Optoelectronic Devices and Circuits 1 2013
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Optoelectronic Devices and Circuits I Jürgen Werner Institut für Photovoltaik ©JHW 1 Photovoltaik [email protected] Contact Jürgen Werner, Institut für Photovoltaik Room No.: 1.215 Phone: 685-67140 [email protected] Jürgen Köhler, Institut für Photovoltaik Room No.: 1.235 Phone: 685-67159 j k hl @i i t tt td ©JHW 2 juergen.koehler@ipv.uni-stuttgart.de
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Transcript of Optoelectronic Devices and Circuits 1 2013
Optoelectronic Devices and Circuits I
Jürgen Werner
Institut fürPhotovoltaik
©JHW 1
Contact
Jürgen Werner, Institut für PhotovoltaikRoom No.: 1.215Phone: [email protected]
Jürgen Köhler, Institut für PhotovoltaikRoom No.: 1.235Phone: 685-67159j k hl @i i t tt t d
©JHW 2
Institut für Photovoltaik
a) Structure of ipv
Employees: ~ 33Annual Turnover: ~ 2.0 Mill. €Annual Turnover: 2.0 Mill. €PhD Students: ~ 10Research Groups: 5Student Works: 15 - 20 per Year
b) Main Focus of ResearchMicro- and OptoelectronicsS T h l d Ph l i
©JHW 3
Sensor Technology and PhotovoltaicsSemiconductor Technology and Semiconductor PhysicsSolar Cells and Thin Film Technology
0. Introduction1. Basic Physics
1.1 Simple equations
Table of contents
1.2 Reflectance, absorptance, transmittance1.3 Refraction and total internal reflection
1.4 Reflectance rΦ, transmittance tΦ for Θi = 0
2. Thermal Radiation2.1 Black body radiation 2.2 Grey body radiation2 3 Selective body radiation of a semiconductor
©JHW 4
2.3 Selective body radiation of a semiconductor
3. Coherence3.1 Definition3.2 Temporal coherence3.3 Spatial coherence3.4 Emission of photons
4. Semiconductor Basics4.1 Energy bands and Fermi function 4.2 The wave vector 4.3 The band structure 4.4 Limited range of -values, the Brillouin zone
kg ,
4.5 The crystal momemtum
4.6 Impulse pe4.7 Direct and indirect band gap semiconductors
5. Excitation and recombination processes insemiconductors5.1 Introduction
©JHW 5
5.2 Absorption of radiation in semiconductors 5.3 Carrier recombination in semiconductors
6. Light emitting diodes6.1 Working principle of an LED6.2 The spectrum emitted by an LED 6.3 Materials for LEDs (and lasers)6.4 Emission efficiency of LEDsy
7. Semiconductor Lasers7.1 Working principle and compounds of lasers 7.2 General lasing conditions7.3 Lasing conditions for semiconductor lasers7.4 Laser modes7.5 Radiation amplification in a semiconductor laser 7 6 Semiconductor laser configurations
©JHW 6
7.6 Semiconductor laser configurations7.7 Light guiding in semiconductor lasers7.8 Modern semiconductor lasers
8. Glass Fibers8.1 Configurations and optical properties8.2 Step-index fibers 8.3 Graded-index fibers 8.4 Mono-mode fibers 8.5 Dispersion in glass fibers8.6 Attenuation in glass fibers
9. Photodetectors9.1 Introduction, general considerations 9.2 Properties and specifications of photodetectors 9.3 Photoconductors9 4 Photodiodes
©JHW 7
9.4 Photodiodes9.5 Photodiodes with internal gain: Avalanche
photodiodes (APDs) 9.6 Materials and detector configurations
0. Introduction
©JHW 8
What is optoelectronics?
geometrical physiologic
semiconductortechnology
optoelectronicsintegrated
optics
optics
communicationtechniques
physiologicoptics
©JHW 9
Fig. 0.1: Overlap of optoelectronics with classic areas.
physicaloptics
quantumoptics
radio frequencytechniques
© JHW
Optoelectronics =
generation and communication ofelectromagnetic radiation from optical regime
++conversion of this radiation into electrical signals
Optical regime =
100 nm (UV) to 1 mm (far IR)
©JHW 10
100 nm (UV) to 1 mm (far IR) (glass fibers use “light” of 800 - 1500 nm)
glass fiber hl
Fig. 0.2: Scheme of an optical communication system
© JHW
electrical signal electrical signal
glass fiber
optical signal
photodetector
laserdiode
©JHW 11
light = small visible part of the optical regime between 380 nm and 780 nm
What is light?
Fig. 0.3:
The sun’s spectrum: only tiny part of the optical regime.
dΦdA
2.0
1.5
1.0
IEC standard 904 (AM 1.5G)
integrated radiation density =1kW/m2
λal r
adia
tion
dens
ity
[Wm
-2nm
-1]
© JHW
Vi ibl t
©JHW 12
0.5
0.0400 600 800 1000 1200 1400
wavelength λ [nm]
dΦ
dAd
λ
spec
tra
only small part of the sun’s spectrum.
Visible part:
1. Basic Physics
©JHW 13
1.1 Simple equations
a) wavelength λ, frequency ν, and velocity c of light:
νλ== cc 0 (1 1)νλ==rnc 0 (1.1)
with c0 = vacuum light velocity = 2.998 × 108 m/s,
nr = refraction index.
At interfaces between media of different nr:
©JHW 14
changes by a change of wavelength λ (not of frequency ν !).
velocity c
Max Planck: Radiation = stream of particles (photons);
energy: E = hν
b) particle properties of radiation:
E hhc
= = =ν ω λ . (1.2)
energy: E hν ,
related to wavelength λ of the radiation by
particle wave
©JHW 15
particle wave
h = 4.14 x 10-15 eV⋅s = 6.62 x 10-34 Js = Planck’s constant.
Impinging power on a surface due to monochromatic photons with
number nphot :
photdnhΦ ( ).photh
dtΦ ν= (1.3)
c) conversion of energies into wavelengths λ and frequencies ν :
1 24
©JHW 16
1.24[eV]
[μm]E
λ= (1.4)
ν [THz] = 242 E [eV] (1.5)
Table 1.1: The regime of light (visible radiation)
violet green dark red violet green dark red
λ [nm] 380 500 780
E [eV] 3.26 2.48 1.59
ν [THz] 789 600 385
©JHW 17
ν [THz] 789 600 385
1.2 Reflectance, absorptance, transmittance
Φ0
reflectance, reflection coefficient
(1.6 a)rΦ
0
Φr =
Φ
Fig. 1.1: Reflected, absorbed, and transmitted radiation.
Φr
0
Φa
Φt
© JHW (1.6 c)transmittance, transmission factor
tΦ
0
Φt =
Φ
absorptance, absorption factor
aΦ
0
Φa =
Φ
(1.6 b)
1
©JHW 18
rΦ , aΦ , tΦ depend on frequency ν, polarization, angle of incidence,
(and on temperature T).
1Φ Φ Φr +a +t =
a) Refraction: Change of light velocity, Snell’s law
1.3 Refraction and total internal reflection [1]
> nint
Refraction: ray bends towards the normal.
Fig. 1.2:Refraction for two different angles of i id t li ht
it
a)
ni
nt
Θ t
Θr Θ i
b)
ni
nt
Θr Θi
Θ t
©JHW 19
incident light.
i i t
t t i
sinΘ c n= =
sinΘ c n(1.7) i i t tn sinΘ = n sinΘ (1.8)
a) b)© JHW
b) Total internal reflection:
>ni nt
Θt
a)ni
nt
Θi Θr
Θt
b)
Θr
Θ t
Θi
©JHW 20
Fig. 1.3 a + b: Partial internal reflection for two
different angles Θi
Θi Θr© JHW
d)
=Θ Θ iΘ Θ
For Θt = 90°:
sinΘc ;i
t
n
n=c)90°
c)
> =Θc Θ iΘi Θr
Fig. 1.3 d: Total internal reflection.
i Θ 1 (1 9)F Θ 90° d 1
Θc = critical angle
i
Fig. 1.3 c: Critical angle
=Θc=ΘcΘi =ΘcΘr
©JHW 21
Total internal reflection requires radiation coming from
the side with the higher optical density (ni > nt).
sinΘc = ni-1 (1.9)For Θt = 90° and nt = 1:
Table 1.2: Critical angle for total internal reflectionin optoelectronic materials
16.8 °17.1 °14 6 °
material index ofrefraction ni
critical angle Θc
glass 1.5 - 1.7 35 ° - 41 °Si 3.45GaAs 3.4Ge 3 9
©JHW 22
14.6 Ge 3.9
1.4 Reflectance rΦ, transmittance tΦ for Θi = 0
2t irΦ
0 t i
n - nΦr = = ( )
Φ n +n(1.10)
a)ntni
0Φnt > ni
Due to the quadratic dependence,
rΦ is the same for a) and b)!!!
0 t i
Φ Φt = 1- r (1.11)
b)nt ni
0Φ
rΦt i
©JHW 23
Fig. 1.4: Reflectance rΦ for perpendicularincidence of radiation
rΦ© JHW
nt < ni
Table 1.3: Perpendicular reflectance for different interfaces
4 % 96 %
30 % 70 %
interface reflectance rΦ transmittance tΦglass/air
GaAs/air
©JHW 24
1.5 Internet Links
1. Refraction of Light (Applet): http://OLLI.Informatik.Uni-Oldenburg.DE/sirohi/refraction.html
2. Total Internal Reflection in Water (Applet): http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=43
3. Snell's Law (Applet): http://www.phys.ksu.edu/perg/vqm/laserweb/Ch-1/F1s1t2p3.htm
©JHW 25
1.6 Literature
1. E. Hecht, Optics 3rd edition (Addison Wesley, Reading, MA, 1998), p. 121
©JHW 26
2. Thermal Radiation
©JHW 27
2.1 Black body radiation [1,2]
2.1.1 What is a black body?
To human eyes:
A body appears as black, if all radiation in the visible regime, i.e. all
light is absorbed!
Consequently:
©JHW 28
aΦ (hν) = 1 in this regime of the electromagnetic spectrum.
Ideal black body: aΦ (hν) = 1 for all frequencies.
Ideal black bodies do not exist;
but some systems are close to the ideal one:
What is a black body?
y
very thick non-reflecting bodies
a tiny hole in a black shoe box
the old stove of your great grand parents
©JHW 29
T
What is a black body?
T
©JHW 30
© JHW
Fig. 2.1: Absorption and emission by walls of temperature T:
Thermal equilibrium between radiation field and walls.
Personal experience:
What is a black body?
Black body (black jeans) absorbs more radiation than a non-black body
(blue jeans).
However: Black body emits also more radiation than a non-black body!
©JHW 31
The stronger a body absorbs radiation,
the stronger it must emit radiation.
Rule is a consequence of the following requirement:
What is a black body?
q g q
Rates of absorbed and emitted energy are equal at T = constant.
Unequal rates: Temperature change.
Strongly absorbing body: must get rid of the energy.
Body must also have strong emission
©JHW 32
Body must also have strong emission
(or explode of radiation overflow....).
2.1.2 Kirchhoff’s radiation law
Emitted power Prad from a body with absorptance aΦ :
2( ) ( ) ( ) [ ].BB
e e
WL a Lλ λλ λ λΦ= (2.1)
Leλ(λ) = radiated power per wavelength interval (µm) and steradian (sr)
emitted per surface element (m2) = f(aΦ(λ)).
2( ) ( ) ( ) [ ]e e m sr mλ λ μΦ
Note: Absorptance aΦ (number between 0 and 1) depends on
surface (color, texture, roughness etc.) and on wavelength λ;
©JHW 33
( , , g ) g ;
Measurement of aΦ allows calculation of Leλ(λ).
= emitted power spectrum of a black body = universal function(λ)BBeL λ
2.1.3 Planck’s radiation law
Power spectrum of a black body:
23
/( )5
2 1
10
BB 0e hc kT
hcdL
dAd d eλ λΦΩ λ λ
= = − 2
W
m sr μm(2.2)
©JHW 34
Power spectrum of a black body:
srm
)μ
-2-1
-14x107 © JHW
tra
lde
nsi
tyL
(Wm
-
eλBB
1x107
2x107
3x107
3000 K
4000 K
5000 K
T = 6000 K
©JHW 35
Fig. 2.2: Spectrum of a black body.
spe
c
0.0 0.5 1.0 1.5 2.00
wavelength λ (μm)
View into the door of an oven:
Cold oven: everything appears black
Increase of T: Spectrum shifts to shorter wavelengths.
Upon heating: red color;
higher temperature: yellow color;
green and blue colors ??
Radiation law of Planck
©JHW 36
= mathematical description of color
of burning fire, heated oven!
2.1.4 Wien’s displacement law
a) wavelength of maximum’s position
(2.3)µmKT
1
108978.2 3max ×=λ
λmax ≅ 500 nm
λmax ≅ 10 μm
Examples:
Sun temperature Tsun ≅ 6000 K ⎯→Earth temperature Tearth ≅ 300 K ⎯→
C l i t li ht ith hi h “ l t t ” i i d!
(2.4)T
1max ∝λ
©JHW 37
Color pictures: light source with high “color temperature” is required!
b) height of the maximum (for Ω = 2π sr)
552
17max 106.2)( T
µmKm
WLBB
e ×=λλ(2.5)
Integrated power emitted per surface element (and Ω = 2π) of a
2.1.5 Stefan-Boltzmann law
black body:
4BB BBtotal e
dP L d d T
dAλΦΩ λ σ= = =
(2.6)
= Stefan constant.42
81067.5W−×=σ
©JHW 38
42Km
Emission of the sun: Tsurface = 5800 K:
Stefan-Boltzmann law: Example:
16 m2 sun surface make up one nuclear power plant of 1 GW power!!!
64 MW/m2!!
©JHW 39
Emission of the earth (T = 300 K): 500 W per m2 surface area.
2.2 Grey body radiation
Black body: Absorptance aΦ = 1 for all λ.
Grey body: aΦ < 1, but independent of λ!
aΦ (λ) = constant < 1
Power emission (Kirchhoff‘s law!) equal to black body,
(2.7)
©JHW 40
but reduced by a constant factor (aΦ) for all wavelengths.
2 3 1 Selective body radiation
2.3 Selective body radiation of a semiconductor
(2 7)2.3.1 Selective body radiation
Selective body: Absorptance aΦ < 1, but dependent on λ.
Power emission (Kirchhoff‘s law!!)
not only reduced by a constant factor (as for the grey body),
(2.7)
©JHW 41
but dependent also on wavelength λ.
a) Absorptance of a semiconductor of gap Eg:
2.3.2 Radiation from a semiconductor
) p g p g
Simplest model: no light absorption for hν < Eg and
complete light absorption for h Egν ≥ .
