EE-612:
Lecture 31:
Heterostructure Fundamentals
Mark Lundstrom
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
Fall 2006
NCN
www.nanohub.org
Lundstrom EE-612 F06
1
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
2
reference
This presentation is based on a set of notes, which
contains detailed derivations of the results
presented here.
“Heterostructure Fundamentals,” by Mark
Lundstrom, Purdue University, Fall, 1995.
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3
review: pn homojunctions
potential must
decrease
EC
EC
E FN
EI
EI
E FP
EV
EV
(
qVBI = E FN − E FP
)q
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4
built-in potential
EC
EC
δN
E FN
EI
EI
E FP
δP
EV
(
qVBI = E FN − E FP
(
)
)
EV
N D = NCe
−δ N / kBT
qVBI = EC − EV − δ N − δ P
N A = NV e
qVBI = EG − δ N − δ P
qVBI = k BT ln N A N D ni2
−δ P / kBT
(
Lundstrom EE-612 F06
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)
review: pn homojunctions
E ( x ) = dEC ( x ) dx
EC
EC (x)
EI
EF
EV
E I (x)
E ( x ) = dEV ( x ) dx
EV (x)
0
Lundstrom EE-612 F06
x
6
reference for the energy bands
E0
field-free vacuum level
χN
ΦN
ΦP
χP
EC
EC
E FN
EI
EI
E FP
EV
EC = E0 − χ
EV = E0 − χ − EG
(
qVBI = Φ P − Φ N
)
Lundstrom EE-612 F06
EV
7
local vacuum level
qVBI
E0
()
qV x
El ( x )
EC
EC (x)
EI
EF
EV
E I (x)
EC ( x ) = E0 − χ − qV ( x )
EV ( x ) = E0 − χ − qV ( x ) − EG
EV (x)
0
Lundstrom EE-612 F06
x
8
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
9
reference for the energy bands
E0
field-free vacuum level
χN
ΦN
ΦP
χP
EC
EC
E FN
EI
EI
E FP
EV
EC = E0 − χ
EV = E0 − χ − EG
(
qVBI = Φ P − Φ N
)
Lundstrom EE-612 F06
EV
10
Al0.3Ga0.7As : GaAs (Type I HJ)
E0
field-free vacuum level
χ1
EC
χ2
ΔEC = χ 2 − χ1
≈ 0.23 eV
EI
EV
EG ≈ 1.80 eV
EC
‘electron affinity
rule’
EG ≈ 1.42 eV
EI
EV
ΔEV ≈ 0.25 eV
ΔEG = ΔEC + ΔEV
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11
N-Al0.3Ga0.7As : p-GaAs (Type I HJ)
E0
field-free vacuum level
χ1
EC
EG ≈ 1.80 eV
χ2
E FN
EC
qVBI = E FN − E Fp
E FP
EG ≈ 1.42 eV
EV
EV
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N-Al0.3Ga0.7As : p-GaAs (Type I HJ)
field-free vacuum level
E0
qV (x)
qVBI
El
ΔEC = χ 2 − χ1
χ1
EC
EG ≈ 1.80 eV
EV
χ2
EC
V jp
V jN
EG ≈ 1.42 eV
EV
ΔEV
Lundstrom EE-612 F06
VBI = V jN + V jp
13
N-Al0.3Ga0.7As : p+-GaAs
E0
field-free vacuum level
χ1
EC
χ2
E FN
EC
EG ≈ 1.80 eV
EV
EG ≈ 1.42 eV
E FP
Lundstrom EE-612 F06
EV
14
N-Al0.3Ga0.7As : p-GaAs
‘band spike’
V jN ≈ V j
EC
EG ≈ 1.42 eV
EC
EG ≈ 1.80 eV
ΔEV
EV
EV
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N+-Al0.3Ga0.7As : i-GaAs
E0
field-free vacuum level
χ1
EC
χ2
E FN
EC
EG ≈ 1.80 eV
E FP
EG ≈ 1.42 eV
EV
EV
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16
N-Al0.3Ga0.7As : i-GaAs
‘modulation doping’
‘2D electron gas’
EC
EG ≈ 1.42 eV
EV
EC
EG ≈ 1.80 eV
EV
depletion layer
inversion/accumulation layer
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17
N-Al0.3Ga0.7As : n-GaAs
E0
field-free vacuum level
χ1
EC
χ2
E FN
E Fn
EG ≈ 1.80 eV
