EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
EE529 Semiconductor
Optoelectronics
Semiconductor Basics
1.
2.
3.
4.
Semiconductor materials
Electron and hole distribution
Electron-hole generation and recombination
p-n junction
Reading: Liu, Chapter 12, Sec. 13.1, 13.5
Reference: Bhattacharya, Sec. 2.1-2.2, 2.5-2.6, 4.2
Lih Y. Lin
EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Energy Bands in Semiconductors
Origin: Periodic lattice
structure in the crystal.
E-k diagram details the
band structure.
k: Electron wave vector
k = 2π/λ
Indirect bandgap
Si
Direct bandgap
Near the band edge:
2 k 2
E
= Ec +
2me*
2 k 2
E
= Ev −
2mh*
me*,h :
Effective masses
of electrons/holes
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Semiconductor Materials
Elementary: e.g., Si and Ge
Binary
Ternary
Quaternary
Lih Y. Lin
EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
III-V Compound Semiconductors
Solid curve: Direct bandgap
Dashed curve: Indirect bandgap
AlxGa1-xAs closely lattice
matched to GaAs
In1-xGaxAsyP1-y lattice
matched to InP for 0 ≤ y ≤ 1
and x = 0.47y
Matching lattice constant is important when depositing one semiconductor on another.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Electron-Hole Generation
How do electrons get to the conduction band (and leave holes in the valence band)?
Free electrons and holes can be generated by:
• Thermal excitation (thermal equilibrium)
• Optical excitation (quasi-equilibrium)
• Current injection (quasi-equilibrium)
It’s important to understand the carrier (electrons and holes) distribution as a function of energy.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Carrier Distribution in the Energy Band
Concentration of electrons (holes) versus
energy in the conduction (valence) band =
Density of states
(density of allowed energy levels)
Analogy: Density of available office spaces
X
Probability of occupancy
(probability that each of these levels is
occupied)
Analogy: People’s desire of occupying these
spaces
A skyscraper without elevators
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Probability of Occupancy
Fermi-Dirac function
f (E) =
1
 E − Ef
exp
 k BT

 + 1

f(E) = probability of occupancy by an electron
1 - f(E) = probability of occupancy by a hole (in valence band)
(Electrons like to sink to the bottom, holes like to float to the top.)
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Density of States ― Bulk Material
k2
ρ(k ) =
π2
(2me* )3/2
1/2
ρ=
E
−
E
(E)
(
)
C
C
2π 2  3
(2mh* )3/2
1/2
ρ
=
E
−
E
(E)
(
)
V
V
2π2  3
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Density of States ― Quantum Wells
Green laser diode
Nature Photonics 3, 432 - 434 (2009)
Discrete energy levels
 2 (qπ / d )
Eq =
, q = 1, 2, ...
2m
2
Density of states
 me*
, E > EC + Eq1

2
ρC ( E ) =
 π d1
0,
E < EC + Eq1

Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Quantum Wires and Quantum Dots
Quantum wires
 (2 / d1d 2 )( me* / 2π)
, E > EC + Eq1 + Eq 2

1/2
ρC ( E ) =
 ( E − EC − Eq1 − Eq 2 )

0,
otherwise

(Gudiksen et al., Nature, 2002)
Quantum dots
ρc ( E ) = 2δ( E − Ec − Eq1 − Eq 2 − Eq 3 )
1
d1d 2 d3
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Thermal-Equilibrium Carrier Concentration
Bulk Semiconductor
(a)
(b)
(c)
(d)
1/2
E g(E) ∝ (E–Ec) E
Ec+χ
∞
 E − EC 
n0 =
N C (T ) F1/2  F
∫ ρC ( E ) f ( E )dE =

E
 k BT 
c
E
 E − EF 
p0 =
NV (T ) F1/2  V
∫ ρV ( E )[1 − f ( E )]dE =

−∞
 k BT 
E
v
[1–f(E)]
CB
Area
nnEE((E)dE
E )dE == nn
Area == ∫
For
electrons
Ec
Ec
nE(E)
EF
EF
Ev
Ev
For holes
pE(E)
Area = p
VB
We can simplify this more …
0
g(E)
fE)
nE(E) or pE(E)
(a) Energy band diagram. (b) Density of states (number of states per unit energy per
unit volume). (c) Fermi-Dirac probability function (probability of occupancy of a
state). (d) The product of g(E) and f(E) is the energy density of electrons in the CB
(number of electrons per unit energy per unit volume). The area under nE(E) vs. E is
the electron concentration.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Carrier Concentration and Mass Action Law
For non-degenerate semiconductors, ( E
C
− EF ) / k BT ≥ 3.6
( EF − EV ) / k BT ≥ 3.6
 ( E − EF ) 
n0 NC exp  − C
=

