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Chapter 4 Optical Source
4.1 Semiconductor physics
- Energy band
- Intrinsic and Extrinsic Material
- pn Junctions
- Direct and Indirect Band Gaps
4.2 Light Emitting Diodes (LED)
- LED structure
- Light source materials
-Quantum Efficiency and power
- Modulation of LED
4.3 Laser Diodes
- Laser diodes modes and thershold
- Rate Equations
- External Quantum Efficiency
- Resonant Frequencies
- Single mode lasers
- Laser modulation
- Outline
4.1 Semiconductor physics
- Energy band
¾ Semiconductor: Conduction properties lies somewhere between
those of conductor (metal) and insulator
¾ Intrinsic Semiconductor: Pure crystal (such as Si, Ge) Æ group IV
I
II
IIIb
1
H
Li
Na
K
Rb
Cs
Fr
2
3
Be
Mg
Ca Sc
Sr
Y
Ba La*
Ra Ac**
Lanthanides *
Actinides **
IVb Vb VIb VIIb
4
5
6
7
VIIIb
8
Ib
IIb III
IV
V
VI VII
9
0
10 11 12 13 14 15 16 17 18
He
B C N O F Ne
Al Si P S Cl Ar
Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
Uuq
Uuh
Uuo
Rf Db Sg Bh Hs Mt Uun Uuu Uub
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
4.1 Semiconductor physics
- Energy band
¾ Energy-band diagram:
- conduction band EC
- valence band EV
- band gap Eg= EC - EV
¾ Carrier: electrons / holes
¾ Concentration :
- free electron concentration n
- hole concentration p
- intrinsic carrier concentration ni
n = p = ni = K exp(−
Eg
2kBT
) (4.1)
K = 2(2π kBT / h2 )3/ 2 (me mh )3/ 4
4.1 Semiconductor physics
- Energy band
¾ Doping: Conduction can be greatly increased by adding traces of impurities
from Group V or Group III
¾ Doping Group V Æ donor impurity (P, As, Sb; 5 electrons)
Æ free-electrons Æ n-type material
¾ Doping Group III Æ acceptor impurity ( Al, Ga, In, Boron )
Æ free-holes Æ p-type material
Semiconductor physics
- Intrinsic and Extrinsic Material
¾ Intrinsic material : A perfect material containing no impurities is called ~.
¾ Extrinsic material : Doped semiconductor is called ~.
¾ Thermal generation Æ produce electron-hole pairs
( for intrinsic material: equal concentration Æ n = p = ni )
¾ Recombination : a free electron releases its energy and drops into a free
hole in the valence band.
¾ For extrinsic material: concentration of p and n is different, and follow
the mass-action law: Æ pn = n 2
n : intrinsic carrier concentration
i
i
¾ Majority carriers : refers to electrons in n-type material, and holes in p-type material.
¾ Minority carriers: refers to holes in n-type material, and electrons in p-type material
Semiconductor physics
- Intrinsic and Extrinsic Material
Example 4-2
Consider an n-type semiconductor which has been doped with a net
concentration of ND donor impurities. Find electron and hole
concentrations (nN, pN).
¾ Let nN and pN be the electron and hole concentrations, respectively, where
the subscript N is used to denote n-type semiconductor characteristics.
pN
¾ Total hole concentration pN (only from thermal excitation):
¾ Total electron concentration nN (from doped and thermal excitation): n N = N D + p N
¾ Mass-action law:
pN =
− N D + N D 2 − 4ni 2
2
n N ⋅ p N = ni2
N
= D ( −1 + 1 − 4ni 2 / N D 2 )
2
( n D + p N ) p N = ni2
ni n D
p N ni 2 / nD
n N nD
Semiconductor physics
- pn Junction
¾ Doped n- or p-type semiconductor material by itself serves only as a conductor.
¾ Only pn junction is responsible for the useful electrical characteristics of a
semiconductor device
¾ Barrier potential : prevents
further movement of charges
¾ Depletion region :
¾ External battery can be connected
to the pn junction, by reverse-bias
or forward bias.
¾ Reverse–biased Æ for application in photodiode
¾ Forward–biased Æ for application in laser diode
Semiconductor physics
- pn Junction
¾ Forward-biased pn junction:
Creates barrier potential, which prevent holes and electrons to move to junction
region, but when pn junction is applied forward voltage, if eV >= Wg, the electrons
and holes will move into junction region, and recombine, which will create Photons.
¾ Reverse-biased pn junction:
The width of the depletion region will
increase on both the n side and p side. (will
talk in next chapter for photodiode)
Operating Wavelength:
hf = Wg , λ = hc / Wg
Note: for direct bandgap material
Semiconductor physics
¾ Direct band gap material:
no change of wavevector
Æ efficient
For example: GaAs
¾ Indirect band gap material
with change of wavevector
- Direct and Indirect Band Gaps
Light-Emitting Diodes
- LED Structure
¾ To achieve high radiance, high quantum efficiency, carrier confinement and
optical confinement are necessary.
