Recombination - nanohub.orgnanohub.org/groups/ece606lundstrom/File:Recombination.pdf · Carrier...
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Lundstrom ECE-606 S13
Notes for ECE-606: Spring 2013
Carrier Recombination
Professor Mark Lundstrom
Electrical and Computer Engineering Purdue University, West Lafayette, IN USA
1 2/19/13
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carrier recombination-generation
Lundstrom ECE-606 S13
n0 =1017 cm-3
p0 = ni2 n0 = 103 cm-3
N-type, equilibrium
(low-level injection)
Δp t = 0( ) >> p0
Δn t = 0( ) ≈ Δp t = 0( ) << n0Δp t = 0( )
expect: Δp t( ) = Δp t = 0( )e− t τ Δp may be either positive or negative.
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how can excess carriers recombine?
EC
EV
EG
∂n∂t b−b
= −B np − ni2( )
1) Band-to-band recombination
E = hf ≈ EG
energy given to photon
(Note that this is zero in equilibrium – as it should be.)
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excess carriers
EVfilled states
EC
EV
EF
n = n0 + Δn
p = p0 + Δp
Δn ≈ Δp
n0 =1017 cm-3
p0 = 103 cm-3
example: Δn = 108 cm-3 << n0
Δp ≈ Δn = 108 cm-3 >> p0
“low level injection”
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low level injection
The term “low level injection” means that the excess carrier concentration is orders of magnitude smaller than the equilibrium majority carrier concentration but orders of magnitude larger that the equilibrium minority carrier concentration.
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Low level injection in a p-type semi
Lundstrom ECE-606 S13
EC
EV
E = hf ≈ EGEG
∂n∂t b−b
= −B np − ni2( )
n = n0 + Δn ≈ Δn
p = p0 + Δp ≈ NA
np ≈ Δn NA >> ni2
∂Δn∂t b−b
≈ −BNAΔn = − Δnτ bb
τ bb =1
BNA
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excess carrier concentration vs. time
Lundstrom ECE-606 S13
∂Δn∂t bb
= − Δnτ bb
Δn t( ) = Ce− t /τbb
Δn 0( ) = C
Δn t( ) = Δn 0( )e− t /τb−b
τb-b = “minority carrier lifetime”
τ b−b =1
BNA
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how can excess carriers recombine?
EC
EV
EG
∂n∂t Auger
= Cnn np − ni2( ) +Cpp np − ni
2( )
2) Auger recombination energy given to a second
electron
(Note that this is zero in equilibrium – as it should be.)
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low level injection in a p-type semi
Lundstrom ECE-606 S13
∂n∂t Auger
= −Cpp np − ni2( )
n = n0 + Δn ≈ Δn
p = p0 + Δp ≈ NA
np ≈ Δn NA >> ni2
∂Δn∂t Auger
≈ −CpNA2Δn = − Δn
τ Auger
EC
EV
EG
energy given to a second electron
τ Auger =1
CpNA2
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how can excess carriers recombine?