Absorptance aΦ
©JHW 42
(ratio of absorbed to incident radiation, see chapter 1.2)
= step function.
© JHW
a) b) c)
©JHW 43
Fig. 2.3: a) absorption in a semiconductor,
b) step function of absorptance vs. photon energy,
c) versus wavelength.
Simplest model (step-like absorption, no reflection):
spectrum similar to black body spectrum, however, cut off for λ > λg;
T = 300 K: maximum of black body radiation at about 11 µm;
b) Emitted spectrum of a semiconductor:
variations of T: only weak change of spectrum.
Fig. 2.4:Black body spectrum near room temperature.
Semiconductor with E =
L(W
msr
m)
μ- 2
el
BB
- 1- 1
60
80
100
120
140
T = 500 K
© JHW
©JHW 44
Semiconductor with Eg = 0.31 eV: same spectrum
as black body for λ < 4µm.
spec
tral
de
nsity
0 5 10 15 200
20
40
60
200 K300 K
400 K
wavelength λ µ( m)
c) Absorptance aΦ and absorption constant α :
We go back to Fig. 1.1: Reflected, absorbed, and transmittedradiation (see chapter 1.2):
reflectance, refl. coefficient
Φr
Φ0
Φa
Φt(1 6 c)
absorptance, absorp. factor
aΦ
0
Φa =
Φ
transmittance, tΦt
(1.6 b)
(1.6 a)rΦ
0
Φr =
Φ
©JHW 45
© JHW (1.6 c)transm. factor
tΦ
0
t =Φ
Now we assume: rΦ = 0: Φa = Φ0 - Φt
Transmitted intensity within a semiconductor at depth x:
-αxt 0Φ (x)=Φ e
Definition of absorption constant α :
(2.8)
The absorptance aΦ is thus
(2.9)
00
taa 0)(λΦ e1 x)( λα−−=ΦΦ−Φ
=ΦΦ
=
with Φ0 = incident intensity.
t 0( )
©JHW 46
If w is the thickness of the sample:
The absorption constant is discussed in chapter 5.2.1.
(2.10)a 1)(λ we )( λα−Φ −=
2.4 Internet Links
1. Black body Radiation (Applet): http://100-online.ipe.uni-stuttgart.de/applets/planck/Planck.html
2. Black body Radiation (Applet): http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html
©JHW 47
2.5 Literature
1. E. Hecht, Optics 3rd edition (Addison Wesley, Reading, MA, 1998), p. 578
2. H. G. Wagemann and H. Schmidt, Grundlagen der optoelektronischen Halbleiter-bauelemente (Teubner, Stuttgart, 1998), p. 60.
©JHW 48
3. Coherence
©JHW 49
Two waves are coherent when their phase difference is constantin time.Only in this case, interference is observable, because interference isthe result of phase differences between waves.
3.1 Definition
the result of phase differences between waves.
coherent:
incoherent:
monochromatic, very (infinitely) long wavetrains ofsame frequency (e.g. Laser)
light with different wavelengths (e.g. light from a fluorescent lamp)
©JHW 50
Interference is only observable with coherent light!
Wavetrains as long and as monochromatic as possibleare needed in order to observe interference.
Correlation between the phases of a travelling wave separated by a delay time τ at the same location.
3.2 Temporal coherence
l
short coherence time infinitely long coherence time
lc
©JHW 51
Relation between coherence time tc and coherence length lc:
lc = tc cwith c = speed of light
3.3 Spatial coherence
Correlation between the phases of a travelling wave at different locationsat the same time.
locations of constant phase
small spatial coherence infinite spatial coherence
constant phase
Interference experiments:
wave train splits into two parts which traverse different distances, difference of distance must never exceed the coherence length lc.
R = 100 %
©JHW 52
R = 50 %
R = 100 %
detector
Example:Michelson interferometer
radiation: photon emission via spontaneous transition of electrons in atoms from excited (E1) to lower energy state (E0).
3.4 Emission of photons
e-
hν energy E and frequency ν of photon:E1ΔE1
(3.1)
(3.2)
E = E1 – E0 = h νE0
excited state has finite average lifetime ΔtHeissenberg´s uncertainty relation: ΔE Δt > h,
with h = Planck´s constant
energy of photon not exactly defined: ΔE > h/Δt
©JHW 53
frequency of photon not exactly defined:from (3.1): Δ(h ν) = h Δν = ΔE and (3.2): Δν > 1/Δt
many atoms emit many photons with different frequencies ν +/- Δν:
resulting wavetrain not monochromatic
Example:
Ne gas discharge lamp, λ = 632 nm, Δt = 10-8 s
ν = c0/λ = 4.7 x 1014 Hz
Δν = 1/Δt = 108 Hz
spontaneously emitted light of a hot body (grey, black, etc.):
from excited independently emitting atoms not coupled or
©JHW 54
from excited, independently emitting atoms, not coupled orsynchronized;
every atom emits photon with different frequency
superposition incoherent light.
lasers:
rely not on spontaneous but on stimulated emission
Only lasers are able to emit really coherent light!
rely not on spontaneous but on stimulated emission
(chapter 7.1).
e-
2 hνE1
E0
hν
©JHW 55
If you are interested in more information about different lightsources and lasers (fluorescent lamps, solid state lasers etc.)then visit the lecture “Lasers and Light Sources” during thewinter term.
3.5 Internet Links
1. http://en.wikipedia.org/wiki/coherence_(physics)
©JHW 56
3.6 Literature
1. H. Weber and G. Herziger, Laser – Grundlagen und Anwendung(Physik-Verlag Weinheim, 1972), p. 11.
2. C. Gerthsen, H. O. Kneser, and H. Vogel, Physik 16. Auflage(Springer, Berlin, 1989), p. 457.
3. E. Hecht, Optics 3rd edition (Addison Wesley, Reading, MA, 1998), pages 308-311.
4. H. G. Wagemann and H. Schmidt, Grundlagen der optoelektronischen Halbleiter bauelemente (Teubner Stuttgart
©JHW 57
optoelektronischen Halbleiter-bauelemente (Teubner, Stuttgart, 1998), pages 39-43.
4. Semiconductor Basics
©JHW 58
Crystal: electrons cannot take arbitrary positions and energies.
semiconductor: allowed energy bands, separated by band gap(energetically forbidden band).
4.1 Energy bands and Fermi function
Highest occupied band at low T: valence band
lowest unoccupied band at low T: conduction band
conduction band
valance band
band gap
©JHW 59
EF = Fermi energy.
f Ee
E EkT
F( ) ,=
+−
1
1
(4.1)Fermi function
for all temperatures T; probability to find electron at certain energy E:
4.2 The wave vector k
Spatially periodic crystal lattice:
probability ΨΨ*dx to find an electron in a certain interval dxis also spatially periodicis also spatially periodic.
Wave function : solution of (time independent) Schrödingerequation:
Ψ( )r
2
pot(r) (E E (r)) (r).2m
− ΔΨ = − Ψ (4.2)
©JHW 60
Spatial periodicity of : direct consequence of spatial periodicityof
Ψ( )r
E rpot ( ).
2m
Probability to find an electron in a certain (crystallographic) direction isspatially periodic.
Wave functions : Bloch functions, spatially modulated
sin-functions.
Ψ( , ) r k
Wave functions are characterized by wave vectork .
©JHW 61-4
4.3 The band structure E k( )
= total energy of electrons in a certain state (Bloch wave) with
wave vector k.
Total energy = sum of kinetic and potential energy.
E k( )
conduction E(eV)
3210-12
band
valenceband
Allowed bands: separated by band gap.
©JHW 62
Fig. 4.1: Band structure of silicon (seen in [100]-direction) withlowest conduction band and highest valence band.
-2
2aπ 4
aπ 6
aπ2
aπ4
aπ6
aπ [100][100]
© JHW
Crystals with face centered cubic structure (fcc) and two atoms in baseof the lattice (Si, GaAs, etc):
Energy periodic in k with periodicity 4π/a, where a = lattice constant.
4.4 Limited range of -values, the Brillouin zonek
Consequence: usually only values for -4π/a < k < 4π/a(first Brillouin zone) are shown.
Symmetry with respect to k = 0 for cubic semiconductors: only half of this zone is mostly given.
©JHW 63
4.5 The crystal momentum pk
conserved in many processes („k-conservation“).
Electron in state : crystal momentumk
p kk = . (4.3)
4.6 Impulse pe
different from crystal momentum: p m ve eff e= (4.4)
2with the effective mass
0
2
eff 2
2k k
mE
k =
=∂∂
(4.5)
for an electron in a certain statek
and the velocity
v E ke k k k
= ∇=
1
0
( ) (4.6)
©JHW 64
Note: Notpe is conserved, but
pk !!!
k-conservation means: Periodicity of Bloch function is conserved!
for an electron in a certain state k0.
4.7 Direct and indirect band gap semiconductors
direct (band gap) semiconductor:
maximum of valence band (and mini-mum of conduction band) at same
k
indirect semiconductor: maximum of valence band (and mini-mum of conduction band) at different
kmum of conduction band) at different k
E E
©JHW 65
Fig. 4.2: Direct and indirect band gap semiconductors.
k© JHW
k
4.8 Internet Links
1. AlGaAs band diagram and E-K diagram (Applet):http://www.acsu.buffalo.edu/~wie/applet/students/mcg/ternary.html
2. SiGe band diagram and E-K diagram (Applet):http://jas2.eng.buffalo.edu/applets/education/semicon/SiGe/index.htp j g ppml
3. Carrier Concentration vs. Fermi Level (Applet):http://www.acsu.buffalo.edu/~wie/applet/fermi/fermi.html
4. Carrier Concentration vs. Fermi Level and Density of States(Applet):http://www.pfk.ff.vu.lt/lectures/funkc_dariniai/sol_st_phys/fermi_level_applet2.htm
©JHW 66
5. Fermi Function and Localized Energy States:http://www.acsu.buffalo.edu/~wie/applet/fermi/functionAndStates/functionAndStates.html
6. 3D Solid State Crystal modelshttp://www.ibiblio.org/e-notes/Cryst/Cryst.htm
7. Oscillating 3D Crystalhttp://www.physics.uoguelph.ca/applets/Intro_physics/kisalev/java/anon/index.html
©JHW 67
5. Excitation and recombinationprocesses in semiconductors
©JHW 68
5.1 Introduction
Light emitting diodessemiconducting lasers back bone of optoelectronicssemiconductor detectors
for generation detection and amplification of light and other non
being too wide-banded non-coherence too low an intensity
for generation, detection, and amplification of light and other non-visible radiation.
Black, grey, and selective radiation not appropriate for optoelectronics:
©JHW 69
Emission of light from semiconductors (visible and non-visible):• not based on heat,• no Planck´s spectrum (chapter 2.3),• but based on luminescence!
Luminescence
What is luminescence ?
= generation of optical radiation by non-thermal processes
excitation term example
Table 5.1: Examples for luminescence
light photoluminescence
voltage, injection
electron beam
h i l it ti
electroluminescence
cathodoluminescence
h l i
fluorescence tubes
LEDs, lasers
TV, image tubes
l
©JHW 70
chemical excitation chemoluminescence glow-worms
Optoelectronics makes use of electroluminescence.
5.2.1 Beer’s absorption law
5.2. Absorption of radiation in semiconductors
hνhνabsorbing
Power of impinging wavedamped by absorption:
.α-x/L-αx0 0Φ(x)=Φ e =Φ e (5.1)
hν
Φ 0
Φ tΦ (x)
absorbingbody
©JHW 71
Fig. 5.1: Absorption of radiation.
distance x0 d
© JHW α = absorption constantor -coefficient
Low-doped and defect free semiconductor:
• absorbs radiation only for hν ≥ E ;
Absorption coefficient α,absorption length Lα = α-1 depend on photon energy hν.
absorbs radiation only for hν ≥ Eg;
• absorption requires k-conservation.
Important consequence of conservation of crystal momentum (or quasi
momentum) for absorption constant α:α higher for direct than for indirect band gap semiconductors.
©JHW 72
Absorption process in indirect semiconductors:
• difference in k-value (Δk) of conduction and valence band edge;
• Δk cannot be overcome by absorbed photons alone (see chapter 5.2.2).
5.2.2 Crystal momentum and momentum (impulse) of photons
Position of conduction band minimum of Si:
a) The indirect band gap of Si
(5.2)2
0.85 [100],mincondk
a
π=
Position of valence band maximum: 0.max
valencek =
Excitation of electrons from valence band maximum to conduction band minimum:
required momentum change:
© JHW
a
©JHW 73
Fig. 5.2: Indirect band structure of
silicon with Eg = 1.12 eV
required momentum change:
with,k Si Sip kΔ Δ= (5.3)
20.85 .Sik aΔ π= (5.4)
Lattice constant a of silicon: a = 0.54 nm;
which has to be overcome in the absorption process of a photon.
,101108.9108.9285.0 181719 −−− ×≈×=×==Δ cmcmmakSiπ
b) Conservation of energy E and k-value
Transitions of electrons between different states in a semiconductor crystal:
conservation of energy E and k-value (i.e. crystal momentum) required according to
E = E ± ΔE (5 5)
©JHW 74
where E1, are energy and k-values before, and E2, after thetransition.
1k
2k
2 1
2 1
E = E ± ΔE,
k = k ± Δk,
(5.5)
(5.6)
Excitation of electrons across Eg by interaction with photons:
energy conservation not a problem:photon energies of visible regime: hν ≥ Eg,condition ΔE = Eg easily fulfilled;
however: k conservation is a problem:however: k-conservation is a problem:
k-value of photons too small to fulfill k-conservationin case of an indirect semiconductor.
c) Energy E and k-value of photons:
Photon energy: Ephoton = hν, and c = νλ. Consequently,
©JHW 75
photonphoton kcc
hE ==λ
(5.7)
linear relationship between photon energy Ephoton and
wave number (absolute value of wave vector) kphoton.
Table 5.1: Energy and k-value for photons
Ephoton [eV] λ[nm] kphoton[cm-1]
1 1240 0.5x105
52 620 1x105
3 413 1.5x105
10 124 0.5x106
2000 0.62 1x108
Photons with hν ≈ 1 eV: k ≈ 105 cm-1 i e a factor of 1000 below
©JHW 76
Photons with hν 1 eV: k 10 cm , i. e. a factor of 1000 below
ΔkSi = 1x108 cm-1.
Electronic transitions not possible with such low-energy photons;
allowed transitions more or less “vertical”;
in Si strong absorption of light only for hν > 3.4 eV.
E(eV) E(eV)
3
2
1
IkI-1 0-2 1x10 cm5 -1
( )
© JHW
2πa
2πa
3
2
1
IkI
CB
?
