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Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED

EE529 Semiconductor Optoelectronics

Optical Processes and Light Emitting Diodes

• 1. Band-to-band optical transitions
• 2. Absorption spectrum and mechanisms
• 3. Spontaneous emission
• 4. LED principles and efficiency
• 5. Frequency response and modulation bandwidth

Reading: Liu, Sec. 13.2, 13.4-13.5, 13.7 Reference: Bhattacharya, Sec. 3.1-3.4, 5.4-5.5, 5.8

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 2

Schrödinger’s Equation and Probability





2 2 2 Potential energy Total energy Kinetic energy

2 i V t m x ∂Ψ ∂ Ψ = − + Ψ ∂ ∂      

Schrödinger’s Equation:

( , ) x t Ψ

: Wave function of a matter wave

2

| ( , ) |

b a

x t dx Ψ ∫

: Probability of finding the particle between a and b at time t

Position operator: x Momentum operator:

i x ∂ ∂ 

Energy operator: i

t ∂ ∂ 

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 3

Solving Schrödinger Equation

• 1. Solve the time-independent Schrödinger Equation.

2 2 2

2 d V E m dx ψ − + ψ = ψ 

Obtain ψn(x) with associated energy En.

• 2. Initial wave function:

1

( ,0) ( )

n n n

x c x

∞ =

Ψ = ψ ∑

Cn can be obtained by matching initial conditions.

• 3. Wave function at subsequent time t:

Assume time-independent potential V(x).

/ 1 1

( , ) ( ) ( , )

n

iE t n n n n n n

x t c x e c x t

∞ ∞ − = =

Ψ = ψ = Ψ ∑ ∑



Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 4

Infinite Square Well Potential

a x V(x)

2 ( ) sin n x x a a π   ψ =    

2 2 2 2 2 2

2 2

n n

k n E m ma π = =  

Eigenfunction: Quantized energy:

E1 E2 E3

Wave function:

2 2 2

( /2 ) 1

2 ( , ) sin

i n ma t n n

n x t c x e a a

π

π

∞ − =

  Ψ = ∑    

 2 1|

|

n n n

H c E

∞ =

= ∑

Expectation value of the energy:

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 5

Direct Bandgap vs. Indirect Bandgap

E CB k –k Direct Bandgap (a) GaAs Eg Ec Ev Photon VB

Direct bandgap

• Recall that photon momentum is very small,

compared to electron momentum.

• Momentum conservation can be satisfied in

direct bandgap semiconductors.

• Can be good photon absorbers and emitters.

( ) p k = 

E CB VB Indirect Bandgap, Eg k –k kcb (b) Si E k –k Phonon (c) Si with a recombination center Ec Ev kvb VB CB Er Ec Ev

Indirect bandgap

• Momentum conservation

cannot be satisfied with photon-only process in indirect bandgap semiconductors.

• Can be good photon

absorbers, but not good photon emitters.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 6

ph

ε ω = 

e g p

ω ε ε = + 

a g p

ω ε ε = − 

Band-to-Band Absorption and Recombination

Momentum of photons and electrons p = h/λ → Photon momentum << Electron Momentum

Absorption and emission in direct bandgap Emission in indirect bandgap Absorption in indirect bandgap

Defect center e.g. GaAs, InP Involves emission of a phonon e.g. Si, Ge Involves absorption or emission of a phonon

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 7

Exercise: Determine bandgap and phonon energies from absorption experiment

The figure below shows the absorption for Ge at 300K and 77K. Analyze the 300K data to

• btain the value of the direct bandgap, the

indirect bandgap, and the phonon energy participating in the indirect transitions.

Note: The band structure of Ge shows possibilities of indirect and direct transitions. The values we obtain from this exercise will be different from the calculated data at 0K.

(Source: Kittel, “Introduction to Solid State Physics”)

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 8

Absorption Spectrum of a Typical Semiconductor

(Source: Wolfe, Holonyak, and Stillman, “Physical Properties of Semiconductors)

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED

Exciton Absorption

9

Q: Exciton absorption peaks are normally seen in very pure semiconductors at low temperatures. Higher degree of confinement in the semiconductors also greatly helps observing these, e.g., quantum wells. Why?

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 10

Calculated Absorption Spectrum due to Franz-Keldysh Effect

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 11

Quantum-Confined Stark Effect (QCSE)

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 12

Relation between Absorption and Spontaneous Emission

2 2 / 2

( ) 8 ( ) 1

B

sp h k T

n R c e ν α ν π ν ν = −

Roosbroeck-Shockley relation

Spontaneous emission spectrum

• f GaAs at 300K

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 13

Light Emitting Diodes

Principles

hυ - Eg Eg (b) V (a) p n+ Eg eVo EF p n+

Electron in CB Hole in VB

Ec Ev Ec Ev EF eVo Electron energy Distance into device

A p-n junction diode typically made from a direct bandgap semiconductor. Electron-hole pair recombination results in the emission of a photon. Spontaneous emission; emitted photons in random direction.

Device Structures

L ig h t o u tp u t In su lato r (o xid e ) p n+ E p itax ial la ye r L ig h t o u tp u t p E p itax ial la ye rs (a) (b) n+ S u b strate S u b strate n+ n+ M etal ele ctro de

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 14

Heterojunction LEDs

2 eV 2 eV

eVo Holes in VB Electrons in CB

1.4 eV

No bias With forward bias Ec Ev Ec Ev EF EF

(a) (b) (c)

(d) p n+ p ∆Ec GaAs AlGaAs AlGaAs p p n+

~ 0.2 µm

AlGaAs AlGaAs

(a) A double heterostructure diode has two junctions which are between two different bandgap semiconductors (GaAs and AlGaAs) (b) A simplified energy band diagram with exaggerated features. EF must be uniform. (c) Forward biased simplified energy band diagram. (d) Forward biased LED. Schematic illustration of photons escaping reabsorption in the AlGaAs layer and being emitted from the device.

GaAs

Avoid re-absorption of photons along the emission path.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED 15

Exercise: LED Efficiency

An AlGaInP/GaP LED with peak emission wavelength of 636 nm has an external quantum efficiency of 25%. (a) To achieve 5 mW of output

• ptical power, what should be the injected current?

(b) If this corresponds to forward-biasing the device at 2 V, find its power conversion efficiency. (c) Find its luminous efficiency and luminous flux.

Lih Y. Lin EE 529 Semiconductor Optoelectronics – Optical Processes and LED

Photon Escape Efficiency

16

The intensity distribution of the LED radiation is

• Lambertian. At the semiconductor-air interface, if the

incident angle is greater than the critical angle 𝜄𝑑= sin−1(𝑜𝑏𝑏𝑏 𝑜𝑏 ⁄ ), total internal reflection occurs and the light is trapped inside the semiconductor. Possible solutions: (a) Shape the semiconductor surface as a hemisphere (expensive). (b) Encapsulate the LED in a transparent dome with higher refractive index than air to increase 𝜄𝐷.