Light-Emitting Diodes

445.664 445.664

Light

Light- -Emitting Diodes Emitting Diodes

Chapter 2.

Chapter 2.

Radiative and Non

Radiative and Non- -radiative Recombination radiative Recombination

Euijoon Yoon Euijoon Yoon

2 0 0 law

action

Mass n p = n_i

p p

p n

n

n =

₀

+ Δ and =

₀

+ Δ

•Under equilibrium conditions,

n₀ equilibrium electron concentration p₀ equilibrium hole concentration n_i intrinsic carrier concentration

• Under excitation conditions,

(absorption of light, injection of current)

n free electron concentration

Δn excess free electron concentration p free hole concentration

Δp

Electron and Hole Concentrations

Electron and Hole Concentrations

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Recombination of Carriers Recombination of Carriers

dt Bnp dp dt

R dn

equation rate

r

Bimolecula = − = − =

R recombination rate

( 10^-11~ 10^-9 cm³/s for III-V semiconductors) B bimolecular recombination coefficient

• The number of recombination events will be proportional to the concentration of electrons and holes

Radiative Recombination for Low

Radiative Recombination for Low- -level Excitation level Excitation

• Electrons and holes are generated and annihilated in pairs.

• For the case of low-level excitation, the photogenerated carrier concentration is much smaller than the majority carrier

concentration.

state - steady under

) ( ) (

Δ

nt =

Δ

pt

⇒

[ ][ ] ( )

excess

i

R R

t n p n B Bn t

p p t n n B R

) (

) ( )

(

0

0 0 2

0 0

+

=

+ +

= +

+

=

Δ Δ Δ

0

0

p p

n

n and << <<

⇒ Δ Δ

R₀ equilibrium recombination rate R_excess excess recombination rate

)

( )

( ) (

0

0

G

excess

R R

excess

G R dt G

t

dn = − = + − +

G₀ equilibrium generation rate G_excess excess generation rate

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Carrier Concentration as a Function of Time Carrier Concentration as a Function of Time

( )

n(t) R B

(

n p

)

n(t)

dt n(t) d dt n

n(t) d dt

d

excess

= − + Δ

−

= Δ

= Δ +

=

⇒

_{0 0 0}

• t = 0(light off) Î G_excess = 0

• Equilibrium Î G₀ = R₀

( )

Δ

_τ

Δ

Δ n(t) = n

₀

e

^{−B n0+ p0 t}

= n

₀

e

^−t

( )

[

_{0 0}

]

¹

lifetime

carrier

τ

= B n + p ⁻

tors semiconduc type

- n for 1

1

tors semiconduc type

- p for 1

1

0 A

B N B n

B N B p

=

=

=

= τ τ

Carrier Lifetime

Carrier Lifetime

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Minority carriers

• Termination of photoexcitation

ÎThe carrier concentration decays exponentially with a characteristic time constant. (minority carrier lifetime, τ) Majority carriers

• The carrier concentration also decays with the same time constant.

• However, only a very small fraction of the majority carriers disappear by recombination.

Carrier lifetime can be extracted from minority carrier concentrations.

p-type semiconductors Î electron carrier lifetime n-type semiconductors Î hole carrier lifetime

Minority Carrier Lifetime Minority Carrier Lifetime

Minority Carrier Lifetime of p

Minority Carrier Lifetime of p- -type GaAs type GaAs

• In nominally undoped materials, minority carrier lifetime as long as 15 µs have been measured in GaAs at room temperature.

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Radiative Recombination for High

Radiative Recombination for High- -level Excitation level Excitation

• Electrons and holes are generated and annihilated in pairs.

• For the case of high-level excitation, the photogenerated carrier concentration is larger than the equilibrium carrier

concentration.

state - steady under ) ( ) (

Δn t = Δp t

⇒

) (

n >> n

₀

+ p

₀

⇒ Δ

[

₀ ( )

][

₀ ( )

]

² ( )²

B n nt p pt Bn B nt

R = +

Δ

+

Δ

= _i +

Δ

(

₀

)

1 0

0

1 )

( n( ) n

n t t B

n

Δ Δ

Δ Δ

=

= + ₋

2

n(t) B n(t)

dt

d Δ = − Δ

⇒

Înon-exponential carrier decay

Carrier Lifetime for High

Carrier Lifetime for High- -level Excitation level Excitation

) ) (

(

) ( 1 )