k
T
B


N=
2(2πme* k B T / h 2 )3/2 : Effective density of states at the conduction band edge
C
 ( E − EV ) 
p0 NV exp  − F
=

k
T
B


*
2 3/2
N=
m
k
T
h
2(2
/
) : Effective density of states at the valence band edge
π
V
h B
The location of the Fermi level energy EF is the key.
 Eg 
 2πk BT 
*
* 3/2
2
=
n0 p0 4 
m
=
m
exp
−
n
Mass action
law:
(
)


e
h
i

2
 h 
 k BT 
3
→ Knowing one carrier concentration, you can determine the other (no matter intrinsic or extrinsic)
Lih Y. Lin
12
EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Intrinsic and Extrinsic Semiconductor
N 
EC + EV 1
− k BT ln  C 
2
2
 NV 
EFi
Intrinsic:=
n-type semiconductor
p-type semiconductor
Electron energy
B atom sites every 106 Si atoms
Electron Energy
Ec
x Distance
into crystal
CB
As+
e–
~0.05 eV
h+
Ec
Ed
B–
+
As
+
As
As+
+
As
Ea
Ev
Ev
B–
h+
B–
B–
B–
~0.05 eV
VB
x
i
6
i
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Exercise: Fermi Levels in Semiconductors
a)
b)
c)
d)
Where is the Fermi level of intrinsic bulk Si at room temperature?
What kind of dopants can make it n-type?
If the donor concentration Nd is 1016 cm-3, where will the Fermi level be?
If the wafer is compensation-doped with boron (Na = 2 x 1017 cm-3), where
will the Fermi level be?
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Quasi Equilibrium
― What happens to the Fermi levels during photon absorption
Probability of occupancy for electrons:
Probability of occupancy for holes:
fC ( E ) =
1
 E − EFc
exp 
 k BT
fV ( E ) =
1
E −E
exp  Fv
 +1
k
T
 B

How to calculate the quasi Fermi levels?

 +1

Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Quasi-Fermi Levels
The figure below shows positions of quasi-Fermi levels as a function of photogenerated electron-hole pair density. The semiconductor is n-type GaAs with
ND = 1015 cm-3.
Q: Why does εfp decrease gradually with increasing density while εfn shows a
sudden increase?
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Exercise: Carrier Lifetime and Internal
Quantum Efficiency in GaAs
A n-type ( n0 = 5 × 1018 m −3 ) GaAs is under optical excitation generating excess carrier
concentration N =n − n0 =p − p0 . It has the followsing recombination coefficients:
A= 5 × 105 s −1 , B= 8 × 10−17 m3s −1 , and C =Ce + Ch =5 × 10−42 m 6 s −1 . Assume that
C=
C=
C / 2 . (1) Find the range of N where each of the three different recombination
e
h
processes (Shockley-Read, bimolecular, Auger) dominates. (2) Plot the spontaneous
carrier lifetime τ s as a function of N for 1018 ≤ N ≤ 1026 m −3 . (3) Assume only the
bimolecular recombination process is radiative, plot the internal quantum efficiency vs N.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Bimolecular Recombination and
Steady-State Concentration
Rate of recombination R = Bnp ( 1
cm3 ⋅ sec
)
In thermal equilibrium, generation = recombination
G0 = Bn0 p0 ( 1
cm3 ⋅ sec
)
With electron-hole injection (by external current or photon)
Net radiative recombination rate
R = Bnp − G0 =
N
τrad
1
τrad =
B ( N + n0 + p0 )
In steady sate:
dN
N
=
G−
=
0
dt
τrad
→ Determines Ν if G is known, and therefore the quasi-Fermi levels.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Exercise: Carrier Injection at Steady State
Electron-hole pairs are injected into n-type GaAs at a rate G = 1023 /cm3s . At room
temperature GaAs has the following parameters: Eg = 1.42 eV, ni = 2.33× 106 cm-3,
=
N C 4.35 × 1017 cm-3, and =
NV 9.41 × 1018 cm-3. The thermal equilibrium concentration
of electrons is n0 = 1016 cm-3. Assume bimolecular recombination process dominates
and the coefficient for bimolecular recombination B= 8 × 10−11 cm3s −1 . Determine:
(a) The thermal equilibrium concentration of holes p0.
(b) The steady-state excess carrier concentration Ν.
(c) The recombination lifetime τrad .
(d) The separation between the quasi-Fermi levels EFc – EFp.
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
The p-n Junction
p
n
Build-in field
eN aW p
eN W
E0 =
− d n =
−
ε
ε
As+
Bh+
(a)
e–
M
M
Metallurgical Junction
Eo
Neutral p-region
E (x)
Neutral n-region
–Wp
0
Built-in potential
Wn
x
(e)
(b)
–Eo
V(x)
M
log(n), log(p)
Wp
Wn
Vo
Space charge region
(f)
Depletion widths
ppo
x
nno
(c)
ni
pno
npo
Wn =
PE(x)
eVo
Hole PE(x)
x
x=0
ρnet
x
M
eNd
Electron PE(x)
–Wp
x
Wn
-eNa
1
V0 =
− E0 (Wn + W p )
2
k T N N 
= B ln  a 2 d 
e
 ni 
(d)
–eVo
(g)