0.5 0.5
CB
1x10 cm8 -1-1x10 cm8 -1
( )
©JHW 77
Fig. 5.3: Band structure for a) photons and b) electrons in Si
VB
a) b)
5.2.3 Phonons
Phonons = energy quanta of lattice vibrations.
Interaction of phonons: momentum conservation possible during absorption (or emission) of a photon in an indirect semiconductor.
Simple cubic lattice:
The phonon momentum is largest, when the wavelength
2π /p pkλ =
is smallest.
p pp k=
The phonon energy Ep = hωp is small: Ep ≈ 10 … 50 meV << Eg.
©JHW 78
smallest wavelength λmin equal to lattice constant a, i.e. λmin = a.
Therefore: 2π 2π / .maxp p mink k aλ≤ = =
Momentum of phonons spans same range as momentum ofelectrons!
Phonons: supply (or take over) large momentum and small energy.
Absorption process of photons with energy close to band gap in indirectsemiconductors:
5.2.4 Light absorption / light emission
Direct semiconductors: no phonons necessary for transition of electrons between conduction band and valence band
photon supplies energy (and almost no momentum)phonon supplies momentum (and almost no energy)!
©JHW 79
electrons between conduction band and valence band.
absorb light better (higher absorption const. α);
emit light also easier (higher constant B for radiative
transitions, see chapter 5.4).
5.2.5 Fundamental absorption in semiconductors [1]
Fundamental absorption = absorption at the band edge.
Absorption behavior different for direct and indirect semiconductors;- Absorption behavior different for direct and indirect semiconductors;
- different dependence of absorption constant α on photon energy hν.
- Vice versa: measurement of α(hν):
distinction between direct and indirect semiconductor.
©JHW 80
5.2.5.1 Direct band gap semiconductors
Exactly parabolic band:
E
Fig. 5.4: Absorption in a direct band gap semiconductor Consequently,
.)( 21
gdirdir EhA −= να (5.8)
).(2gdir Eh −∝ να (5.9)
k© JHW
©JHW 81
)( gdir
Constant Adir:
with n = refraction index
er
he
he
dir mchn
mm
mmq
A2
23
2 2
+
= (5.10)
with nr = refraction index,
me, mh = (effective) masses of electrons and holes, and
q = elementary charge.
For me ≈ mh ≈ m0 (free electron mass) and nr ≈ 4:
(5.11)2/14 ])[(102 eVEh gdir −×≈ να
©JHW 82
This equation yields for
hν = Eg+ 1 eV αdir ≈ 2×104 cm-1
Lα = 1/αdir ≈ 5×10-5 cm = 0.5 µm
5.2.5.2 Indirect band gap semiconductors
Fundamental absorption requires
1. absorption of a photon+ emission of a phonon, or
2. absorption of a photon
© JHW
p p+ absorption of a phonon
(5.12a)
1.: photon needs energy above Eg:
hνabs,1 = Eg + Ep
(5.12b)
2.: photon needs energy below Eg:
hνabs,2 = Eg – Ep
©JHW 83
Fig. 5.5: Fundamental absorp-tion in indirect semiconductors
phonon emission = lattice vibrations become stronger,
phonon absorption = latticevibrations become weaker.
, g p
Absorption constant αind of an indirect semiconductor
= sum of processes of phonon absorption (αabs) and emission (αemi):
Number of phonons = f(T): αabs and αemi = f(T).
f
(5.13)( ) ( ) ( )ind abs emih h hα ν α ν α ν= +
Low temperatures T: only few phonons can be absorbed.
Phonon absorption: strong temperature dependence,
phonon emission: weak temperature dependence.Both processes depend on statistics of phonons.
E
pgindabs p
EEhA
+−=
)( 2να low T: αabs 0 ;
strong T dependence(5.14a)
©JHW 84
Aind = constant.
kTE
pgindemi
kT
p
p
e
EEhA
e
−
−
−−=
−
1
)(
12ν
α
strong T-dependence
low T: αemi prevails;
weak T-dependence(5.14b)
For constant temperature T we get
Pre-factors depend on temperature T.
Fi 5 6 T d d f
(5.15)2 2( ) ( )( ) ( )( ) .abs emiind ind g p ind g phv A T h E E A T h E Eα ν ν= − + + − −
Fig. 5.6: T-dependence ofphoton absorption;
low T: only phonon emission.
Only those photons with
E = Eg + Ep are absorbed.
Extrapolation to α = 0:
©JHW 85
Extrapolation to α 0: two axis intercepts
at Eg ± Ep..© JHW
© JHW
©JHW 86
Fig. 5.7: Comparison of absorption constant for direct and indirect
semiconductor of same band gap (at high T).
© JHW
5.2.5.3 Absorption via impurity to band transitions
Useful for detection of very low energy photons (hν ≈ 50 meV).
For that purpose: cooling of semiconductor (to T < 20 K for Si);
shallow donors (or acceptors) in n-type (p-type) Si are occupied with
l t (h l )electrons (holes);
photons excite electron (hole) from shallow donor (acceptor) to
conduction (valence) band;
increased conductivity.
Fig. 5.8:EcEF
e
Ec
©JHW 87
Light absorption inshallow level forinfrared detection.
FED
Ev
+
hν
© JHW
EAEFEv
h+
hν
5.3. Carrier recombination in semiconductors
5.3.1 Classification of recombination processes
©JHW 88
Fig. 5.9: Recombination in semiconductors
© JHW
Recombination = recovery of equilibrium
Figure 5.9 distinguishes between transition between bands (1) level transitions (2) intra band transitions (3)( ) Auger transitions (4)
(1) Transitions between bands (inter band transitions):
1a) direct transitions
1b) indirect transitions (with phonon emission/absorption)
(2) Level transitions:
©JHW 89
(2) Level transitions:
2a) level to band transitions (for example from donor D)
2b) donor acceptor/transitions (often radiatively)
2c) phonon cascade transitions and/or multi phonon transitions.
Energy dissipation of electrons and holes via emission of phonons,
i.e. they excite lattice vibrations.
(4) Auger transitions:
(3) Intra band transitions:
(4) Auger transitions:
Electron and hole recombine over the band gap;
excess energy given either to electron (in n-type material)
or to hole (in p-type material);
carrier excited to high energies in conduction band (valence
band);
©JHW 90
band);
looses its energy finally via excitation of lattice vibrations,
i.e. process No. (3).
Processes for the generation of radiation: 1a) to 2b)
Only process 1a) is important for LEDs and lasers.
Lifetime due to radiative recombination: calculated below.
Process No. 2c):
Recombination via deep traps “deadly” for most optoelectronic
devices (Shockley-Read-Hall-recombination, SRH).
Process No. 4):
Limits the lifetime of carriers in silicon.
©JHW 91
5.3.2 Carrier lifetime due to radiative recombination
Measures to avoid recombination processes of Fig. 5.9:
• Use of crystals without defects (to avoid SRH-recombination),
• low doping (to avoid donors/acceptors)• low doping (to avoid donors/acceptors),
• low temperatures (to suppress phonons).
However: one process cannot be suppressed by principle:
Radiative recombination.
P li it lif ti f i t fi it l
©JHW 92
Process limits lifetime of excess carriers to a finite value.
Gedanken-experiment:
For understanding radiative recombination:
“Gedanken”-experiment: Semiconductor of band gap Eg and
temperature T,in a black shoe box of same temperature and closed lid:in a black shoe box of same temperature and closed lid:
carrier concentration of electrons and holes in the bands?
time dependence of these concentrations?
Is the semiconductor in thermodynamic equilibrium? Why and how?
a) Thermodynamic equilibrium
©JHW 93
Requirement of thermodynamic equilibrium:
Balanced exchange of energy between semiconductor and its
environment (the shoe box);
for T = constant: no net energy stream from or to semiconductor.
Energy stream from black shoe box to semiconductor:black body radiation of inner walls of the shoe box;
Absorbed by the semiconductor:
only photons with energy hν > Eg.
Condition for thermal equilibrium:semiconductor has to emit the same energy as it absorbs.
For Tsemic = Tshoe box:emitted radiation spectrum of semiconductor
= absorbed spectrum (otherwise Tsemic ≠ Tshoe box!).
©JHW 94
recombination of electrons and holes.
There is only one source for radiation of semiconductor:
Thermodynamic equilibrium: Tsemic = Tshoe box:
constant stream of photons onto semiconductor:
continuous excitation of electrons from valence band intoconduction band,i e continuous generation of excess electron/hole pairs;i.e. continuous generation of excess electron/hole pairs;
recombination of e/h-pairs radiation emitted by the semiconductor.
Dynamic equilibrium between excitation and recombination.
Mean electron and hole concentration = constant:
.2innp = (5.16)
©JHW 95
Equilibrium generation rate G0 of e/h-pairs:
must depend on absorption properties of the semiconductor,
i.e. on band gap Eg and on absorption constant α(see information sheet).
Thermodynamic equilibrium:
recombination rate R0 of recombining thermal generation rate
e/h-pairs within the semiconductor G0 due to the black bodyper cubic centimeter and second radiation
b) The equilibrium recombination rate R0
=per cubic centimeter and second radiation,
1.: G0 = f(optical properties of semiconductor via absorption constant α);see information sheet;
2.: R = f(concentration of electrons and holes)because their recombination must supply the radiation:
.00 RG = (5.16)
©JHW 96
because their recombination must supply the radiation:
and for thermodynamic equilibrium:
BnpR = (5.17)
.2000 iBnpBnR == (5.18)
B = radiative recombination constant,characteristic value for a particular semiconductor.
20
20
ii n
G
n
RB == (5.19)Since
and G0 = f(absorption constant α),
B depends on band structure of the semiconductor under
consideration via G0 (α) and ni.
The higher the absorption constant α, the higher is also B.
ii nn
©JHW 97
Direct semiconductors with large α-values have stronger radiative recombination.
(examples given below)
c) The non-equilibrium case
Starting condition:
- semiconductor in black shoe box, thermal equilibrium,
- G0 of e/h pairs within the semiconductor as before.
Change to non-equilibrium state:
- constant injection of electrons and/or holes into semiconductore.g. by application of bias voltage to contacts:
non-equilibrium concentrations n, p with in the steady state,and
2innp ≥
BnpR = (5.17)
©JHW 98
R larger than equilibrium rate R0 = G0.
Net radiative recombination rate Urad (number of disappearing carriersper second):
0 0 0radU R G B[ np n p ]= − = − (5.20)
(5.20)
0 0 0 0
0 0 0 0 0 0
0 0
0 0
[( )( ) ]
[ ]
( )
, ,
B n n p p n p
B n p np pn n p n p
B p n p due to n p
and n p n p
= + Δ + Δ −= + Δ + Δ + Δ Δ −≈ Δ + Δ = Δ
Δ Δ <<
Radiative lifetime τr
= mean lifetime of excess carrier until it recombines:
(5.21)1 2
ir
rad 0 0 0 0 0
nΔpτ = = =
U B(n + p ) R (n + p )
©JHW 99
maximum of τr for minimum of n0 + p0;
minimum of n0 + p0 for intrinsic semiconductor,
i. e. n0 = p0 = ni and n0p0 = ni2;
1
2 2 2
2undoped dopedi ir r
0 i 0 i
n nτ = = = > τ
R n R Bn
Dependence of carrier lifetime on doping concentration due to pureradiative recombination:
(5.22)
τr ∝ 1/n0 ≈ 1/ND , with ND = doping concentration
Tab. 5.2: Radiative lifetime of minorities and majorities for severalintrinsic semiconductors.
quantity Si Ge GaAs
B[ 3/ ] 2 10-15 3 4 10-14 7 10-10
©JHW 100
B[cm3/s] 2 x 10 15 3.4 x 10 14 7 x 10 10
ni[cm-3] 1.04 x 1010 1.84 x 1013 2.04 x 106
Eg[eV] 1.12 0.67 1.45
τrundoped 6.6 h 0.79 s 350 s
5.3.3 Emitted spectrum under non-equilibrium due to band/band recombination
see information sheet
©JHW 101
5.3.4 Other radiative recombination processes
Processes 2a, 2b in Fig. 5.9 also radiative;
• Process 2a: in GaP diodes with isoelectronic nitrogen centers,
green/yellow emission. Nowadays: GaP replaced by InGaAsP;
• Process 2b, donor/acceptor transitions:
often used for material analysis of doped direct semiconductors;
luminescence very weak, not useable for light generation.
©JHW 102
5.3.5 Non-radiative recombination processes
Processes 2c to 4 in Fig. 5.9: non-radiative;
• no general closed-form expression available describing carrier lifetime;
• Auger processes and recombination via deep traps (SRH-re-g (combination): lifetime modeling relatively simple (see lecturePhotovoltaics);
• processes involving phonons: modeling difficult.
Process 2c: phonon cascades (= subsequent emission of phonons)
Phonons: only small energies (10 to 50 meV)
©JHW 103
Phonons: only small energies (10 to 50 meV),
energetically closely spaced levels required;
phonon cascades only important for recombination into shallow
levels.
Process 2c: multi phonon emission (= simultaneous emission of phonons)
Process very improbable for energy dissipation of electrons;however: important for lattice relaxation of deep levels.
Process 3: intra band transitions
Within bands: available energy levels are continuous;
energy dissipation of electrons and holes via phonon cascades.
Process 4: Auger recombination
Most important effect for recombination in (pure) indirect
©JHW 104
p (p )semiconductors;
Mechanism: transfer of excess energy of recombining electron/hole-
pairs to either a third partner electron (in n-type material) or hole(in p-type material).
© JHW
©JHW 105
Fig. 5.10: Auger-effects in a direct semiconductor
5.4 Internet Links
1. Indirect recombination via an energy state in the band gaphttp://www.acsu.buffalo.edu/~wie/applet/recombination/indirect.html
©JHW 106
5.5 Literature
1. J. I. Pankove, Optical Processes in Semiconductors (Dover Publications, New York, 1971), p. 35 ff.
©JHW 107
6. Light emitting diodes
©JHW 108
Mechanism: spontaneous emission due to radiative band/bandrecombination of electrons and holes.
Spontaneous emission: inverse process of absorption.
6.1 Working principle of an LED
Figure 6.1 compares the two processes for a general two-level system.
Fig. 6.1:Absorption and spontaneous emission for a
absorption
E
before
E
2
1
hν
spontaneous emission
E
E
2
1
©JHW 109
emission for a system with two discrete electron
levels E1, E2.after
E
E
E
2
1
1 © JHW
E
E
E
2
1
1
hν
Energies E1, E2 mono-energetic: radiation emission with hν = E2 – E1.
Semiconductor: E1, E2 correspond approximately (but not exactly) tovalence band edge EV and conduction band edge EC.
Emitted photon energy hν of luminescence diodes with band gap Eg:
hν ≈ EC - EV = Eg.