( 1

0 ) ( 0

dt n(t) d

t t n

n(t) t e t

dt n n(t) d

e n n(t)

t t t

Δ Δ τ

Δ τ Δ τ

Δ

Δ Δ

τ τ

−

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ −

=

=

−

−

Low-level excitation

High-level excitation

(

₀

)

¹

)

( = − = t + B n

⁻

dt n(t) d

t Δ Δ n(t) Δ

τ

time-dependent

carrier lifetime : definition of carrier lifetime

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Luminescence Decay Luminescence Decay

• The carrier decay in semiconductors can be measured by the decay of the luminescence after a short optical excitation pulse.

• luminescence intensity ∝ recombination rate ( R )

• All non-exponential decay functions can be expressed by an exponential function with a time-dependent time constant.

τ

τ Δ

0 /

) (

n e

^t

dt t

R = − dn =

⁻

(

0 ¹

)

²

) (

= − = − +

−

n Bt

B dt

t R dn

Δ

for low excitation

for high excitation

1 0) ( 0 ) ( /

0

)

(

^{+ −}

−

=

−

=

^{t B n}

t t

t

n e

e n t

n Δ

^τ

Δ

^Δ

Δ

Îstretched exponential function

Radiative Recombination for High

Radiative Recombination for High- -level Excitation level Excitation

•For sufficient long times, low-level excitation conditions will be reached and τwill approach the low-level value.

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Non- Non -radiative Recombination in the Bulk radiative Recombination in the Bulk

Two basic recombination mechanisms in semiconductors Radiative recombination

: One photonwith energy equal to the bandgap energy is emitted.

Non-radiative recombination

: The electron energy is converted to vibrational energy of lattice atoms (phonon). Î Heat generation

Cause for Non

Cause for Non- -radiative Recombination Events radiative Recombination Events

• The most common cause for non-radiative recombination events : Defects in the crystal structure

foreign atoms – interstitial, substitutional

native defects – vacancies, antisite defects, interstitials dislocations

complexes of defects

• All such defects have energy level structures that are different from substitutional semiconductor atoms.

ÎOne or several energy levels within the forbidden gap of the semiconductor

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Electron and Hole Concentrations Electron and Hole Concentrations

(a) Recombination via deep levels (b)Auger recombination

(c) Radiative recombination

⎟⎠

⎜ ⎞

⎝

⎛ −

⎟ =

⎠

⎜ ⎞

⎝

⎛ −

= kT

E N E

kT p E N E

n_{0 c}exp ^{F c} _{0 v}exp ^{v F}

Electron and Hole Concentrations Electron and Hole Concentrations

E_F energy of Fermi level

E_c energy of lowest conduction band level E_v energy of highest valence band level

N_c effective density of levels for conduction band N_v effective density of levels for valence band

n_i density of electrons in an intrinsic semiconductor n₀ density of electrons at equilibrium condition p₀ density of holes at equilibrium condition

2 0

0 _{c v} exp ^{v c} n_i

kT E N E

N p

n ⎟=

⎠

⎜ ⎞

⎝

⎛ −

=

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⎟⎠

⎜ ⎞

⎝

⎛ −

⎟ =

⎠

⎜ ⎞

⎝

⎛ −

= kT

E N E

kT p E N E

n_{1 c}exp ^{T c} _{1 v}exp ^{v T}

Electron and Hole Concentrations (

Electron and Hole Concentrations ( E E

_FF

= E = E

_{T T}

) )

E_T energy of the trap level in the bandgap

n₁ density of electrons in the conduction band for E_F = E_T p₁ density of holes in the valence band for E_F = E_T

0 0 2 1

1 exp n n p

kT E N E

N p

n _{c v} ^{v c}⎟= _i =

⎠

⎜ ⎞

⎝

⎛ −

=

Shockley

Shockley- -Read Recombination Read Recombination

p n

p n

Nt

σ σ

ν ν

and

and

: the concentration of the traps

: the electron and hole thermal velocities : the capture cross sections of the traps

n n t n

p p t p

N C

N C

σ ν

σ ν

=

=

Ref) W. Shockley and W. T. Read

probability per unit time that a hole in the valence bandwill be captured at the traps filled with electrons probability per unit time that an electron in the conduction bandwill be captured at the empty traps