EC
EG ≈ 1.42 eV
EV
EV
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N-Al0.3Ga0.7As : i-GaAs
‘isotype heterojunction’
EC
EC
EG ≈ 1.42 E
eV
V
EG ≈ 1.80 eV
EV
EV
depletion layer
accumulation layer
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19
P-Al0.3Ga0.7As : n-GaAs
E0
field-free vacuum level
χ1
χ2
EC
E Fn
EG ≈ 1.80 eV
EV
EC
EG ≈ 1.42 eV
E FP
Lundstrom EE-612 F06
EV
20
P-Al0.3Ga0.7As : n-GaAs
field-free vacuum level
E0
χ1
EC
qV (x)
qVBI
El
V jP
χ2
ΔEC
EG ≈ 1.80 eV
EF
V jn
EV
EC
EG ≈ 1.42 eV
ΔEV
depletionLundstrom
layer EE-612depletion
layer
F06
EV
21
P-GaSb : n-InAs (Type III)
E0
field-free vacuum level
χ1
EC
χ2
EG ≈ 0.72 eV
EV
E FP
E Fn
EC
EG ≈ 0.36 eV
EV
Lundstrom EE-612 F06
22
P-GaSb : n-InAs (Type III)
ΔEC = 0.87 eV
EC
EG ≈ 0.72 eV
EC
EF
EG ≈ 0.36 eV
EV
EV
accumulation layer
accumulation layer
Lundstrom EE-612 F06
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outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
24
geometry
p-type
n-type
− xP
0
Lundstrom EE-612 F06
xN
x
25
space charge density
ρ(x)
qN D
− xP
0
xN
x
− qN A
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26
electric field
E(x)
− xP
0
x
xN
( )
( )
ε P E 0− = ε N E 0+
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27
electrostatic potential
V (x)
V jN
− xP
V jP
0
Lundstrom EE-612 F06
xN
x
28
key results
( )
E (0 )= −qN
E 0− = −qN A x P ε P
+
p-type
N
ND + εPNA
)
V jN = VBI ε P N A ε N N D + ε P N A
)
V jP = VBI ε N N D
n-type
xN ε N
(ε
N
(
x
−xP
0
xN
V jP V jN = ε N N D ε P N A
x P = 2ε PV jP qN A
x N = 2ε NV jN qN D
Lundstrom EE-612 F06
29
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
30
Poisson’s equation for homostructures
(
q
d 2V
+
−
=
−
p
(x)
−
n
(x)
+
N
(x)
−
N
(x)
o
o
D
A
2
ε
dx
po ( x ) = ni e
( EI − EF ) / k B T
no ( x ) = ni e
)
( E F − E I )/ k B T
()
E I ( x ) = C − qV x
(
d 2V −q
− qV / k B T
qV / k B T
+
−
=
n
e
−
n
e
+
N
(
x
)
−
N
( x)
i
i
D
A
2
ε
dx
)
(Poisson-Boltzmann equation)
Lundstrom EE-612 F06
31
Poisson’s equation for heterostructures
(
q
d 2V
+
−
=
−
p
(
x
)
−
n
(
x
)
+
N
(
x
)
−
N
( x)
o
o
D
A
2
ε
dx
)
d 2V
d ⎛ dV ⎞
−ε 2 =→ ⎜ −ε
dx ⎟⎠
dx ⎝
dx
must relate n(x) and p(x) to V(x)
po ( x ) = NV ( x )e
( EV − E F )/ k B T
no ( x ) = N C ( x )e
()
( E F − EC ) / k B T
EV ( x ) = EC ( x ) − EG ( x )
EC ( x ) = E0 − χ ( x ) − qV x
(
)
E I ( x ) = E0 − χ ( x ) − EG ( x ) 2 + k BT 2 ln ⎡⎣ NV ( x ) N C ( x ) ⎤⎦ − qV ( x )
Lundstrom EE-612 F06
32
Poisson’s equation for heterostructures
po ( x ) = nir e
(
− q (V ( x ) − V p ( x )) / k B T
)(
)
qV p (x) = χ (x) − χ ref − EG (x) − EGref + k BT ln ⎡⎣ NV (x) NVref ⎤⎦
no ( x ) = nir e
(
q (V ( x ) +Vn ( x )) / k B T
)
qVn (x) = χ (x) − χ ref + k BT ln ⎡⎣ N C (x) N Cref ⎤⎦
d ⎛ dV ⎞
⎛ n e− q (V −V p )/ kB T − n eq (V +Vn )/ k B T − N + + N − ⎞
=
q
−
ε
ir
D
A⎠
⎝ ir
dx ⎜⎝
dx ⎟⎠
Lundstrom EE-612 F06
33
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
34
abrupt
V (x)
−x N
V jp
V jN
EC
EG ≈ 1.80 eV
xP
x
xP
x
EC
EG ≈ 1.42 eV
EV
ΔEV
χ (x)
