1

EE232 Lecture 3-1 Prof. Ming Wu

EE 232 Lightwave Devices Lecture 3: Basic Semiconductor Physics

and Optical Processes

Instructor: Ming C. Wu

University of California, Berkeley Electrical Engineering and Computer Sciences Dept.

EE232 Lecture 3-2 Prof. Ming Wu

Optical Properties of Semiconductors

•  Optical transitions –  Absorption: exciting an electron to a higher energy level

by absorbing a photon –  Emission: electron relaxing to a lower energy state by

emitting a photon

EC

EV

Absorption Emission

Interband Transition

EA

Impurity-to-Band Transition

Intraband Transition (Free-Carrier Absorption)

2

EE232 Lecture 3-3 Prof. Ming Wu

Band-to-Band Transition

•  Since most electrons and holes are near the band-edges, the photon energy of band-to-band (or interband) transition is approximately equal to the bandgap energy:

•  The optical wavelength of band-to-band transition can be approximated by

λ =cν=hcEg

≈1.24Eg

λ : wavelength in µmEg : energy bandgap in eV

hv = Eg

EE232 Lecture 3-4 Prof. Ming Wu

Energy Band Diagram in Real Space and k-Space

EC

VB

E

x

E

k

CB

EV

Real Space K-Space

* 212e C eE E m v= +

* 212h V hE E m v= −

Ee = EC +!2k 2

2me*

Eh = EV −!2k 2

2mh*

Effective Mass Approximation

!k =me*ve

Momentum:

3

EE232 Lecture 3-5 Prof. Ming Wu

Band-to-Band Transition

CB

VB

hν

CB

VB

hν

CB

VB

hν hν

Absorption Spontaneous Emission

Stimulated Emission

Photodetectors; Solar Cells LED Optical Amplifiers;

Semiconductor Lasers

EE232 Lecture 3-6 Prof. Ming Wu

Conservation of Energy and Momentum

•  Conditions for optical absorption and emission:

–  Conservation of energy

–  Conservation of momentum

CB

VB

hν

E2

E1

k k1

k2

E

E2 −E1 = hν

( ) ( )

2 1

2 1

2 1

2, ~

2~

~ 0.5 ~1

h

h

k k k

k ka

k

a nm m

k k

ν

ν

π

πλλ µ

− =

<<

⇒ =Lattice

Constant

Optical transitions are “vertical” lines

4

EE232 Lecture 3-7 Prof. Ming Wu

Direct vs Indirect Bandgaps

•  Direct bandgap materials –  CB minimum and VB

maximum occur at the same k

–  Examples •  GaAs, InP, InGaAsP •  (AlxGa1-x)As, x < 0.45

•  Indirect bandgap materials –  CB minimum and VB

maximum occur at different k

–  Example •  Si, Ge •  (AlxGa1-x)As, x > 0.45

–  Not “optically active”

CB

VB

hν

CB

VB

hν X

Phonon

EE232 Lecture 3-8 Prof. Ming Wu

Absorption Coefficient

•  Light intensity decays exponentially in semiconductor:

•  Direct bandgap semiconductor has a sharp absorption edge

•  Si absorbs photons with hv > Eg = 1.1 eV, but the absorption coefficient is small

–  Sufficient for CCD

•  At higher energy (~ 3 eV), absorption coefficient of Si becomes large again, due to direct bandgap transition to higher CB

xeIxI α−= 0)(

5

EE232 Lecture 3-9 Prof. Ming Wu

Review of Semiconductor Physics

Electron and hole concentrations:

( ) ( )

( ) ( )

Fermi-Dirac distributions:1( )

1 exp

1( )1 exp

: electron quasi-Fermi level : hole quasi-Fermi l

C

V

n eE

E

p h

nn

B

pp

B

n

p

n f E E dE

p f E E dE

f EE Fk T

f EF Ek T

FF

ρ

ρ

∞

−∞

=

=

=⎛ ⎞−

+ ⎜ ⎟⎝ ⎠

=−⎛ ⎞

+ ⎜ ⎟⎝ ⎠

∫

∫

evel

EE232 Lecture 3-10 Prof. Ming Wu

Electron/Hole Density of States (1)

•  Electron wave with wavevector k

•  Periodic boundary conditions

•  An electron state is defined by

•  Number of electron states between k and k + Δk in k-space per unit volume

xL

zL

yL

eik!⋅r!

eik!⋅r!

= eik!⋅ r!+Lx x!

( ) = eik!⋅ r!+Ly y!"

( ) = eik!⋅ r!+Lz z!

( )

kx ,ky ,kz( )↑↓= m 2πLx,n 2πLy,l 2πLz

#

$%%

&

'((↑↓

2 2

2

2 4 ( )2 2 2 k

x y z

k dk k dk k dkV

L L L

πρ

π π π π⋅ = =

k

k k+Δ

xk

yk

zk

Spin Up and

Down

6

EE232 Lecture 3-11 Prof. Ming Wu

Electron/Hole Density of States (2)

•  Number of electron states between E and E + ΔE per unit volume

•  Likewise, hole density of states

E = EC +!2k 2

2me*⇒ dE = !

2kme*dk

k 2

π 2dk = me

*

!2π 22me

* E − EC( )!

dE = ρe (E)dE

ρe (E) =12π 2

2me*

!2!

"##

$

%&&

3/2

E − EC

ρh (E) =12π 2

2mh*

!2!

"##

$

%&&

3/2

EV − E

EE232 Lecture 3-12 Prof. Ming Wu

Electron and Hole Concentrations

n = fn (E)ρe (E)dEEC

∞

∫ =1

1+ expE − FnkBT

!

"#

$

%&

12π 2

2me*

!2!

"##

$

%&&

2

E − EC dEEC

∞

∫

n = NC ⋅F1/2Fn − ECkBT

!

"##

$

%&&

NC = 2πme

*kBT2π 2!2

!

"##

$

%&&

3/2

p=NV ⋅F1/2EV − FpkBT

!

"##

$

%&&

NV = 2πmh

*kBT2π 2!2

!

"##

$

%&&

3/2

( )0

Fermi-Dirac Integral

1 ( 1) 1

Gamma Function

3( )2 2

j

j x

xF dxj e ηη

π

∞

−=Γ + +

Γ =

∫

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EE232 Lecture 3-13 Prof. Ming Wu

Approximation of Electron/Hole Concentration

Fj η( ) = 1Γ( j +1)

x j

1+ ex−η0

∞

∫ dx ≈ eη when η <<1

43η3

π

&

'(

)

*+

1/2

when η >>1

,

-..

/..

When Fn << EC (Boltzmann approximation)

n ≈ NC ⋅e−EC−FnkBT

When Fn >> EC (Degenerate)

n ≈ NC ⋅4 Fn − EC

kBT

&

'(

)

*+

3/2

3 π

Fn − ECkBT

C

nN

Boltzmann Approx

Degenerate

Quasi-Fermi Level Cross Band Edge