The net capture rate of electrons in conduction band

)

1

1

( f n C f n

C

R

_cn

=

_n

−

_t

−

_{n t} The net capture rate of holes in valence band

)

1

1

( f p

C p f C

R

_cp

=

_{p t}

−

_p

−

_t

f

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Non-radiative recombination rate by Shockley-Read Recombination

( ) ( ) [ ]

(

^{n n n}

)

^C

(

^{p p p}

)

C

n p n n p p C C p

p C n n C

n p pn C R C

p n

p n p

n p n

SR Δ Δ

Δ Δ

+ + +

+ +

− +

= + + + +

= −

1 0 1

0

1 1 0

0 1

1

1

1 ) ( )( )

(

2 0 0 1

1p n p ni

n = =

( n n n ) C ( p p p )

C

n p n p p R n

p n

SR

Δ Δ

Δ Δ Δ Δ

+ + +

+ +

+

=

₋

+

₋

1 0 1 1

0 1

0 0

Shockley

Shockley- -Read Recombination Read Recombination

For steady-state conditions,

R

_cn

= R

_cp

= R

_SR

) ( ) 1 (

) ( ) (

1 1

1 1

1 1

p p C n n C

p C n f C

p p C n n C

p C n f C

p n

p n t

p n

p n

t + + +

= + + −

+ +

= +

Non- Non -radiative Lifetime by Shockley radiative Lifetime by Shockley- -Read Read

(

^{n n n}

)

^C

(

^{p p p}

)

C

n n p

n

p Δ Δ

Δ

τ + + + + +

+

= ₋ + ₋

1 0 1 1

0 1

0

1 0

Thenon-radiative lifetimeof excess electrons can be deduced from the equation R_SR = Δn / τ(constant recombination rate) and Δn = Δp.

When small deviation from equilibrium is assumed,

0 1

0

0 , and

p >> n p n<< p

⇒

Δ

n n t n n

σ ν

N C =

=

= 1 1

τ

0

τ

p p t p p

N

C ν σ

τ

τ

1 1

0

=

=

=

p-type semiconductor

0 1

0

0 , and

n >> p n n<<n

⇒

Δ

n-type semiconductor

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Non- Non -radiative Lifetime of Intrinsic Materials radiative Lifetime of Intrinsic Materials

When we assume C_n= C_p and very small Δnand Δp

( )(

n n n

) (

p p p

)

n n N_{t n n} p

Δ Δ

σ Δ

τ ν + + + + +

+

= +

1 0 1

0

0

1 0

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

+ + +

=

0 0

1

1 1

0 p n

n p

τ

n

τ

For the special case of intrinsic materials, ⇒ n₀ =p₀ =ni

( )

( ) ( ( ) )

( )

( ) ( ( ) )

⎥⎦⎤

⎢⎣⎡ + − + − −

=

⎥⎦

⎢ ⎤

⎣

⎡ + − + −

=

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + +

=

2

/ exp

/ 1 exp

2

/ exp

/ 1 exp

1 2

0 0 0

1 1

kT E E kT

E E

n

kT E E n

kT E E n

n n p

Fi T Fi

T n

i

T Fi i

Fi T i

n

i n

i

τ τ τ τ

(

_{0 1}⁰

) (

⁰_{0 1}

)

0

1

p p n n

n p

n + + +

= + τ

•When the trap level is at or close to the mid-gap energy

( E_T– E_Fi= 0 ), the non-radiative lifetime is minimized ( τ= 2τ_n0 ).

ÎMid-gap levels are effective non-radiative recombination centers.

•As T increases, the non-radiative recombination lifetime decreases.

ÎThe radiative band-to-band recombination efficiency decreases at high temperatures.

•Some devices are based on radiative recombination through a deep state. ex) N-doped GaP

ÎThe deep-level recombination rate increases with

⎥⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ −

+

= kT

E E_{T Fi}

n

i 1 cosh

τ 0

τ

Non- Non -radiative Lifetime of Intrinsic Materials radiative Lifetime of Intrinsic Materials

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Visualization of Defects Visualization of Defects

• Clusters of defects or extended defects (dislocations) ÎLuminescence-killing nature

• Luminescence in the vicinity of the defects is reduced due to the non-radiative recombination channels.