EV
()
EC (x) = E0 − χ (x) − qV x
−x N
EV ( x ) = EC ( x ) − EG ( x )
Lundstrom EE-612 F06
35
graded
V (x)
−x N
xP
x
xP
x
EC
EG ≈ 1.42 eV
EC
EG ≈ 1.80 eV
EV
ΔEV
χ (x)
EV
()
EC (x) = E0 − χ (x) − qV x
−x N
EV ( x ) = EC ( x ) − EG ( x )
Lundstrom EE-612 F06
36
graded and quasi-neutral
E0
E
E0
E
χ (x)
χ (x) + qV (x)
EC
EC
EC
EG (x)
EF
EV
EV
EC
EG (x)
EV
x
EV
x
uniformly p-doped
Lundstrom EE-612 F06
37
general, graded heterostructure
E
qV (x)
E0
χ (x)
()
EC
EC (x) = E0 − χ (x) − qV x
EC
EV
EG ( x )
EV ( x ) = EC ( x ) − EG ( x )
EF
EV
x
Lundstrom EE-612 F06
38
quasi-electric fields
dV d χ
Fe = −
=q
+
dx dx
dx
dEC
E
E0
qV (x)
Fe = −qE ( x ) − qEQN ( x )
χ (x)
Fe
EC
EF
dV d χ + EG
Fh = +
= −q
−
dx
dx
dx
EV
Fh = +qE x + qEQP (x)
EC
EG (x)
EV
Fh
()
EC (x) = E0 − χ (x) − qV x
1 dχ
q dx
EQN ≡ −
(
dEV
x
EQP
EV ( x ) = EC ( x ) − EG ( x )
Lundstrom EE-612 F06
()
1 d (χ + E )
≡−
G
q
dx
39
)
graded and quasi-neutral
E
E0
Fe = − qE(x) − qEQN (x)
χ (x) + qV (x)
EC
EF
EG ( x )
EV
EC
dEG (x)
Fe = −
dx
EV
x
uniformly p-doped
Lundstrom EE-612 F06
40
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
41
hole current
J p = pμ p
dFp
dx
E
(
p = NV (x) e
V
)
− Fp / k B T
(
Fp = EV (x) − k BT ln p NV
)
⎡ 1 dp
dEV (x)
1 dNV ⎤
=
− k BT ⎢
−
⎥
dx
dx
⎣ p dx NV dx ⎦
dFp
⎡ dEV (x) k BT dNV ⎤
dp
J p = pμ p ⎢
+
T
μ
−
k
⎥ B p
dx
dx
dx
N
⎣
⎦
V
(
dEV (x) d
⎡⎣ E0 − χ (x) − EG (x) − qV (x) ⎤⎦ = q E(x) + EQP
=
dx
dx
Lundstrom EE-612 F06
)
42
hole and electron currents
⎡
k BT 1 dNV ⎤
dp
J p = pqμ p ⎢ E + EQP +
−
qD
⎥
p
dx
N
dx
q
⎣
⎦
V
‘DOS effect’
⎡
k BT 1 dN C ⎤
dn
J n = nq μn ⎢ E + EQN −
⎥ + qDn
q N C dx ⎦
dx
⎣
quasi-electric fields
EQP
(
1 d χ + EG
≡−
q
dx
)
EQN
1 dχ
≡−
q dx
Lundstrom EE-612 F06
43
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
44
bandgap shrinkage
n+-Si
p-Si
E
EG ≈ 1.0 eV
ni2 (x) = ni2 e ΔG / kB T
EG = 1.1eV
x
Lundstrom EE-612 F06
45
DD equations with bandgap shrinkage
ni2 (x) = ni2 e ΔG / kB T
(
)
⎡
d ⎡⎣ 1 − γ Δ G ⎤⎦ ⎤
dp
⎢
⎥
J p = pqμ p E +
− qD p
dx
dx
⎢
⎥
⎣
⎦
d (γ ΔG ) ⎤
⎡
dn
J n = nq μn ⎢ E −
+
qD
⎥
n
dx
dx
⎣
⎦
(γ = 0.5 is typically assumed)
Lundstrom EE-612 F06
46
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
47
measured offsets
(courtesy Jung
Han,EE-612
Purdue
Lundstrom
F06 Univ., 1995)
48
determining offsets
1) electron affinity rule:
-ΔEC = χ1 - χ2
-χ ~ 4 eV (surface charges, orientations, etc.)
-semi-semi dipole vs. semi-vacuum dipole
2) common anion rule:
-AlGaAs / GaAs
-valence band offset should be smaller than conduction band
3) Tersoff theory:
-gap states produced at interface
-lead to an interface dipole
-bands adjust to minimize dipole
-explains Schottky barrier heights too.
Lundstrom EE-612 F06
49
outline
1. Introduction
2. Energy bands in abrupt heterojunctions
3. Depletion approximation
4. Poisson-Boltzmann equation
5. Energy bands in graded heterojunctions
6. Drift-diffusion equation for heterostructures
7. Heavy doping effects and heterostructures
8. Band offsets
Lundstrom EE-612 F06
50