Generation of visible radiationrequires
λ < 780 nm Eg > 1.60 eV
©JHW 110
Fig. 6.2: Spontaneous emission ina semiconductor.
λ g
λ > 380 nm Eg < 3.26 eV
© JHW
Requirement for efficient generation of radiation: many electrons and holes at same site and same time!
Consequence: semiconductor must be in nonequilibrium!
Equilibrium: np = ni2.
Consequence: in n-type material: n large, p small,
in p-type material: p large, n small,
in intrinsic material: n and p small.
All these cases: emitted radiation low (equilibrium selective body radiation of chapter 2.5.2).
©JHW 111
Requirement for strong radiation: np > ni2,
by injection of carriers, for example across a pn-junction.
Extension of recombination zone:
one diffusion length into bulk of n-type and p-type region.
recombination
pp-type
n-typen
© JHW
©JHW 112
Fig. 6.3: Recombination by carrier injection into junction.
Quasi Fermi levels EFn (for electrons) and EF
p (for holes):
From np > ni2 it follows: EF
n > EFp.
©JHW 113
Fig. 6.4: Band diagrams for LED without and with bias voltage V.
© JHW
6.2 The spectrum emitted by an LED
No black body spectrum described by Planck’s equation.
Exact shape of emitted radiation:
depends on energy distribution of electrons in conduction banddepends on energy distribution of electrons in conduction bandand holes within valence band.
For electron and hole density distribution (see Fig. 6.5b), it holds:
nE = Dc(E)fn(E) = dn/dE (6.1)and
©JHW 114
pE = Dv(E)fp(E)= dp/dE, (6.2)
with the density of states Dc(E), Dv(E), and the Fermi functions
fn(E), fp(E) for electrons and holes.
E
EC
ne
GaAsT = 300 K
arb
.un
it s)
© JHW
C
VE
a)
E
p
g
e
occupiedstates
b)
photon energy hν (eV)1.45 1.50 1.55
2kT
inte
n sit y
(a
c)
©JHW 115
Fig. 6.5: a) Electron energies in a semiconductor, b) occupied states andc) emitted spectrum of an LED.
a) b) c)
Fig. 6.5a shows: Electron and hole recombination not directly fromband edges but between slightly higher energies.
Emitted spectrum: after van Roosebroek and Shockley:
(6.3)2( ) ( ) forgh E
kTg gh h h E e h E
ν
ν ν ν ν−
−Φ ∝ − >
• Energetic width Δ(hν) at room temperature: Δ(hν) ≈ 2kT = 52 meV;
• maximum of the radiation: about 1kT ≈ 26 meV above Eg.
Reason: energetic width of Fermi distribution function of electrons andholes ≈ 2kT.
O l th i ti idth Δ(h ) d t idth Δλ
( ) ( ) og gh h h e hν ν ν ν
©JHW 116
On wavelength axis: energetic width Δ(hν) corresponds to width Δλ.
From λ = c/ν it follows:2 2 2
2 2
1( ).
c ch
c c hc
λ λ λΔλ Δν Δν Δν Δν Δ νν ν ν
∂= = − = − = − = −∂
(6.4)
with Δ(hν) ≈ 2kT it follows for the width Δλ:
• Width Δλ of radiation increases with square of center wavelength λ !
2 2 22 52 meV 1.
1.24 eV 24 µm
kT
hcΔλ λ λ λ= = =
µm(6.5)
q g
• For example, GaAs LED with λ = 870 nm has a width Δλ = 32 nm.
• Diodes emitting at 1.3 µm or 1.5 µm (optimum wavelengths forcommunication via glass fibers):
width Δλ = 70 nm and Δλ = 94 nm, respectively;
too large for optical data communication;
©JHW 117
lasers with typical widths Δλ < 0.1 nm are used.
Frequency band width of GaAs-LED:
λ = 870 nm ⇔ ν = 354 THz
Δλ = 32 nm ⇔ Δν = 12 THz Δν/ν = 3.4x10-2
6.3 Materials for LEDs (and lasers)
6.3.1 III/V-Compounds (GaAs, GaP, InAs etc.)
AlPV)
380 nm3.0
3.5 © JHW
AlP
AlAsindirect
ban
d g
ap
E
(eV
g
direct
visible
780 nm
1300 nm
1500 nm
AlSb
GaSb
GaAs
GaP
Si
Ge0.5
1.0
1.5
2.0
2.5
In Ga As0.53 0.47
InP
I Sb
©JHW 118
Fig. 6.6: Binary and ternary compounds for optoelectronic devices;dashed lines represent indirect band gaps.
lattice constant a (Å)
5.3 5.5 5.7 5.9 6.1 6.3 6.50.0
InAs0.53 0.47 InSb
Fabrication of LEDs: wide range of materials byalloying III/V-compounds.
Example: system GaAs and AlAs completely miscible; AlxGa1-xAs;
variation of x: lattice constant almost unchanged,band gap varies over wide range.band gap varies over wide range.
AlxGa1-xAs grows without lattice defects on GaAs substrates.
GaP and InP also miscible, however, lattice constant changesover wide range.
The following parameters change upon alloying:
©JHW 119
g p g p y g
band gap Eg
lattice constant a
band structure (direct, indirect)
thermal expansion coefficient
Wavelength selection of emitted light: by selection of band gap Eg.
However: Eg coupled to a certain lattice constant a for particular alloy.
Challenge to find an appropriate substrate:
light emitting material usually grown by epitaxylight emitting material usually grown by epitaxy(e. g. any composition in the InGaAs system).
Obstacle: not many materials can be considered as substrate.
Requirements for the substrate:
a ailabilit in large areas (> 3 inches)
©JHW 120
availability in large areas (> 3 inches)
defect free (no dislocations etc.)
lattice matched to epitaxial layer
similar thermal expansion coefficient as epitaxial layer.
Substrate materials:
III/V-compounds: only GaAs, InP and GaP available in required size.
Silicon: most used material in microelectronics,but no fit to lattice constants of III/V-materials.
Figure 6.6 demonstrates the following interesting features:
No direct band gap III/V-semiconductor with Eg > 2.3 eV available. No blue light with these materials; new materials: nitrides and/or organic materials!
G P h hi h t b t i di t b d
©JHW 121
GaP has highest, but indirect band gap;
not very efficient in emission.
For certain mixture, ternary alloys of GaP and InP fit onto GaAs
substrate.
For certain mixture, ternary alloys of GaAs and GaSb (or InAs) fit
onto InP substrate.
Alloy In0.53Ga0.47As = direct, fits on InP and emits at 1.5 µm,
an ideal wavelength for glass fibers.
Semiconductor for 1.3 µm emission:
- growth by ternary alloy not possible,
- requires four instead of three elements (quaternary alloy).
©JHW 122
GaP 2.26Ga(As,P)
indirectGaP
(In,Ga)P
gE (eV)
5.576 Å
5 653 Å
1.8 eV
1.421.35
0.36
InP
InP
GaAs
GaAsGaAsInP
InAs
5.653 Å
5.869 Å
5.960 Å
1.6 eV
1.4 eV
1.2 eV
1.0 eV
0.8 eV
0.6 eV© JHW
©JHW 123
Fig. 6.7: Quaternary alloys in the system In1-xGaxAsyP1-y.
InP
InAs
(In,Ga)AsIn(As,P)
Band gap adjustment:
System In1-xGaxAsyP1-y:
Eg between Eg = 0.36 eV (InAs) and Eg = 2.26 eV (GaP),
corresponds to λ = 3.4 µm to λ = 0.55 µm.
For glass fibers: λ = 1.3 µm (0.95 eV) and λ = 1.5 µm (0.83 eV)
most important.
LED and lasers with these two wavelengths
can be grown on InP substrates.
©JHW 124
On InP substrates, the maximum possible Eg is 1.35 eV (0.918 µm).On GaAs substrates, the maximum is 1.85 eV (0.670 µm) .
6.3.2 Materials for blue light (LEDs and lasers)
Why blue light?
for color displays (RGB) for generation of white light (via LUCOLEDs) f ti l t t ith hi h d it (CD t ) for optical storage systems with higher density (CDs etc.).
Laser scanning of CDs:
• The smaller the wavelength λ, the smaller can be
distance d between two pits on CD;
• Resolution is limited by diffraction of laser beam at edges of lens of scanning system according to
©JHW 125
lens of scanning system according to
1.22 .mindnsin
λΘ
= (6.6)
Materials of the past: SiC, Zn(S,Se)
research essentially given up, due to low luminescence efficiency (hampered by defects)
materials of today: GaN, InN, InxGa1-xN
first good material in 1995
- low sensitivity to defects
- on SiC-substrate (Siemens, Cree)
©JHW 126
- on Al2O3-substrate (Nichia)
more on organic LEDs in “Lasers and Light Sources”
6.4 Emission efficiency of LEDs
External quantum efficiency (EQE)
Internal quantum efficiency (IQE):= number of created photons per recombining e/h-pair: high up to 99 %number of created photons per recombining e/h pair: high, up to 99 %.
External quantum efficiency (EQE):= number of photons leaving crystal per recombining e/h-pair:
low, usually around 3 to 4 % for an LED;most created photons trapped within semiconductor.
Therefore:
EQE = IQE x η (6 7)
©JHW 127
re-absorption reflection total reflection
Three reasons for low optical efficiency ηopt:
EQE = IQE x ηopt (6.7)
a) Re-absorption
Direct semiconductors: high absorption constant only light created directly underneath surface can leave crystal.
Normal pn-junction LED: re-absorption losses 10 to 20 %
(non-)absorption efficiency η b = 0 8 to 0 9
b) Reflection
(non )absorption efficiency ηnon-abs 0.8 to 0.9.
Way out: use of heterostructures.
Reflectivity (to the inner side!) between semiconductor (nsemi) and air(n = 1), according to chapter 1.4:
©JHW 128
Way out: epoxy with n = 1.5.
21
0.3 for 3.51
semisemi
semi
nr n
nΦ −= ≅ ≈ +
(6.8)
c) Total reflection
Photon generation within crystal (Fig. 6.8):surface
cΘ
eΩ
Fig. 6.8:Loss due tototal reflection.
back side
point ofgeneration
0Ω = 4 π
© JHW
©JHW 129
- Equal probability of all directions for emission of photons;
- only photons within cone of half angle Θc can leave crystal;angle Θc = angle of total internal reflection (chapter 1.3);for semiconductors with nsemi = 3.5: Θc = 17°.
Small angle Θc mainly responsible for low optical efficiency of LEDs!
Ratio ηtr of emitted to transmitted light
= ratio of solid angle Ωe (spanned by Θc)
to total spherical angle Ω0 = 4π:
2 (1 )
4e c
tr
Ω cosΘ
Ωη π −= =
π (6 9)
Total optical efficiency:
Way out: encapsulation in epoxy (n = 1.5); increases angle and
(non-)total reflection efficiency to Θc = 25° and ηtr = 9.4 %.
4
(1 ) 2 2.2 %0
c
Ω
- cosΘ /
π= ≈
(6.9)
(6 10)
©JHW 130
usually around 5 %;higher efficiencies: by pre-selection of preferential emission of photons
into the narrow cone of (non-)total reflection.
(6.10)ηopt= ηnon-abs rΦ ηtr
6.5 Internet Links
1. Formation of a PN Junction Diode (Applet): http://www.acsu.buffalo.edu/~wie/applet/pnformation/pnformation.html
2. PN Junction Diode under Bias (Applet): http://fiselect2.fceia.unr.edu.ar/fisica4/simbuffalo/education/pn/biasedPN/index.html
3. Light Emitting Diodes (Color calculation Applet ): http://www.ee.buffalo.edu/faculty/cartwright/java_applets/source/LED/index.htm
©JHW 131
7. Semiconductor Lasers
©JHW 132
laser = light amplification by stimulated emission of radiation
7.1 Working principle and components of lasers
7.1.1 Stimulated emission
c) stimulated emission
E2
E2
E1
hν
b) spontaneous emission
E2
E2
E1
a) absorption
before
E2
E2
E1
hν
©JHW 133
2
E1
hνhν
hν2
E1© JHW
after
2
E1
Fig. 7.1: Principle of stimulated emission.
Lasers: no use of spontaneous emission (in contrast to LEDs; Fig 6 1)
Fig. 7.1: Principle of stimulated emission:
Interaction of photon and excited atom, molecule etc.
emission of second photon amplification.
Lasers: no use of spontaneous emission (in contrast to LEDs; Fig. 6.1),
but stimulated (or induced) emission of photons (Fig. 7.1).
Stimulated emission: induced by resonator or cavity !
LED: individual emission processes independent of each other.
©JHW 134
Laser: stimulated emission resulting in amplification;
synchronization of individual radiation emitting sources.
Result: strong coherence of emitted radiation (see chapter 3).
Emitted radiation (photon) has same
energy
phase
polarization polarization
direction of emission (!)
as incident radiation (photon).
Last point particularly important:
Radiation forced ( ith the help of a resonator) to be emitted
©JHW 135
Radiation forced (with the help of a resonator) to be emitted
into cone not suffering from total reflection (see Fig. 6.8).
Consequence: high external quantum efficiency EQE.
7.1.2 Laser components
Components of (almost) every laser:
active medium (semiconductor, gas, crystal...) resonator (Fabry-Perot, Bragg reflector...) energy pump (bias voltage pump laser ) energy pump (bias voltage, pump laser...)
Fig. 7.2:Laser components.
energy pumpactivemedium
©JHW 136
© JHW
resonator
mirror(semi-transparent)
mirror
7.1.3 The ratio of stimulated to spontaneous emission
Electron transitions from high energy state E2 to low energy state E1:either by stimulated or by spontaneous emission.
Number N2 of electrons leaving state E2:
dN
Φj = photon flux density (i.e. light intensity),σ12 = cross section for stimulated emission,
(7.1).
2
stim 12j
2 AE
spon
dNdt
dN Adt
σ Φ=
©JHW 137
σ12 cross section for stimulated emission,AAE = Einstein coefficient for spontaneous emission.
Important: Sites of high radiation intensity stim. emission also high!!! Sites predetermined by resonator geometry,
which induces standing radiation wave.
7.2 General lasing conditions
Amplification by stimulated emission: must over-compensate losses byabsorption.