2ε N a 
1

 V0
e Nd  Nd + Na 

2ε N d 
1

 V0
e Na  Nd + Na 
W=
Wn + W p
0
Wp =
=
2ε N a + N d
V0
e Na Nd
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Biased p-n Junction
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
p-n Junction Under Forward Bias
Law of the junction
 eV 
pn (0) = pn 0 exp 

k
T
 B 
 eV 
n p (0) = n p 0 exp 

k
T
 B 
Excess minority carrier concentration
 x' 
∆pn ( x ') =
∆pn (0) exp  − 
 Lh 
 x '' 
∆n p ( x '') =
∆nn (0) exp  
 Le 
=
Lh ,e
Diffusion length
Dh ,e τh ,e
Dh ,e : Diffusion coefficient
τh ,e : Minority carrier lifetime
Lih Y. Lin
22
EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Current in Forward-biased p-n Junction
J
p-region
SCL
n-region
J = Jelec + Jhole
Total current
Majority carrier diffusion
and drift current
Jhole
Jelec
Minority carrier diffusion
current
The total current
anywhere in the device is
constant. Just outside the
depletion region it is due
to the diffusion of
minority carriers.
x
–Wp
Wn
Recombination current
Diffusion current
 eDh ni2  
 eV

 exp 
 k BT
 Lh N d  
 
 − 1
 
 eDe ni2  
 eV

 exp 
 k BT
 Le N a  
J diff J D ,hole + J D ,elec
=
 
 − 1
 
J D ,hole
J D ,elec
 eDh
eDe
=
+

 Lh N d Le N a
J diff

 eV
J so exp 
 k BT

 2
 eV
 ni exp 
 k BT
 
=
J recom
eni
2
 W p Wn 
 eV 
+
exp




 2 k BT 
 τe τ h 
A more accurate result:
 
 − 1
 
  eV  
=
J recom J r 0 exp 
 − 1
k
T
2
  B  
 
 − 1
 
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
I-V Characteristics of a p-n Junction
I = Io[exp(eV/ηkBT) − 1]
I
mA
V
Shockley equation
Reverse I-V characteristics of a
pn junction (the positive and
negative current axes have
different scales)
nA
Space charge layer
generation.
  eV  
=
I I 0 exp 
 − 1
η
k
T
  B  
η : Diode ideality factor
η = 1: Diffusion controlled
η = 2: SCL recombination controlled
Lih Y. Lin
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EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
p-n Junction under Reverse Bias
Current due to thermally
generated EHP:
J gen =
eWni
τg
τ g : Mean thermal generation time
Total reverse current
J rev ≅ J s + J gen
 eDh
eDe
=
+

L
N
 h d Le N a
 2 eWni
 ni +
τg

Lih Y. Lin
25
EE 529 Semiconductor Optoelectronics
– Semiconductor Basics
Exercise: GaAs p-n Junction
• Device parameters:
– Cross sectional area A = 1 mm2.
– Na (p-side doping) = Nd (n-side doping) = 1023 m-3.
– Coefficient of recombination Β = 7.21 x 10 -16 m3s-1
– ni = 1.8 x 1012 m-3
– εr = 13.2
– µh (in the n-side) = 250 cm2V-1s-1, µe (in the p-side) =
5000 cm2V-1s-1
– Carrier recombination time in the depletion region =
10 ns
• What is the I-V characteristics? Calculate the diffusion
current and the recombination current under 1V bias.
Lih Y. Lin
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