¾ Structure:
- homojunctions: same material (Wg)
- single and double heterojunctions :
difference bandgap materials
¾ The most effective structure:
Double heterojunction (it could provide
carrier and optical confinements.)
¾ Carrier (electron or hole ) confinement
- bandgap difference of adjacent layer
¾ Optical confinement
- index difference of adjacent layer
¾ Two basic LED configuration
- Surface emitters
- Edge emitters
Put fig. 4-8 here
¾ Surface emitters /lambertian pattern
¾ Edge emitters
unsymmetric radiation
Parallel plane:
Lambertian pattern
Perpendicular plane:
there is beam
confinement
(better coupling)
Light-Emitting Diodes
- Laser Source Materials
¾ III-V materials (Al, Ga, In Æ III group; P, As, Sb Æ V group
¾ Ternary and quaternary combinations
- Ternary alloy Æ Ga1-xAlxAs, spectrum at 800 – 900 nm
- Quaternary alloy Æ In1-xGaxAsyP1-y , spectrum at 1.0 – 1.7 μm
- x, y Æ Lattice constant
¾ Spectrum, Æ full-width-half-maximum (FWHM)
Light-Emitting Diodes
- Laser Source Materials
¾ Relationship between energy E and frequency ν :
E = hν =
hc
λ ( μm) =
1.240
E g (eV )
λ
¾ Relationship between lattice constant (x, y) and band-gap
- For Ga1-xAlxAs :
E g = 1.424 + 1.266 x + 0.266 x 2
(4 − 4)
E g = 1.35 − 0.72 y + 0.12 y 2
(4 − 5)
- For In1-xGaxAsyP1-y :
Example 4-3
Consider a Ga1-xAlxAs laser with x=0.07. Find the diode emission
wavelength.
Example 4-4
Consider the alloy In1-xGaxAsyP1-y (i.e., x = 0.26 and y =0.57), find diode
emission wavelength.
Light-Emitting Diodes
- Internal quantum efficiency
rate equation
¾ When carrier injection stops, carrier density
n = n0e−t /τ
decays:
¾ For constant current flow into LED, an
equilibrium condition will be established.
dn
J n
=
−
dt qd τ
Current
injection
J current density in A/cm2
q electron charge
d thickness of recombination region
¾ Internal Quantum efficiency ηint :
η int =
τ
Rr
1
=
=
Rr + Rnr 1 + τ r / τ nr τ r
τ r = n / Rr
1
τ
=
1
τr
+
τ nr = n / Rnr
1
τ nr
Rr : radiative recombination rate
equilibrium condition
τr : radiative recombination lifetime
n=
Rnr : nonradiative recombination rate
τnr nonradiative recombination lifetime
¾ Pint : internal optical power
Thermal
generation
Pint = η int
I
hcI
hν = η int
q
qλ
Jτ
qd
Light-Emitting Diodes
- Internal quantum efficiency
Example 4-5
A double-heterojunction InGaAsP LED emitting at a peak wavelength of
1310 nm has radiative and nonradiative recombination times of 30 and
100 ns, respectively. The drive current is 40 mA. Compute internal
quantum efficiency and internal optical power.
Light-Emitting Diodes
- External quantum efficiency
External quantum efficiency
n
η ext
1
=
4π
Incidental angle
critical angle
∫
φc
0
T (φ )(2π sin φ ) d φ
φ
−1
φc = π / 2 − θc = sin (n2 / n1 )
Fresnel
transmissivity
for normal incidence
T (0) =
4 n1 n 2
( n1 + n 2 ) 2
Emitted power
Example 4-6
Assuming a typical value of n=3.5 for refractive
index of an LED material, calculate the ηext .
For n1=n, n2=1
η ext ≈
1
n ( n + 1) 2
Pext = η ext Pint =
Pint
n ( n + 1) 2
Light-Emitting Diodes
- Modulation
Modulated output power
P (ω ) =
P0
1 + (ωτ i )
2
(4.18)
P0 power emitted at DC
ω modulation frequency
τi carrier lifetime
P (ω3dB ) 1
=
P ( 0)
2
•
Optical 3-dB modulation bandwidth :
•
Detected current is linearly proportional to optical power :
•
Detected electric power :
•
Therefore, 3-dB electrical loss corresponds to 1.5-dB optical loss; in other
words, 3-dB optical loss corresponds to 6-dB electrical loss.
f 3− dB (electrical ) =
p (ω ) = I 2 (ω ) R
1
f 3− dB (optical ) = 0.707 f 3− dB (optical )
2
P (ω ) I (ω )
=
P (0) I (0)
Laser Diodes
- Principles
¾ Types of Laser : lasing medium Æ gas, liquid, solid state (crystal), semiconductor.