7.2.1 The gain of a laser (first general lasing condition)
p
Spatial dependence of radiation intensity Φ :
(7.2)
12 2abs ind
12 1 12 2
dΦ dΦ dΦ(x)= + = -αΦ + σ N Φ
dx dx dx
= -σ N Φ +σ N Φ
(N N )Φ( )
stim
©JHW 138
Integration:
l12 1 2 g x-σ (N -N )x0 0Φ(x)=Φ e =Φ e . (7.3)
12 1 2= -σ (N - N )Φ(x)
Quantity gl, (differential) gain of laser:
( ) ( )( ) 1 1 .2 2l 12 1 2 12 1
1 1
N Ng N N N N Nσ σ α= − − = − = − (7.4)
First lasing condition (holds for any laser):
Requirement for light amplification in a laser:
increase of Φ(x) with increasing x, i.e. Φ(x) > Φ0.
t b > 0 d th f
©JHW 139
gl must be > 0 and therefore:
N2 > N1 ! population inversion (7.5)
7.2.2 The resonator (second general lasing condition)
Laser: optical amplification necessary feedback by resonator required.
Resonator supplies feedback by generating standing light wave.
O ti l lifi ti !
eEC
E
h
Optical amplification necessary!
©JHW 140
Fig. 7.3: Light amplification in a laser.
hEV
h ν
© JHW
Fabry-Perot, the simplest resonator (cavity)
Fabry-Perot: two parallel mirrors with high reflection coefficient.
Standing wave between the two mirrors;
Requirement for amplification: second lasing condition for mechanical
length d:
m = integerλm = vacuum wavelengthnr = refraction index
Purpose of cavity:
r
mnmd λ=2 (7.6)
d
8 half waves
©JHW 141
Fig. 7.4: The Fabry-Perot.
Purpose of cavity:
selection of only one wavelength
for amplification;
mirrors have to be
extremely parallel.
7 half waves
mirror mirror
© JHW
Example: GaAs laser, typical length d = 200 µm, λ = 850 nm, nr = 3.5:
m ≈ 1647.
Wavelengths for different m: so-called (longitudinal) modes.
Sites with spatial distance of λ/2 within resonator:
intensity of wave goes with a frequency ν through a maximum.
High light intensity: more photons created by stimulated emission
- provided electrons available at required energy.
This local generation of photons = basic mechanism of amplification
©JHW 142
This local generation of photons = basic mechanism of amplification.
Note: Certain sites present within cavity not contributing to
stimulated emission and therefore not amplifying!
7.3 Lasing conditions for semiconductor lasers
Request for occupation inversion:separation of quasi-Fermi levels of electrons and holesby more than band gap [1] at recombination sites:
7.3.1 The first lasing condition
by more than band gap [1] at recombination sites:
Requirement of equation: at least one of the two Fermi levels
to be within a band.
Note:
.νhEEE gpF
nF ≈≥− (7.7)
©JHW 143
Thermodynamic equilibrium: both Fermi levels equal;
lasing: non-equilibrium conditions required by injection of carriers;
doping of semiconductor: high doping of both sides, both Fermi levels
within band (degenerate semiconductors).
E
V = 0
C E =EF Fn p
LEDE
E
V > 0
F
F
n
p
Figure 7.5 compares LED and laser:
n pEV
EFnLASER
EF
ECE =EF F
n p
©JHW 144
Fig. 7.5: LED and semiconductor laser. Laser: quasi-Fermi levels haveto be separated by at least the band gap value.
EFp
© JHW
n++ p++
EV
, , const. 1V C n pg( h n p ) D (E)D (E + hv)[f (E + hv) f (E)- ]dEν ≈ + (7.8)
Gain in semiconductor laser:
depends on population of conduction and valence band:
DV, DC = density of states of valence and conduction band,
fn = occupation probability of conduction band with electrons,
fp = occupation probability of valence band with holes.
A iti i > 0 i
, , V C n p
E
g( p ) ( ) ( )[f ( ) f ( ) ]
©JHW 145
A positive gain g > 0 requires
.01)()( >−++ EfhvEf pn(7.9)
For the occupation functions it holds
1,
11
,1
nF
pF
n (E-E )/kT
p (E -E)/kT
f (E)e
f (E)
=+
=
(7.10a)
(7.10b)
The inequality 1 0,n pf (E + hv)+ f (E) − > (7.11)
11 1
1 1 .1 1
F
p pF F
(E E)/kT
p (E -E)/kT (E-E )/kT
e
f (E)e e
+
− = − =+ +
(7.10c)
©JHW 146
q y
holds then only for
,n pf ( ) f ( )
.n pF F Fh E E Eν Δ< − = (7.12)
EC
EFn
Eg EF
p EV
Δ
©JHW 147
Fig. 7.6: Only electron levels in the energy regime between ΔEF and
Eg yield a positive gain g.
EFp V
© JHW
Consequence:
• light amplification only for energies hν below distance of quasi-Fermi
levels;
• light has energy hν larger than band gap Eg;
condition for amplification:
For hν = Eg and hν = ΔEF: gain g = 0;
peak between these two values.
.g FE h Eν Δ< < (7.13)
©JHW 148
7.3.2 The second lasing condition
Task of resonator: amplification of light.
Condition for resonator of length d: Φ(2d) > Φ0.
In general: dΦ= (g α)Φdx or (7 14a)g
Amplification: gain g has to - overcome absorption losses and- compensate reflection losses.
After length 2d: two reflections at mirrors with reflectivity R R
(g-α)x0
dΦ= (g -α)Φdx, or
Φ(x)=Φ e .
(7.14a)
(7.14b)
©JHW 149
After length 2d: two reflections at mirrors with reflectivity R1, R2,
light intensity after 2d:
22 .(g-α) d0 1 2Φ( d)=Φ R R e (7.15)
1 1
2 1 2
g lnd R R
α
> +
(7.16)
Solution for g with requirement Φ(2d) > Φ0:
High quality lasers require
small α large d large R1, R2
© JHW
©JHW 150
Fig. 7.7: Laser cavity with mirrors.
absorption coefficient αdifferential gain g
mirror(semi-transparent)
mirror
7.4 Laser modes
Condition for standing light wave within cavity of length d:
a) Longitudinal (axial) modes
/2 nmd λ= (7 17)
not only one, but many m and many λm fulfill Eq. (7.17);
each wavelength represents one mode.
From c = νm λm, we obtain:
,/2 rm nmd λ= (7.17)
.2 dn
cm
cm ==
λν (7.18)
©JHW 151
Frequency distance Δν = νm+1 - νm of modes:
,1
2 dn
c
r
=Δν (7.19)
2 dnrmλ
which corresponds to wavelength distance
Energy distance of longitudinal modes:
1 1.24 eVµm,
2 2lmr r
hcΔ (hν)
n d n d= = (7.20)
Requirement for good separation of modes (Δλ large):
short lasers in case of Fabry-Perot structures.
1.
2
-1 2
rn d
ν λΔλ Δνλ
∂ = = − ∂ (7.21)
©JHW 152
However: the shorter the laser, the smaller the volume for emission and,
therefore, the intensity.
Long high-intensity Fabry-Perot lasers: many modes;
solution for mode reduction: DFB and DBR laser
Example for longitudinal modes:
InGaAsP-laser, λ ≈ 780 nm, nr = 3.6, and length d = 170 µm:
number of (longitudinal) half waves of one mode within cavity:
2 340 μm 3 6dn
Reality: not only one, but about 60 modes, with m = 1540 ... 1600,
and wavelength separation of Δλ ≈ 0.5 nm;
value in accordance with Eq. (7.8)!
2 340 μm 3.61569
0.78 µmrdn
mλ
= = = (7.22)
©JHW 153
equal energetic distance after Eq. (7.23):
1 1.24 eVµm 1.24 eVµm1.01 meV
2 2 2 3.6 170 µmlmr r
hcΔ (hν)
n d n d= = = =
⋅ ⋅(7.23)
Interpretation of observation:
Emission line of LED: (mean) width Δ(hν) ≈ 2kT ≈ 50 meV at roomtemperature (see chapter 6.2, Fig. 6.5);
ithi thi i l ti f i ith Δ (h ) 1 Vwithin this energy regime: selection of energies with Δlm(hν) ≈ 1 meVby cavity of Fabry-Perot laser for emission;
here: ≈ 50 modes separated by ≈ 1 meV;
so-called super luminescent regime below threshold current density: laser emits all these lines;
at currents above threshold current density:
©JHW 154
at currents above threshold current density:many of side modes die out.
r b. u
nit s
)
2kT
i nt e
nsit y
( ar
© JHW
©JHW 155
Fig. 7.8: Longitudinal modes in InGaAsP laser of 170 µm length belowthreshold. Mode distance about 1 meV.
1.60photon energy h (eV)ν
1.65 1.70
b) Transversal modes
Finite width of laser cavity:
not only modes with different m, but also of different optical lengths
transversal modes, see Fig. 7.9. , g
Transversal modes: suppressed by narrow cavity.
©JHW 156
Fig. 7.9: Transversal modes in a waveguide.
© JHW
7.5 Radiation amplification in a semiconductor laser [5,6]
Spectrum of Fig. 7.8: still spectrum of an LED;
Lasing: - one (or a few) lines have to be amplified,
- first lasing condition (population inversion) has to be fulfilled;
- intensities of lines in Fig. 7.8 have to be multiplied by gaincurve egl(hν);
- gain g must be positive and has to exceed optical losses.
Figure 7.10: Calculated gain curve [5] for laser with GaAs-layer,
hi hl + d d ith + 1 1019 3
©JHW 157
highly p+-doped with p+ = 1x1019 cm-3,
hole Fermi level position EV - EFp = 12 meV below EV;
injection of different concentrations of electrons
into GaAs-layer by application of bias voltage.
From certain electron concentration n on:
gain gl positive for energies between Eg and ΔEF of the quasi-Fermi levels.
amplificationfor n4
Fig. 7.10:Gain curve for GaAs laser with band gap Eg = 1.424 eV.1.424
n1Eg
n2
0
gain
g
αi n3 n4
1.441 1.4511.466
©JHW 158
0 17 27 42
photon energy hν (eV)
difference hν - Eg (meV)
© JHW
Table 7.1: Values for Fig. 7.10.
Too low injection: distance ΔEF of quasi-Fermi levels for electrons and
holes smaller than band gap Eg
gain gl negative (absorption).
electr. conc.
[cm-3]
EFn - EC
[meV]
ΔEF - Eg
[meV]
n1 = 2.2 x 1017 -15 -3
n2 = 4 0 x 1017 5 17
©JHW 159
n2 4.0 x 10
n3 = 5.6 x 1017 15 27
n4 = 8.2 x 1017 30 42
Gain gl in Fig. 7.10: depends strongly on the injected carrier (electron)
concentration n.
For amplification: a) gain gl must be positive and
b) has to exceed threshold value αi in order to
compensate intrinsic losses;
intrinsic losses: e. g. absorption by free carriers,
not contained in the fundamental absorption constant α,
characterized by an additional absorption coefficient αi .
Above threshold (i.e. for g > αi):
©JHW 160
Above threshold (i.e. for g αi):
emission lines of Fig. 7.8 are amplified with gain curve from Fig. 7.10;
the higher the current (the injected carrier density n), the less laser lines
survive;
spontaneousi i
stimulatedi i
Fig. 7.11 shows light output versus current density:
emission
light
outp
ut
jth
emission
g > αi
© JHW
©JHW 161
Fig. 7.11: Above threshold current jth, gain gl exceeds threshold value αi.
current density j
Fig. 7.12 shows example [8] for emission spectrum:
b) 75 mA2 3 mW
d) 85 mA6 mW
e) 100 mA10 mW
Please note that the vertical scales of the diagrams are different!
a) 67 mA1.2 mW
c) 80 mA4 mW
2.3 mW 6 mW
©JHW 162
Fig. 7.12: Modes of AlGaAs/GaAs double heterostucture laser of length
d = 250 µm and width w =12 µm for various currents at 300 K [8].
836 832 828
Wavelength λ (nm)824 820 816836 832 828
Wavelength λ (nm)824 820 816
© JHW
7.6 Semiconductor laser configurations
Heterojunctions: two materials of different chemical nature,
e.g. SiGe/GaAs or InGaAsP/InP.
7.6.1 Heterojunctions, heterostructures
Lasers and LEDs: use band gap difference between two (or more)
materials in order to achieve
Carrier confinement: based on offsets between conduction (ΔEC)
carrier confinement
optical confinement
©JHW 163
and/or valence bands (ΔEV).
Light confinement: based on Moss’ law:
const.4r gn E ≅ (7.24)
To distinguish:
isotype heterojunctions (same type of doping, Nn-, Pp-
Total reflection of light from material with smaller band gap Eg.
Quantum wells (Fig. 7.13): ideal for confinement.
Fig. 7.13:Quantum well for optical
yp j ( yp p g, , pstructure)
anisotype heterojunctions (different type of doping, Np- or Pn-structure)
©JHW 164
pand carrier confinement.
© JHW
Offsets ΔEC , ΔEV at conduction and valence band between two
materials:
derived from same principles as for homojunctions.
7 6 2 Homojunction (anisotype)7.6.2 Homojunction (anisotype)
Two equal principles / assumptions of contact formation for
homojunctions and heterojunctions (leading to Anderson’s rule):
Homojunction (see Fig. 7.14):
Fermi level EF across junction is constant (flat)
©JHW 165
Vacuum level Evac across junction is continuous
(no change of electron affinities χ, no offset at conduction band edges,
whereas work functions Φw change).
n-type p-type
qχ1 qχ2qΦ(1)
qΦ(2)
Evac
interface
q
q
χ
χ
1
2
EF
EV
EF
EC
a) before contact
2qΦw qΦw
© JHW
b) after contact
1
EF
©JHW 166
Fig. 7.14: Homojunction formation:
a) before contact
b) after contact: Fermi level EF flat, vacuum level Evac
continuous across interface.
7.6.3 Heterojunction (anisotype)
N-type p-typeEvac
interface
© JHW
q χ2
EF
EE
EV
EC
gg(2)(1)
EV
EF
EC
q χ1q χ2
qΦw(1)
qΦw(2)
E g
(2)
(1)
EF
EVEV
EC
Δ
Δ EC
χχ
12
Eg
©JHW 167
Fig. 7.15: Heterojunction formation according to Anderson’s rule forconduction band offset ΔEC.
a) before contact b) after contact
Requirement: continuous vacuum level Evac across interface.
Discontinuities ΔEC , ΔEV in conduction and valence band edge;
Anderson rule for the discontinuities ΔEC and ΔEV:
see Fig. 7.15!
Very large discontinuities potential well at interface, see Fig. 7.16.
C 2 1
(1) (2)V g g C g C
ΔE = q(χ - χ )
ΔE = (E - E )- ΔE = ΔE - ΔE (7.25b)
(7.25a)
©JHW 168
Electrons in potential well:
• only discrete energies (subbands),
• localized perpendicular to interface with respect to movements.
Formation of two-dimensional electron gas (2DEG).
2DEG
© JHW
EC
C
FE
subbandsEΔ
©JHW 169
Fig. 7.16: Two-dimensional electron gas (2DEG) at interface of ahetero-structure.