Lundstrom ECE-606 S13
EC
EV
EG
energy released as thermal energy
ET
∂n∂t SRH
= ∂p∂t SRH
=− np − ni
2( )τ p n + n1( ) + τ n p + p1( ) (steady-state)
3) SRH recombination (defect assisted recombination)
“donor like” or “acceptor like”
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SRH recombination
Lundstrom ECE-606 S13
EC
EV
EGET
∂n∂t SRH
= ∂p∂t SRH
=− np − ni
2( )τ p n + n1( ) + τ n p + p1( )
n1 = nieET −Ei( ) kBT p1 = nie
Ei−ET( ) kBT
τ n =1
cnNT
τ p =1
cpNT
(steady-state)
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SRH recombination (low injection)
Lundstrom ECE-606 S13
∂n∂t SRH
= ∂p∂t SRH
=− np − ni
2( )τ p n + n1( ) + τ n p + p1( )
Δn >> n0Δp << p0
∂Δn∂t SRH
= − Δnτ n
τ n =1
cnNT
p-type n-type Δp >> p0Δn << n0
∂Δp∂t SRH
= − Δpτ p
τ p =1
cpNT
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SRH recombination (high injection)
Lundstrom ECE-606 S13
∂n∂t SRH
= ∂p∂t SRH
=− np − ni
2( )τ p n + n1( ) + τ n p + p1( )
Δn >> n0 Δp >> p0
∂Δn∂t SRH
= − Δnτ n +τ p
τ n =1
cnNT
Δn ≈ Δp
τ p =1
cpNT
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SRH recombination (depla]etion)
Lundstrom ECE-606 S13
∂n∂t SRH
= ∂p∂t SRH
=− np − ni
2( )τ p n + n1( ) +τ n p + p1( )
τ n =1
cnNT
τ p =1
cpNT
np << ni2 , n << n1, p << p1
∂n∂t SRH
= ∂p∂t SRH
= + niτ p + τ n
low level injection (summary)
15 Lundstrom ECE-606 S13
∂Δn∂t b−b
= − Δnτ b−b
∂Δn∂t SRH
= − Δnτ n
τ n =1
cnNT
τ b−b =1
BNA
∂Δn∂t Auger
= − Δnτ Auger
τ Auger =1
CpNA2
band-band
Auger
SRH
∂Δn∂t tot
= − ∂Δn∂t b−b
− ∂Δn∂t Auger
− ∂Δn∂t SRH
= − Δnτ eff
1τ eff
= 1τ b−b
+ 1τ Auger
+ 1τ SRH
low level injection (summary)
16 Lundstrom ECE-606 S13
∂Δn∂t tot
= − Δnτ eff
1τ eff
= 1τ b−b
+ 1τ Auger
1τ SRH
Δn t( ) = Δn 0( )e− t τ eff
recombination-generation
17 Lundstrom ECE-606 S13
R = X np − ni2( )
X = B
X = Cnn +Cpp
X = 1τ p n + n1( ) +τ n p + p1( )
band-to-band radiative
Auger
SRH
np > ni2( )
np < ni2( )
net recombination
net generation
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summary
Lundstrom ECE-606 S13
1) Band-to-band radiative recombination
2) Auger recombination
3) SRH recombination
dominates in direct gap semiconductors makes lasers and LEDs possible
dominates when the carrier densities are very high (heavily doped semiconductors or lasers)
dominates in high quality, indirect gap semiconductors and in low quality direct gap semiconductors
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other types of generation
Lundstrom ECE-606 S13
1) Impact ionization
2) Optical generation
“Inverse of Auger recombination”
“Inverse of band-to-band recombination”
photoelectric effect
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EF
EVAC
ΦM
E = hf > ΦM
Einstein, in 1905, when he wrote the Annus Mirabilis papers
http://en.wikipedia.org/wiki/Photoelectric_effect
how many photons can be absorbed?
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Example: Silicon EG = 1.1 eV. Only photons with a wavelength smaller than 1.13 µm will be absorbed.
λ < hcEG
solar spectrum (AM1.5G)
wasted energy
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Energy is lost for photons with energy greater than the bandgap.
hf > EG
Electron is excited above the conduction band.
However, extra energy is lost due to thermalization as electron relaxes back to the band edge.
absorption coefficient
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Incident flux: Φ0
Flux at position, x : Φ x( ) = Φ0e−α λ( )x
optical absorption coefficient:
α λ( ) > 0 for E > EG λ < hc EG( )
Generation rate at position, x :
G x( ) = −dΦ x( )dx
= Φ0α λ( )e−α λ( )x
α λ( )
direct vs. indirect
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p
Econduction
band
valence band
EG
Direct gap: strong absorption (high alpha)
p
Econduction
band
valence band
EGneed
phonon
Indirect gap: weaker absorption lower alpha)
E =p2
2m
light absorption vs. semiconductor thickness
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The direct bandgap of CIGS allows it to absorb light much faster than Silicon. A layer of silicon must be 104 microns thick to absorb ~100% of the light, while CIGS need only be about 2 microns thick.