Luminescence from Deep

Luminescence from Deep- -level Transition level Transition

• While most deep-level transitions are non-radiative, some deep-level transitions are radiative.

• n-type GaN / sapphire

365 nm : band-to-band transition

~ 550 nm : deep-level transition (related to V )

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Auger Recombination Auger Recombination

• The electron-hole recombination energy ( ~ E_g)

ÎExcitation of a free electron high into the conduction band or a hole deeply excited into the valence band

• The highly excited carriers will subsequently lose energy by multiple phonon emissionuntil they are close to the band edge.

p n C R

np C R

n Auger

p Auger

2 2

=

= p-type semiconductor ( 2 holes + 1 electron ) n-type semiconductor ( 1 holes + 2 electron ) C_p Auger coefficient in valence band

C_n Auger coefficient in conduction band ÎOwing to the differences in conduction and valence band

structure in semiconductors, C_pand C_nare generally different.

Auger Recombination Auger Recombination

• In the high-excitation limit, the non-equilibrium carriers have a much higher concentration than equilibrium carriers.

p n ≈

⇒

(

^{C C}

)

^{n3 Cn3}

R_Auger = _p+ _n =

C : Auger coefficient

10^-28~ 10^-29cm⁶/s for III-V semiconductors

Î Auger recombination reduces the luminescence efficiency at very high excitation intensity.

Î At low carrier concentrations, the Auger recombination rate is very small and negligiblefor practical purposes.

n3

R_Auger∝

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Non- Non -radiative Recombination at Surfaces radiative Recombination at Surfaces

Band diagram model

• This model is based on the strict periodicity of a crystal lattice.

• Surfaces are strong perturbation of the periodicityof a crystal lattice.

Î The modified band diagram at a semiconductor surface.

(the addition of electronic states within the forbidden gap)

Chemical point of view

• Atoms at the surface cannot have the same bonding structure as bulk atoms due to the lack of neighboring atoms.

• Some of the valence orbitals do not form a chemical bond.

Îdangling bonds

• These partially filled electron orbitals are electronic states that can be located in the forbidden gap.

Non- Non -radiative Recombination at Surfaces radiative Recombination at Surfaces

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Surface Recombination Surface Recombination

( )

⎥ ⎦

⎢ ⎤

⎣

⎡

+

− − +

= +

=

_∞

S L

L x n S

n x n n x n

n n

n n

τ Δ τ

Δ ( ) 1 exp /

)

(

_{0 0}

Assuming

p-type semiconductor

uniform steady-state generation rate ex) illumination

S surface recombination velocity x distance from semiconductor surface L_n carrier diffusion length

0 0

) 0 ( , For

) 0 ( , 0 For

n n S

n n n S

→

∞

→

+

→

→ Δ _∞

Non-radiative recombination at the surface

Æreduced luminescence efficiency & heating of the surface 10¹cm/s Si

10³cm/s InP

10⁶cm/s GaAs

S

Demonstration of Surface Recombination

Demonstration of Surface Recombination

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Concentration of Native Defects Concentration of Native Defects

•Any semiconductor crystal will have some native defects.

ÎEven though the concentration of these native defects can be low, it is never zero.

•E_a : energy to create a specific point defect in a crystal lattice Probability that such a defect indeed forms at a specific lattice site

= exp ( – E_a/ kT)

•Concentration of specific point defects

( E kT )

N

N

_defect

= exp −

_a

/

N : concentration of lattice sites

Internal Luminescence Efficiency Internal Luminescence Efficiency

•In the 1960’s,

~ 1 %of the internal luminescence efficiencies (RT)

•At the present time,

> 90 %of the internal luminescence efficiency (RT) Î improved crystal quality

reduced defect and impurity concentration quantum well structures

1 1 1

⁻ ₋ ⁻

−

= τ

_rad

+ τ

_{non rad}

τ

1 1

1

int −

−

−

−

= +

rad non rad

rad

τ τ

η τ

Internal quantum efficiency

τ carrier lifetime

τ_rad radiative carrier lifetime τ_non-rad non-radiative carrier lifetime η_int internal quantum efficiency

relative probability of radiative recombination