7.6.4 Band engineering with heterostructures of type I, II, III
a) The relative position of bands for semiconductors
Heterostructures allow manipulation and engineering of electron, hole
and light behavior within semiconductors.
However, band adjustment not always according to Anderson’s rule
(adjustment of vacuum levels):
• other reference levels needed for understanding: charge neutrality
levels, dielectric mid-gap levels, energy of mean dangling bond;
©JHW 170
• these levels: derived from three-dimensional band structure of
individual semiconductors.
• Anderson rule just a crude prediction.
• Further – rather complicated – theories:
- charge neutrality level (Tersoff),
- dielectric mid-gap energy (Cardona and Christensen),
- dangling bonds (Lanoo).
• Finally: particular energies for particular semiconductors;
to be matched upon formation of contacts.
b) Carrier confinement in heterostructure types [2]
Figures 7.17 and 7.18: three types of heterostructures;
©JHW 171
• type I: Double layer confines electrons and holes;
• type II: Only hole confinement;
• type III: Holes from one material in direct contact with electrons from
second semiconductor.
E
type straddling
I type staggered
II type misaligned
III
EC
E
EC
EV
EE
EC
C
V
EV
©JHW 172
Fig. 7.17: The three types of heterostructures.
EV
© JHW
In Ga AsInP In Al As 0.530.52In Ga As0.53 0.47 0.470.48
0.25 0.470.47
0.75 1.35 1.44 0.75
0.34 0.16 0.22E (x)V
E (x)CE (eV)=g
E (eV)=CΔ
E (eV)=VΔ
0.26
0.16V
InAs InAsGaSb AlSb
1.35
0.73 1.58
0.500.88E (eV)=CΔ
©JHW 173
Fig. 7.18: Lattice-matched compositions for InGaAs/InAlAs/InP show type I.System InAs/GaSb/AlSb shows all three types.
0.360.36
-0.51 0.35 -0.13© JHW
E (x)C
E (x)VE (eV)=VΔ
E (eV)=g
c) Optical confinement in heterostructures
b) c)
E (x)C
a)
E (x)C
19 f f f f
SCH GRINSCH
n (x)r
E (x)V
DH
© JHW
E (x)V
n (x)r
©JHW 174
Fig. 7.19: Profile for bands and refraction index for
a) double heterostructure (DH),
b) separately confined heterostructure (SCH), and
c) graded index separately confined heterostructure (GRINSCH).
Modern semiconductor devices:
very small structures quantum wells, quantum boxes (dots)
quantified energy levels; see Fig. 7.16;
size of quantum boxes ≈ 10 nm;
type I hetero-interfaces (see Fig. 7.19a):
- quantum box for carrier confinement,
- high carrier density for population inversion,
- confinement also of light by step-like behavior of
refraction index nr (Moss’ law: nr4Eg = const.)
However: Confinement of light not effective for structures with
©JHW 175
However: Confinement of light not effective for structures with
thicknesses below wavelength of light:
the smaller the structures,
the better the confinement of carriers,
but the higher the losses of light.
Solution of contradicting demands, see Fig. 7.19b:
Light-confining structures need thicknesses not smaller than wavelength of the light,i.e. not smaller than about 1 µm (see also chapter 8).
Different structures for light- and carrier confinement:• inner potential well confines carriers,• outer well - profile of refraction index nr - confines light.
Fig. 7.19b: step function profile of refraction index nr
Fig. 7.19c: graded profile of refraction index nr.
Advantage of graded index profile:
©JHW 176
Advantage of graded index profile:• smaller number of (transversal) optical modes (see chapter 8).
Reality: center zone of structure in Fig. 19b:sometimes not only single layer, but multi quantum well.
7.7 Light guiding in semiconductor lasers
7.7.1 Principle of light guiding
For increasing ratio of stimulated to spontaneous emission (chapter 7.3):
confinement of light to a narrow region in a laser;
for steering emission of photons into a certain direction:
guiding of light within laser;
two methods: Gain guiding and index guiding;
©JHW 177
g g g g;
both methods based on total reflection at interface between
material with higher and material with lower refraction index.
hν
n
n2 < n1
n1
© JHW
Fig. 7.20: Top (!) view of semiconductor laser; principle of light guiding;
relation n2 < n1 must hold for n2 of cladding layer.
Aim of laser design:
mirror
n2
©JHW 178
Aim of laser design:
to achieve zone of low n2 (cladding layer) with simple methods.
In principle:center zone as narrow as possible to avoid (transversal) optical modes.
7.7.2 Gain guiding (active guiding)
However: if inner zone too narrow:
light enters into cladding layer (see chapter 8, structure parameter).
minimum width of the center zone layer: several µm.
Effect occurs in all semiconductor lasers:
refraction index nr increases with increasing carrier concentration(due to higher current) according to [3,4]
( , ) .8
2 2
r r 2e h0 0 r
q n pn n p
m mc
λεε ε
= + + π
(7.26)
©JHW 179
εr and ε0 = relative and absolute dielectric constant.
Gain guiding bases on active wave guiding:
higher refraction index only during operation of laser(increased n, p within space charge region).
7.7.3 Index guiding (passive guiding)
Effect based on step of refraction index built into structure;
typical examples: hetero-structures; e.g. Fig. 7.19.
Index guiding + gain guiding:
lower threshold current density jth (chapter 7.10).
Figures 7.21a,b: two hetero-structure lasers with gain and index
guiding.
oxide
hν
metal
p-GaAs
p-AlGaAsp-GaAs
n-AlGaAs
oxide
metal
4 µm
n-InP
p-InP p-InP
p -InGaAsP
p-InGaAsP
©JHW 180
Fig. 7.21: a) gain guided AlGaAs/GaAs-laser,
b) index guided InGaAsP laser with InP cladding layers.
n-GaAsa)
© JHW
hν
b)n -InP
n -InP-substrate
7.8.1 The stripe contact laser
a) Carrier confinement
7.8 Modern semiconductor lasers
a) Carrier confinement
Stripe contact laser (Figure 7.22): simplest semiconductor laser;
• contact stripe confines carriers to narrow zone of pn-junction;
• Fabry-Perot by splitting wafer along preferential crystallografic
directions, usually along [110];
©JHW 181
directions, usually along [110];
• (111)-split planes then extremely flat. Laser emits light at edges
(“edge emitting” laser)!
Optical confinement: based on gain guiding;
center zone: as narrow as possible to suppress (transversal)
optical modes.
b) Optical confinement
5 µm
type~100 µm
p-type active zone(~0.2 µm)
5 µm
stripecontact
©JHW 182
Fig. 7.22: Side view and cross section of stripe contact laser.
© JHW
n-type
~500 µmcurrent lines
contact
Improvement of current (carrier) confinement in stripe contact lasers:
• use of a V-groove contact,
• hydrogen implantation to form semi-insulating GaAs, or
• by oxide layer; index guiding by oxide with nr ≈ 1.5.
V-groove contact
a) b) c)
oxideH -implant
SiO2
©JHW 183
Fig. 7.23: Confinement of current lines in GaAs stripe contact laser bya) V-groove contact, b) hydrogen implantation, c) oxide layer.
) ) )
active zone
© JHW
7.8.2 The double heterostructure laser
Operation principle: Inversion in center layer by carrier injection from
N-type and P-type layers with the high band gap.
N-type
Evac
EC
EC
Evac
EFEF
EC ΔEC
qχ2qχ1
Eg(2)Eg
(1)
EF
p-type P-type
©JHW 184
Fig. 7.24: Double heterostructure before putting materials together(for example AlGaAs, GaAs, AlGaAs).
EF
EVΔEV
EV EV© JHW
ΔEC
p-typeN-type P-typeEC
EF
Fig. 7.25:Double heterostructure laser consisting of an AlGaAs/GaAs/AlGaAs stack,a) without and b) ith bi
ΔEVa) V = 0EV
EF
electrons
h
©JHW 185
b) with bias.
b) V > 0
holes
EFnEF
p hν
© JHW
oxide
metal
4 µm
n InPp -InGaAsP
hν
b)
n-InP
n -InP
n -InP-substrate
p-InP p-InP p-InGaAsP
Fig. 7.26:InGaAsP double heterostructure laser improves carrier confinement
©JHW 186
G presulting in reduced threshold current density
7.8.3 Quantum well laser
contact layer
confinement layer
ergy
EC
substrate
contact layer
confinement layer
confinement layer
confinement layer
quantum wells
Ene
EV
©JHW 187
Fig. 7.27: Quantum well structure improves carrier and opticalconfinement and reduces threshold current density.
©JHW 188
Fig. 7.28: AlGaAs quantum well laser with two quantum wells andgraded index light guiding structure for separated optical confinement.
7.8.4 Quantum dot laser
©JHW 189
Fig. 7.29: Self-organized growth of InGaAs quantum dots on a GaAssubstrate
homo junction
quantum wells104
105
ens
ity [A
/cm
2 ]
hetero junction quantum dots
101
102
103
1960 2000199019801970 2010
thre
sho
ld c
urr
en
t de
Fig. 7.30: InAs quantumdot laser
©JHW 190
Fig. 7.31: Reduction of threshold energydensity in modern semiconductor lasers
year
7.8.5 Distributed feedback (DFB) laser
Energy distance of (longitudinal) modes in Fabry-Perot cavities; see chapter 7.4, Eq. (7.17) :
a) Mode distance and energy distribution
Figure 7.32: compares mode distance Δlm(hν) and energy widths ΔEn ,ΔEp of carrier distributions;
1 1.24 eVµm,
2 2lmr r
hcΔ (hν)
n d n d= = (7.27)
©JHW 191
p
short cavities: mode distance larger than energy distribution ofelectrons and holes.
electrons can only recombine at a single energydifference.
Ecavity modes carrier distribution
Δlm(hν) ΔEn ne = DC(E)fn(E)EC
short long
ΔEp
EV
ΔEn ≈ ΔEp ≈ 1 kT
pe = DV(E)fp(E)
C
© JHW
©JHW 192
Fig. 7.32: Short cavity: Energy distances of optical modes larger thanenergy widths of electrons and holes; only single modes excited.
short longd
Conditions for single-mode Fabry-Perot laser:
Distributions of electrons and holes about kT ≈ 26 meV wide;
for refraction index nr = 3.6: Δlm(hν) > kT d must be < 6 µm!
Such small Fabry-Perot lasers: too low an intensity.y y
Instead of short Fabry-Perot lasers: Bragg reflectors.
b) Bragg reflector
Λn
©JHW 193
Fig. 7.33:Bragg reflector.
n2
n1
difference 2 Λ© JHW
Principle of Bragg reflector (Figure 7.33):
Waveguide with periodic thickness variation of layer with n1;each hump scatters the light interference effects;
wavelengths with multiple of hump distances constructive inter-ference of individually back-scattered waves.
arrangement of “humps” acts as mirror of high reflectivity, even ifscattering intensity of individual hump is small.
Condition for standing waves with well defined wavelengths λ within structure of Fig. 7.33:
λΛ2 (7 28)
©JHW 194
Similiarity to second laser condition for Fabry-Perot cavity (see chapter7.4)! “Hump” distance Λ replaces cavity length d!
rnm λ=Λ2 (7.28)
Consequence: energy distance of longitudinal modes (standing waves)
in laser with Bragg reflector:
1 1.24 eVµm.
2 2lm
hcΔ (hν) = =
Λ Λ(7.29)
Small Λ: energy distance of modes much larger than kT.
Calculations show: m > 2 strong losses due to radiation;
mode with m = 1 used; needs very small Λ ≈ 150 to 250 nm.
2 2lmr r
( )n nΛ Λ
©JHW 195
mode with m 1 used; needs very small Λ 150 to 250 nm.
For laser with refraction index nr = 3.2, λ = 1.3 µm (1.5 µm):
hump distance must be Λ = 200 nm (234 nm).
c) DBR- and DFB-laser
Fig. 7.28a: Bragg reflector not in electrically active layer, butdistributed among end pieces of crystal.
DBR = Distributed Bragg ReflectorDBR = Distributed Bragg Reflector
In contrast, Fig. 7.34b: Bragg reflector along hole active zone,feedback distributed equally.
DFB = Distributed Feedback Bragg (reflector)
©JHW 196
Figure 7.35, real DFB-laser .
I I
a) Distributed Bragg Reflector b) Distributed Feedback Bragg (reflector)
hν
ARC ARC ARC
DBRDFB
I I
hνactive active
© JHW
©JHW 197
Fig. 7.34: a) DBR- and b) DFB-laser, after Ref. [3]. End faces containanti-reflecting coating (ARC) to avoid Fabry-Perot modes.
ARC
n-InP
p-InP
p-InPn-InP
ARC
hν
p InPp-InGaAsPInGaAsP (active layer)n-InGaAsPDFB
Λ = 200 nm (for λ = 1.3 µm)Λ = 235 nm (for λ = 1.55 µm)
© JHW
©JHW 198
Fig. 7.35: Hitachi HL1541 BF/DL laser with buried heterostructure anddistributed feedback reflector for fiber optic communications;
optical output power 1 mW at laser line, side mode suppression 35 dB.
7.8.6 Vertical surface emitting laser (VCSEL)
Lasers discussed so far: edge emitting lasers.
Str ct re of Fig 7 36
light output
Si/SiO DBR2
Structure of Fig. 7.36: contains vertical stack of Bragg reflectors; vertically standing light
wave; light emission at surface; appropriate for laser arrays
-InP(substrate)
-InP(cladding)
-GaInAsP(active)
-InP
n
n
p
p
©JHW 199
and optical interconnects ofcomputer chips.
Fig. 7.36: Vertically emitting laser.
(cladding)
SiO
active region contact Si/SiO DBR
2
2
p
7.9 Internet Links
1. Population Inversion (Applet): http://stwww.weizmann.ac.il/lasers/laserweb/Ch-2/F2s6p1.htm
2. Creating (Applet): http://stwww.weizmann.ac.il/lasers/laserweb/Ch-3/F3s5p1 htm3/F3s5p1.htm
3. Principle of a Laser: http://www.phys.ksu.edu/perg/vqm/laserweb/Java/Javapm/java/Laser/index.html
4. Laser Cavity: http://webphysics.davidson.edu/Applets/optics4/laser.html
5 Uses of Lasers and comparison (Applet):
©JHW 200
5. Uses of Lasers and comparison (Applet): http://www.colorado.edu/physics/2000/index.pl
6. Quantum Well Calculatorhttp://www.ee.buffalo.edu/faculty/cartwright/java_applets/quantum/numerov/index.html
7.10 Literature
1. H. G. Wagemann and H. Schmidt, Grundlagen der optoelektronischen Halbleiter-bauelemente (Teubner, Stuttgart, 1998), pages 202-205.
