device.
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.126
Drift-diffusion model variables
ρ the charge densityn the electron densityp the hole densityE the electric fieldφ the potentialEi the intrinsic energyFn the electron quasi-Fermi energyFp the hole quasi-Fermi energyJn the electron current densityJp the hole current density
10/15/2008 © B. Van Zeghbroeck Semiconductor Fundamentals 2.127
Drift-diffusion model equations)( −+ −+−= ad NNnpqρ
ερ
=dxdE
E−=dxdφ
Eqdx
dEi =
kTEFi inenn /)( −=
kTFEi
pienp /)( −=
dxdnDqnqJ nnn += Eμ
dxdpDqpqJ ppp −= Eμ
τ1
)cosh(20
21
kTEE
npn
nnpxJ
iti
inq −
++
−−
∂∂
=
τ1
)cosh(20
21
kTEE
npn
nnpx
J
iti
ipq −
++
−−
∂
∂−=