Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom...

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Notes for ECE-606: Spring 2013 Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA [email protected] 1 2/19/13 2 carrier recombination-generation Lundstrom ECE-606 S13 n 0 = 10 17 cm -3 p 0 = n i 2 n 0 = 10 3 cm -3 N-type, equilibrium (low-level injection) Δpt = 0 ( ) >> p 0 Δnt = 0 ( ) ≈Δpt = 0 ( ) << n 0 Δpt = 0 ( ) expect: Δpt () = Δpt = 0 ( ) e t τ Δp may be either positive or negative.

Transcript of Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom...

Page 1: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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

[email protected]

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.

Page 2: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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”

Page 3: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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

Page 4: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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.)

Page 5: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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”

Page 6: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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

Page 7: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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

Page 8: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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

Page 9: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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

Page 10: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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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

Page 11: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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.

Page 12: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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

Page 13: Recombination - nanoHUBRecombination.pdf · Carrier Recombination Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu

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.