2 J H Davies The Physics of Low-Dimensional Semiconductors2. J. H. Davies, The Physics of Low Dimensional Semiconductors(Cambridge University Press, Cambridge, UK, 1998), p. 85.
3. W. Bludau, Halbleiteroptoelektronik: Die physikalischen Grundlagen der LEDs, Diodenlaser und pn-Photodioden (Carl Hanser, München, 1995), p. 160.
4. see Ref. [3].5. pages 125-132 in Ref. [3].6. W. L. Leigh, Devices for Optoelectronics (Dekker, New York, 1996),
©JHW 201
p. 907. p. 105 in Ref. [6].8. H. Kressel, in Handbook on Semiconductors, Vol. 4 (T.S. Moss and
C. Hilsum editors, North Holland, Amsterdam, 1981), p. 636.
8. Glass Fibers
©JHW 202
Glass fibers: dielectric waveguides;
- core with refraction index n1 and cladding with refraction index n2;
8.1 Configuration and optical properties
8.1.1 Advantages of glass fibers
core with refraction index n1 and cladding with refraction index n2;
- total reflection keeps light in core;
- today: power losses below 0.16 dB/km, bandwidths around1 GHz km (i.e. 1 ns dispersion/km);
- high capacity (1 Gbit/s), cheap, potential free, and light.
cladding
©JHW 203
Fig. 8.1: Glass fiber for optical data communication.
n1
n1n2 <g
coreb a
© JHW
Three major fiberoptic configurations [1]:
8.1.2 Fiber configurations
Fig. 8.2:g 8Configurations and index profiles. a) multi-mode step-
index fiber, b) multi-mode
graded-index fiber,c) single-mode step-
©JHW 204
c) single-mode step-index fiber.
© JHW
Light carried by total internal reflection due to small discontinuous fractional step:
8.2 Step-index fibers
1 2n
1
n n
nΔ −= (8.1)
Fig. 8.3: Step-index fiber.
© JHW
©JHW 205
Refraction index n1: in the range n1 = 1.44 to 1.46,varied by doping SiO2 with Ge, Ti, B;
value for Δn: between 0.1 and 2 %;
typical diameter of cladding: 2b = 125 µm; of core: 2a = 8 - 100 µm.
8.2.1 Ray guiding
Core: guides rays with angle Θ > Θc.
Critical angle Θ for total reflection (chapter 1 3):
© JHW
Fig. 8.4: Ray guiding in a step-index fiber.
©JHW 206
Critical angle Θc for total reflection (chapter 1.3):
11 .1 n2
cc n1 1
n ( )nsin cos
n n
ΔΘ Δ Θ−= = = − = (8.2)
= complementary critical angle. cΘ
For Δn = 1%: Θc = 81.9° and 090c cΘ Θ= − = 8.1 °;
90 .cΘ Θ Θ= ° − <Total reflection requires
Ray guiding only within cone of acceptance (half) angle Θa (Fig. 8.4).
With na = refraction index outside fiber:
numerical aperture NA:
8.2.2 Acceptance cone, numerical aperture NA
©JHW 207
p
.a aNA n sinΘ= (8.3)
Maximum acceptance angle Θa = f (Θc):
from Snell’s law (chapter 1.3) it follows:
1 2NA Θ Θ Θ1
1
2 .
2c ca a 1 1
2
2 221 1 2
1
1 n
NA n sin n sin n cos
nn n - n
n
n
Θ Θ Θ
Δ
= = = −
= − =
≈
(8.4a)
(8.4c)
(8.4b)
©JHW 208
Example: na = 1, n1 = 1.46, Δn = 1 % ( 8.1 ) :cΘ = °
Θa = 11.9° and NA = 0.206 (typical value for glass fibers).
Small Θa and small NA: consequences of small Δn.
Larger Δn easier light coupling.
Example: un-cladded fiber with n1 = 1.46, n2 = 1:
Θa = 90° and NA = 146.8 ,cΘ = °
rays guided from all directions,
however: large NA larger number of modes.
Θ
Θasmall NA
©JHW 209
Fig. 8.5: Fibers with different numerical apertures NA.Angles are exaggerated.
Θalarge NA© JHW
Geometrical optics: all angles Θc < Θ allow ray communication.
Wave optics: only certain angles Θc < Θ < 90° allowed.
Each angle defines a mode.
Condition for allowed (transversal !) modes:
8.2.3 Modes in a step-index fiber
Condition for allowed (transversal !) modes:
Θ1
claddingA
phase fronts
n1
n2
λ
©JHW 210
Fig. 8.6: Wave optic condition for modes. Reflected and original wavehave to interfere constructively.
12
B cladding
core
n1
n2© JHW
Condition for constructive interfere:
Original waveWave reflected at points A and B
Condition fulfilled, if distance between equivalent points 1 and 2 is
must have same phase!
a multiple of λ !
“Equivalent” points 1 and 2: distance 1⇔A = distance B⇔2
For fixed λ: condition only fulfilled for certain angles Θ;each angle defines 1 transversal mode.
The thicker core, the more modes!
©JHW 211
Calculation of modes: wave optics, Maxwell’s equations [1,2];
solutions (radial energy distribution): Figure 8.7, so-called LPlk –modes,
k = number of rings, l = half of number of spots.
Number of modes: “thousands!”;number = f (fiber thickness, launching angle)
Individual modes:different optical lengths, arrival at end of fiber at different times.
8.2.4 Intermodal dispersion
p g ,
smear-out of light pulse, (inter-) modal dispersion.
©JHW 212
Fig. 8.7: Modes projected onto end face of step-index fiber;LP01 = axial, basic mode; LP21, LP83 = higher order modes [2].
LP01 LP21 LP83
© JHW
cladding
axial
time delay
mode
time t time t
sum
cladding
core
time t
op
t.p
owe
r
© JHW
©JHW 213
Fig. 8.8: Different modes in step-index fiber;inter-modal dispersion; smear-out of pulses.
Mdis: characterizes time delay between fastest (axial) and slowest rayper unit length of fiber.
Fastest ray: along fiber axis of length Lf , arrival after minimum time tmin:
Mdis8.2.5 Mode dispersion coefficient
y g g f min
Slowest ray: largest length Lslow , reflected with critical angle Θc:
./f
mino 1
Lt
c n= (8.5)
©JHW 214
y g g slow , g c
from chapter 8.3.1: sin Θc = n2 /n1;
from Fig. 8.9: Lslow/Lf = l/lf = 1/sin Θc .
Therefore:
1slow f
2
nL L
n= (8.6)
LL (8 7)ffast LL = (8.7)
©JHW 215
Fig. 8.9: Length Lslow of slowest ray: Lslow = Lf /sin Θc .
© JHW
Propagation time of the slow ray:
1 2ff 12
0 0 2
1
nLL L nnslowt = = = ,max c c n
nv
(8.8)
time difference between slowest and fastest ray:
Mode dispersion coefficient of the step-index fiber:
1 .f 1 1 n1mode max min f
0 2 0
L n nnt t t L
c n c
ΔΔ
= − = − ≈
(8.9)
mode 1t nM
Δ Δ (8 10)
©JHW 216
Example: n1 = 1.46, Δn = 1 %, c0 = 3 x 108 m/s: Mdis = 49 ns/km.Spreading out of pulse after 1 km: to width of 49 ns,
≙ 10 m in space!
.mode 1dis n
f 0
ML c
Δ= ≈ (8.10)
Fig 8 10:
49ns
input signal
time t
(10m)
Fig. 8.10: Intermodal dispersionspreads input pulses;
spatial delineation
Pulses need minimum distancedepending on mode
(49ns)1.0 km
10m© JHW
©JHW 217
depending on modedispersion coefficient Mdis.
49ns
output signal
time t
(10m)
Example in Fig. 8.10: width of light pulse = 10 m: required separation of 2 subsequent light pulses:
≈ 20 m before the fiber, corresponding to Δt = 98 ns.
bandwidth Bfiber = 1 /(98 ns) = 10 MHz.
Quantity Bfib :Quantity Bfiber:
Bandwidth-distance product:
1
2fiberBtΔ
≈ (8.11)
1
2fiber fdis
B LM
= (8.12)
©JHW 218
Bfiber = 10 MHz and BfiberLf = 10 MHz·km: small value. Step-index fibers not useful for data communication
over long distances.
Figure 8.11 demonstrates dispersion of a series of pulses.
© JHW
©JHW 219
Fig. 8.11: Dispersion destroys clear separation between pulses.Closely spaced pulses degrade more quickly.
Intermodal dispersion:due to optical lengths difference ΔLopt = nrΔL for rays(with ΔL = geometric length difference).
Solution of problem: “acceleration” of long rays by smaller refraction
8.3 Graded-index fibers
p g y yindex (higher light velocity) smaller optical length Lopt = nr L;see Fig. 8.12.
©JHW 220
Fig. 8.12: Graded-index fiber.
© JHW
Refraction index nr (power law):
Very large g ≫ 10: step index fiber
( )( ) 1 .g
r 1 nrn r n aΔ = −
(8.13)
Very large g ≫ 10: step-index fiber.
According to theory [3,4]: minimum of intermodal dispersion
for optimum goptimum = 2 - 2Δn. For Δn ≈ 1% goptimum ≈ 2;
Since nr = f (λ) (usually called dispersion):
compensation of intermodal dispersion by tailored refraction index
the profile is parabolic.
©JHW 221
compensation of intermodal dispersion by tailored refraction index
profile only possible for a single wavelength λ !
Note: Term “dispersion” has two meanings:
1. In the whole field of optics: Wavelength dependence nr(λ),resulting in a dependence of light velocity on refraction index;
Table 8.1: Comparison of multi-mode step-index and graded-indexfibers;
2. In the field of fiber optics: The spread out of pulses, a consequenceof modes (different path lengths).
fibers;graded-index fiber: smaller dispersion coefficient,bandwidth Bfiber ≈ 100x higher than for step-index fiber;BfiberLfiber ≈ 1 GHz km.
fiber change of nr light guiding dispersion coefficient
©JHW 222
Mdis [ns/km]
step-index step-like total reflection 25 - 50
graded-index continuous diffraction 0.2 - 0.5
Table 8.2: Typical values for a graded-index fiber (after Ref.[5]).
parameter value
core radius a cladding radius b
50 µm 125 µm
numerical aperture NA acceptance half angle Θa
0.2 ± 0.02 11.5°
mode dispersion coefficient Mdis (for 1300 nm)
0.5 ns/km
bandwidth-length product Bfiber Lf = 1/ (2Mdis)
1 GHz km
©JHW 223
(for 1300 nm) attenuation 850 nm
1300 nm 1550 nm
2.5 dB/km 0.5 dB/km 0.4 dB/km
Dependence nr = nr(λ):
tailored refraction index yields maximum bandwidth(minimum intermodal dispersion) for well-defined λ.
Fiber in Fig 8 13: maximum tailored to appear at λ ≈ 1300 nm Fiber in Fig. 8.13: maximum tailored to appear at λ ≈ 1300 nm(λ-regime of minimum attenuation; see table 8.2).
Advantage of graded-index fibers: relatively high aperture.
simple plugs for cable interconnection.
However: low bandwidth length product (≈ 1 GHz·km)
©JHW 224
However: low bandwidth-length product (≈ 1 GHz·km)
restriction to short distances (local area networks, LAN).
1.0z km
) © JHW
1.2 1.6 2.0
0.5
l th ( )λ0.8
B
L
(G
Hz
fiber
f
©JHW 225
Fig. 8.13: Compensation of intermodal dispersion in a graded-indexfiber.
wavelength (µm)λ
8.4.1 Structural parameter (V-parameter)
8.4 Mono-mode fibers
Finite radius a of core: modes in step-index fibers.
For 2a < 6 8 µm: only one mode (ground mode)
ba core
2a < 6...8µm0
-a
c
For 2a < 6...8 µm: only one mode (ground mode).
In general: number of modes = f (structural parameter V ).
©JHW 226
Fig. 8.14: Structure of mono-mode fiber; ac = critical core radius.
cladding-a-b
n2 n1
© JHW
Definition of structure parameter [6]:
V = f (core radius a, wavelength λ, and profile of refraction index nr(r)),
NAakNAaV λλπ == 2
(8.14)
f ( g p r( ))
where nr(r) defines numerical aperture NA.
Requirement for mono-mode operation (from wave-optics):
single mode operation by
V < Vc = 2.405 (8.15)
©JHW 227
reducing core radius a,
increasing wavelength λ , reducing aperture NA.
single-mode operation by
8.4.2 Number of modes for V > Vc
Number Nmod of modes for V ≫ Vc:
.2 2
2
mod
V gN
g=
+(8.16)
g = power of index profile (see chapter 8.4);
2 2g +
2
2
mod
VN = for step-index fiber with ,∞→g and (8.16a)
2VN for graded index fiber with g = 2 (8 16b)
©JHW 228
Equal V-parameters:
graded-index fibers: only half the number of modes as step-index fibers!
4modN = for graded-index fiber with g = 2. (8.16b)
Example 1:
Fiber with n1 = 1.447, Δn = 1%: NA = n1 (2Δn)1/2 = 0.205.
With a = 25 µm and λ = 850 nm: V = 37.8;
Nmod = 714 (step-index fiber), Nmod = 357 (graded-index fiber).
Example 2:
Requirement of mono-mode operation: V < Vc = 2.405,
reduction of V by factor 37.8/2.405 ≈ 16.
fiber of example 1 = mono-mode for λ = 16 x 850 nm = 13.6 µm.
Alternative: - reduction of NA (smaller step Δn) and/or
©JHW 229
- reduction of a.
However: - penetration of wave into cladding layer and
- waveguide dispersion (see chapter 8.7)!
Reduction of a limited to a ≈ λ !
Example 3:
Fiber of example 1: n1 = 1.447, Δn = 1 %, NA = 0.205:
operates as mono-mode fiber at λ = 1.3 µm for a < 2.43 µm.
Reduction of Δn to Δn = 0.25 %: single-mode operation (V < 2.405)
for larger core radius a < 4.86 µm.for larger core radius a 4.86 µm.
8.4.3 Cut-off wavelength
Condition for mono-mode fiber: V < Vc.Fixed geometry: mono-mode behavior for λ > cut-off wavelength λc
(λc = 13.6 µm in example 2).
8 4 4 Cladding penetration
©JHW 230
8.4.4 Cladding penetration Geometrical optics: light transport within core by total reflection.However, wave optics: total reflection,
but part of optical power penetrates into cladding!Example in Fig. 8.15.
b
radius r
a
cladding
coreoptical power Popt
-a
-b© JHW
©JHW 231
Fig. 8.15: Distribution of optical power for modes across fiber diameter.Wave-optics: part of power penetrates into the cladding,even when condition for total reflection is fulfilled. Waveguide dispersion.