¾ Laser action is the result of 3 key process: photon absorption, spontaneous
emission, and stimulated emission.
¾ Photon absorption: When a photon of energy hν12 impinges on the system, an
electron in ground state E1 can absorb the photon energy and be excited to state E2.
¾ Spontaneous emission : The electron in state E2 falls down to state E1 by itself
quite spontaneously, and emits a photon of energy hν12 in random direction.
¾ Stimulated emission : The electron in state E2 falls down to state E1, induced by a
coming photon of energy of hν12, and emits a photon of energy hν12 in the same
direction.
Laser Diodes
- Principles
¾ In thermal equilibrium : The density of exited electron is very small
¾ Population inversion : Population of excited states > that of the ground state Æ
stimulated emission will exceed absorption
¾ Pumping techniques : Æ obtain population inversion
For a semiconductor laser, population inversion can be achieved by injecting
electrons, or another pumping laser for solid state laser (crystal)
Laser Diodes
- Modes
¾ Laser cavity : to convert the device into an oscillator and provide optical
feedback to compensate the optical loss in the cavity
- Fabry-Perot (FP) laser: mirrors, cleaved facets
- Distributed feedback (DFB) laser : Bragg reflector
¾Modes: Longitudinal mode; Lateral mode; Transverse mode
- spectral characteristics (resonant frequencies) / longitudinal modes
- spatial characterisitics depend on / lateral and transverse modes
Laser Diodes
- Threshold conditions -1
Optical field intensity in longitudinal direction
Amplitude condition
I ( z , t ) = I ( z )e j (ω t − β z )
I ( 2 L ) = I ( 0)
Ra Rb e 2 L ( Γg ( hν ) −α ( hν )) = 1
I(z) optical field intensity
ω optical radian frequency
β propagation constant
with
Γg th = α +
I ( z ) = I (0)e[ Γg ( hν ) −α ( hν )] z
cavity
threshold
gain
Γ optical confinement factor
g gain coefficient depended on optical frequency
α effective material absorption coefficient
1
1
ln(
) = α + α end
Ra Rb
2L
cavity cavi
Material mirr
loss
loss
Phase condition
Fabry-Perot laser cavity
Reflecting
mirror
e− j 2β L = 1
Decided by laser cavity dimension !
Ra
Rb
Amplitude during one round trip
I ( 2 L ) = I (0) Ra Rb e
Gain medium
n1
n2
z
0
Amplitude condition
Phase condition
L
2 L ( Γg ( hν ) −α ( hν ))
Ra Rb mirror Fresnel reflection
n1 − n2 2
coefficients
R=(
n1 + n2
)
Laser condition: Laser occurs
when the gain is sufficient to
exceed the optical loss during
one round trip
I ( 2 L ) = I ( 0)
Laser Diodes
- Threshold conditions -2
Example 4-7
For GaAs, R1=R2=0.32 for uncoated facets (i.e. 32% of the radiation is
−1
reflected at a facet) and α 10cm −1 . This yields Γg th = 33cm for a
laser diode of length L = 500μm .
Optical power vs. drive current
Threshold current density
g th = β J th
Jth Threshold current density
gth Threshold gain
β constant depended on
device construction
J th = I th / A
- spontaneous radiation below threshold
- threshold current
Ith Threshold current
A Area
Laser Diodes
- Rate equation -1
¾ Relationship between optical power and drive current can be determined
by two rate equations:
¾ For photon density Φ ; For electron density n
1
1
rate equation for photon density
≥
=
n
n
Cn −
≥0
th
C τ ph
τ
ph
dΦ
Φ
= CnΦ + Rsp −
dΦ
Electron density
τ ph
dt
≥0
stimulated
emission
spontaneous
emission
photon
loss
dt
with Rsp → 0
Φ : photon density
Photon density should be
C coefficient for optical absorption and emission interactions in increasing mode towards
lasing with negligible
Rsp rate of spontaneous emission into lasing mode
spontaneous emission
τph photon lifetime
rate equation for electron density
dn
J
n
=
−
− CnΦ
dt qd τ sp
injection
spontaneous
recombination
stimulated
emission
injection current density
τsp spontaneous recombination lifetime
J
dn
=0
dt
with Φ = 0 n = nth
must exceed a
threshold value in
order for photon
density to increase
n th
τ sp
=
J th
qd
If electron density is at threshold level, injected
electrons are just fully consumed by spontaneous
recombination without light emission
Laser Diodes
- Rate equation -2
Steady-state solution for rate equations
Φ
dΦ
= Cnth Φ s + Rsp − s = 0
τ ph
dt
+
dn
J nth
n th
J th
=
−
− Cnth Φ s = 0
=
dt qd τ sp
Φs steady-state photon density
τ sp
qd
Φs =
τ ph
qd
( J − J th ) + τ ph Rsp
Photons resulting
from stimulated
emission
Spontaneously
generated
photons
Laser Diodes
- External quantum efficiency
External quantum efficiency ηext is defined as the number of photons emitted per radiative
electron-hole pair recombination above threshold.
External quantum efficiency
ηext = ηi (1 −
α
g th
) (4.37)
ηi internal quantum efficiency
~ 0.6-0.7 at room temperature
gth gain coefficient at threshold
Achieved experimentally
η ext =
q dP
dP ( mW )
= 0.8065λ ( μm)
E g dI
dI ( mA)
Εg band-gap energy
λ emitted wavlength
Laser Diodes
- Resonant frequencies
Phase condition for lasing
e
− j 2β L
=1
ν =m
2β L =
c
2n1 L
c
Δν =
2n1 L
2πν
n1L = 2mπ
c
Optical resonant frequencies
(or longitudinal modes)
or Δλ =
λ2
2n1 L
Frequency or wavelength spacing between modes
(or free spectral range FSR)
These modes describe the possible resonant
optical frequency, if lasing really happen at
these frequencies or not, still depends on
the laser gain profile.
If many modes are allowed for lasing under
the gain spectral profile, it is a multi-mode
laser e.g. Fabry-Perot laser
Laser Diodes
- Resonant frequencies
Laser spectral gain profile
g ( λ ) = g ( 0) e
−
( λ − λ0 ) 2
2σ 2
g(0) maximum gain proportional to population inversion
λ0 wavelength at the spectrum center
σ spectral width of the gain
Example, 4-12
A GaAs laser operating at 850 nm has a 500 μ m length and a refractive
index n=3.7. What are the frequency and wavelength spacings. If, at the
half-power point, λ – λ 0 = 2 nm, what is the spectral width σ of the gain ?
Laser Diodes
- Single mode lasers
- reduce cavity length L to increase frequency interval (FSR) between modes
until there is only one mode falls within the laser gain bandwidth (not practical
2
due to its dimension and low optical power)
- distributed-feedback (DFB) laser
- two 0-order modes will be degenerated to
single mode due to imperfect AR coating
DFB laser cavity
λB =
2ne Λ
k
n1
n2
Anti-reflection
coating
k
ne
Λ
order of the grating
effective refrative index of the mode
period of corrugation
λ = λB ±
λB
1
(m + )
2ne Le
2
Laser Diodes
- Modulation
¾ Internal (direct, current) modulation ; External modulation
¾ Modulate the laser above the threshold
- Spontaneous radiative lifetime τsp ~ 1 ns
- Stimulated carrier lifetime τst ~ 10 ps
- photon lifetime τph ~ 2 ps Æ sets the upper limit to the direct modulation capacity
¾ Modulation frequency also can not be larger than the frequency of the relaxation of
laser field f
f =
Laser injection current
P
1
2π
1
τ spτ ph
I
−1
I th
Laser output power
IB Ip+ IB
I
t
d
Laser carrier density
Fig. 4-30 Example of the relaxationoscillation peak o a laser diode
P195, 4-9
a) A GaAlAs laser diode has a 500 μ m cavity length which has an effective
absorption coefficient of 10 cm-1. For uncoated facets the reflectivities
are 0.32 at each end. What is the optical gain at the lasing threshold ?
b) If one end of the laser is coated with a dielectric reflector so that its
reflectivity is now 90 percent, what is the optical gain at the lasing
threshold ?
c) If the internal quantum efficiency is 0.65, what is the external quantum
efficiency in cases of (a) and (b)?
P196, 4-12
A GaAs laser emitting at 800 nm has a 400 μ m cavity length with a
refractive index n=3.6. If the gain g exceeds the total loss α t throughout
the range 750 nm < λ < 850 nm, how many modes will exist in the laser ?
P195, 4-15
For laser structures that have strong carrier confinement, the threshold current
density of stimulated emission Jth can to a good approximation be related to
the lasing-threshold optical gain gth by gth = β Jth, where β is a constant that
depends on the specific device construction. Consider a GaAs laser with an
optical cavity of length 250 μ m and width 100 μ m . At the normal operating
temperature, the gain factor β = 21x10-3 cm/A and the effective absorption
coefficient α =10 cm –1.
a) If the refractive index is 3.6, find the threshold current density and the
threshold current Ith. Assume the laser end facets are uncoated and the
current is restricted to the optical cavity.
b) What is the threshold current if the laser cavity width is reduced to 10 μ m ?