Reduction of core radii a:
Red. of structural param. V:
These effects restrict the structural parameter to the following values:
increase of cladding penetration.
Note: Penetration of electric field into cladding:
waveguide dispersion (nr(cladding) < nr(core)).
Mono-mode fiber: not all light transported in core,
1.5 < V < Vc = 2.405. (8.17)
©JHW 232
Mono mode fiber: not all light transported in core,
dispersion of light pulse.
Remaining dispersion even for a mono-mode fiber (see chapter 8.5.3);
Table 8.3: typical values for mono-mode fiber.
© JHW
©JHW 233
Fig. 8.16: Electric field E of ground mode;Gaussian shape of width 2ω0;
small a (and V): Field penetration into cladding.
© JHW
©JHW 234
Fig. 8.17: Radial dependence of electric field E of ground mode fordifferent structural parameters V;V < 1.5: large power losses into cladding (see also Fig. 8.18).
1.0
0.8
0.6 /P
ing
core
1211
© JHW
0.4
0.2
0.00 2
structure parameter V
ratio
P
cla
dd
2.405
4 6 8 10 12
01
0302
©JHW 235
Fig. 8.18: Penetration of optical power into cladding [7].
Table 8.3: Typical values for a mono-mode fiber.
parameter value
core diameter 2a
field diameter 2ω 7 µm
9 ± 1 µmfield diameter 2ω0
cladding radius b
9 ± 1 µm
125 µm
cut off wavelength λc 1100 to 1300 nm
chromatic dispersion coefficient Mchr
for 1300 nm
for 1550 nm
3.5 ps/(nm km)
20 ps/(nm km)
©JHW 236
for 1550 nm 20 ps/(nm km)
attenuation
1300 nm
1550 nm
0.4 dB/km
0.25 dB/km
8.5 Dispersion in glass fibers
Three “dispersion” effects of light pulses in glass fibers:
chromatic dispersion
intermodal dispersion (only in multimode fibers) material dispersion
chromatic dispersion waveguide dispersion
8.5.1 Intermodal dispersion and mode mixing
Effects: from different ray lengths; see chapters 8.2.5, 8.3;
implicit assumption: linear increase of Δtmod between
f t t d l t ith l th L f fib
©JHW 237
fastest and slowest ray with length Lf of fiber
(compare chapter 8.2.5):
.1 nmode max min f
0
nt t t L
c
ΔΔ = − = (8.18)
Linear dependence: up to critical fiber length Lc.
L > Lc: intermixing of modes mode coupling.
Reasons: - small imperfections in the fiber (random irregularities at
fiber surface, inhomogeneities of nr in the bulk) or
i f ti t ti b t t fib d- imperfections at connections between two fiber ends.
For L > Lc (Lc = “Kopplungslänge”): smear-out of pulse less than
expected from linear dependence on fiber length Lf.
Instead: Δtmod varies with power law:
( )t = t - t const LΔγ
= (8 19)
©JHW 238
with 0.5 < γ < 1. Very often: γ ≈ 0.7...0.8.
Critical length Lc: between some hundred meters and some kilometers,
depending on perfection of fiber.
( ). ,mod max min ft = t - t const LΔ = (8.19)
8.5.2 Material dispersion
Occurrence: if light source is not perfectly monochromatic.
Figure 8.19a: refraction index nr for fused silica.
In general field of optics: material dispersion = dispersionIn general field of optics: material dispersion dispersion.
Since nr = nr(λ): smear-out of pulse containing different λ.
Facts:
• modulated laser beam or pulse distribution of wavelengths;
• purely monochromatic: only infinitely long wavetrain.
©JHW 239
p y y y g
Pulse (or modulation) of finite length Δtpulse:
frequency spectrum of finite width Δν = 1/Δtpulse and
wavelength spectrum of finite width Δλ (see also chapter 3).
Fig. 8.19:
a) refractive index and group index forfused silica. At λ0 = 1.312 µm 0 µ
refractive index nr
has point of inflection,
group index ngroup
is minimum.
b) material dispersion coefficient Mmat
© JHW
©JHW 240
mat
vanishes at λ0 .
Light transport across fiber: by pulses or wave-trains.
Monochromatic waves, centered around wave with frequency ϖ
( wave-train or pulse):
a) The group refraction index ngroup
traveling with phase velocity c(λ) determined by nr(λ).
refractive index =
Maximum of pulse: different speed: group velocity cgroup;
different group index n (λ)
(8.20)0
0 )(
)(
1
)()(
c
n
cc
cn r
r
λλλ
λ ==
©JHW 241
different group index ngroup(λ).
group index = (8.21)
0
0)(
)(
1
)()(
c
n
cc
cn group
groupgroupgroup
λλλ
λ ==
(8.22a)
For each of the monochromatic waves:phase velocity, i.e. their speed:
ωπνλπνλ k
c ===2
/2)/(1/1
In contrast, group velocity [8]:
(8.22b)ωd
dkcgroup =/1
©JHW 242
From chain rule and k = w/c:
ω
ω
ωω
ω d
dc
cc
cc
c
d
dc
cc
d
dkcgroup
0
200
2/1/1
−=−== (8.22c)
with:
λπω 02 c= λ
λπω d
cd
202−=
and:and:
c
cnr
0= rdnc
cdc
0
2
−=
8.22c results in:
0 dncdk rλ−
©JHW 243
.0
0
cdc
d
dk rλλ
ω−
= (8.22d)
From 8.20, 8.21 and 8.22d:
.λ
λd
dnnn r
rgroup −= (8.23)
for small λ decreases ith λ (normal dispersion)
Traveling time of pulse maximum with group velocity c along fiber
b) The material dispersion coefficient Mmat [9]
for small λ, ngroup decreases with λ (normal dispersion),
for larger λ, ngroup increases with λ (anomalous dispersion).
(as shown in Fig. 8.19)
©JHW 244
Traveling time of pulse maximum with group velocity cgroup along fiber
with length Lfiber:
fiber group fiber fiberrgroup r
group 0 0
L n L Ldnt = = = n - .
c c d cλ
λ
(8.24)
Time spread Δtmat for waves making up the pulse:
.groupmat
dtt
dΔ Δλ
λ= (8.25)
The last two equations yield
Material dispersion coefficient Mmat: obtained from normalizing with
respect to fiber length and spectral width of light source:
.2
fiberrmat mat fiber2
0
Ld nt M L
d cΔ λ Δλ Δλ
λ= − = (8.26)
©JHW 245
respect to fiber length and spectral width of light source:
2
2.mat r
matfiber 0
t d nM
L d c
Δ λΔλ λ
= = − (8.27)
Material dispersion coefficient for fused silica, Fig. 8.19b:
for λ = 1312 nm: curve goes through zero.
Optical data communication:
development of light sources for λ = 1312 nm.
Material dispersion coefficient: Material dispersion coefficient:
tailoring by composition of glass (see Fig. 8.20).
8.5.3 Waveguide dispersion [10]
Small structural parameters V: light penetrates into cladding (seeFigs. 8.15 to 8.18). “acceleration” of light due to lower refraction index of cladding;
©JHW 246
smear-out of light pulse (sum of light from core and cladding).
Pulse widening:
.wave wave fibert M LΔ Δλ= (8.28)
8.5.4 Chromatic dispersionMaterial dispersion and waveguide dispersion:
depend on “width” Δλ of light source.
Both effects together: chromatic dispersion.Chromatic dispersion coefficient Mchr (approximately):
Non mono-mode fibers: chromatic dispersion important,dominated by intermodal dispersion.
Waveguide dispersion in mono-mode fibers: < material dispersion.
Advanced mono-mode fibers: graded-index cores with adjusted profiles
.wavematchr MMM += (8.29)
©JHW 247
(shift of minimum of chromatic dispersion to desired wavelength).
Dispersion-shifted fibers with linearly tapered profile, reduced coreradius, and doping:
shift minimum to larger λ, where attenuation also low (see Fig. 8.20).
30
20
0
-10
-20
30
2a=6.0µm
SiO material dispersion
waveguide dispersion
Mmat
Mwavepers
ion
coe
ffic
. M
,
ma
tM
(ps/
(km
nm
))w
ave
2
SiO +GeO2 2
2a=4.8µm
b
a
30
20
10
0
-10
-20
-30
-40
(ps/
(km
nm
))rs
ion c
oeffi
c. M
chr
chromatic dispersionMchr
disp M
, 2a=4.8µmSiO + GeO2 2
© JHW
0nr
-a
-b
©JHW 248
Fig. 8.20: Dispersion-shifted fiber. Doping and reduction of corediameter shift minimum from 1.3 to 1.5 µm.
-30
-400 1.2 1.4 1.6
wavelength (µm)λ1.8 2.0
dis
pe , 2a=6.0µmSiO 2
Dispersion-flattened fibers (Fig. 8.21): other grading profiles!• minimizing chromatic dispersion for two wavelengths and• reducing effect in between.
© JHW
©JHW 249
Fig. 8.21: Dispersion-flattened fiber. Profile tailored to minimizechromatic dispersion for two wavelengths.
8.6 Attenuation in glass fibers
8.6.1 The attenuation coefficient
Power Φ of light: exponential degradation due toabsorption and scattering.
Definition of effective attenuation coefficient αeff ( see chapter 5.2.1):
[αeff ] = km-1
( ) .eff L0L e αΦ Φ −= (8.30)
1eff ln .
L L0Φα
Φ( )
=
(8.31)
©JHW 250
Instead of αeff : use of quantity αdB:
/10( ) 10 .dB L0L αΦ Φ −= (8.32)
[αdB ] = dB/km
1[km ] 0.23 [dB/km]eff dBα α− = (8.34)
(8.33)1
10 .dB 10logL L
0ΦαΦ( )
Φ0 /Φ(L) = 1, 0.5, 0.1, 0.01, 0.001correspond to 0, -3, -10, -20, -30 dB.
©JHW 251
8.6.2 Attenuation mechanismsFour mechanisms of light attenuation (see Fig. 8.22):
absorption at OH-ions due to water vapor inclusions
UV-absorption at band tails of SiO2
IR-absorption due to molecular excitations (Si-O, Si-Si bonds)
a) UV-absorption: due to electronic transitions between band tails(caused by statistical disorder of amorphous SiO2, band gap = 9 eV);
intrinsic absorption.
b) IR b ti b SiO l l ki th l
absorption at OH ions due to water vapor inclusions,and at metal ions
Rayleigh scattering
©JHW 252
b) IR-absorption: by SiO2-molecules making up the glass;intrinsic absorption.
c) Absorption at OH-ions and metal ions:extrinsic absorption;can be reduced by technological improvements.
IR absorption(molecules)
Rayleighscattering 1/∼ λ
1
3
effi
ccie
nt 4
© JHW
OHabsorption
UV absorption(band tail)
0 1
0.3
1at
tenu
atio
n c
oe
dB
©JHW 253
Fig. 8.22: Attenuation of light in silica glass;local minimum at 1.3 µm, absolute minimum at 1.55 µm.
0.11.21.00.80.6 1.4 1.6
wavelength (µm)λ1.8
d) Rayleigh scattering
Rayleigh scattering = intrinsic effect.
Glass: amorphous, microscopic fluctuations of molecule density.
Spatial fluctuations of refraction index nr,much smaller than wavelength of the light.
Light-scattering; much stronger for “blue” end of spectrumthan for “red” end.
Attenuation for light: decreases with 1/λ4.
Rayleigh scattering at air molecules of atmosphere:
©JHW 254
blue sky
red sunset
y g gresponsible for
Attenuation in glass fibers: limited by
• Rayleigh scattering (on the “blue” side),
• IR-absorption (on the “red” side), and
• OH ion absorption (in between)• OH-ion absorption (in between).
Minima of attenuation at 1.3 and 1.55 µm:
accessible with light sources based on InP-substrates (see Fig. 6.6).
©JHW 255
8.7 Internet Links
1. Fiber Optic (Principle applet): http://webphysics.davidson.edu/applets/Optics/fiber_optics.html
2. Demonstration of Light Guidance in a Step-index fiber (Applets):http://OLLI.Informatik.Uni-Oldenburg.DE/sirohi/guidance.html
3. Fiber Optics - Slab Dielectric Surrounded by Air: http://www.ee.buffalo.edu/faculty/cartwright/java_applets/ray/FiberOptics/index.html
©JHW 256
8.8 Literature
1. B. E. A. Saleh and M. T. Teich, Fundamentals of Photonics (Wiley Interscience, New York, 1991) p. 277-286.
2. O. Strobel, Lichtwellenleiter – Übertragungs- und Sensortechnik(VDE-Verlag, Berlin, 1992), p. 44-45.
3. p. 64 in Ref. [2].
4. p. 295 in Ref. [1].
5. D. Jansen, Optoelektronik (Vieweg, Braunschweig, 1993), p. 171.
6. p. 170 in Ref. [5], p. 45 in Ref. [2], p. 279 in Ref. [1].
©JHW 257
7. p. 46 in Ref. [2].
8. E. Hecht, Optics 3rd edition (Addison Wesley, Reading, MA, 1998), p. 121, see for example p. 297.
9. p. 69 in Ref. [2].
10. p. 301 in Ref. [1].
9. Photodetectors
©JHW 258
Photodetector = radiation absorbing device,
measures flux ( = areal number nphot of photons)
or optical power (n h hν)
9.1 Introduction, general considerations [1,2,3,4]
or optical power (nphot hν)
Energy transfer to atoms, molecules or lattice of solids
Transfer to electronsof gases or solids
C f
A) Thermal detectors B) Quantum detectors
©JHW 259
heating
Signal ~ power
slow
Change of electron distribution
Signal ~ flux
fast
9.1.1 Thermal detectors [5,6]
Various phenomena:
Dependence of contact potential difference on temperature:
thermocouple, thermopile
Dependence of conductivity of metal foil on temperature:
bolometer
Dependence of dielectric constant εr on temperature:
pyroelectric detector
Dependence of volume of enclosed gas on temperature:
©JHW 260
Golay-cell.
In this lecture: Emphasis on photoelectric detectors.
These detectors are based on external and internal photoeffect.
9.1.2 Quantum detectors = photoelectric detectors
External photoeffect:= emission of photo-excited electrons from solid into vacuum
(photon energy to be larger than work function); collection of free electrons.
Internal photoeffect:= generation of electron/hole pairs in semiconductor by
fundamental absorption at band gap (see chapter 5.2).
Further subdivision:- photoconductivity
photoelectric effect at junction or barrier
©JHW 261
- photoelectric effect at junction or barrier(pn- and pin-junctions, Schottky-barriers, avalanche diodes, phototransistors, ...).
9.2.1 External quantum efficiency EQE
EQE (with 0 < EQE < 1) = ratio of photogenerated charge carriers(electrons) contributing to detector current per incident photon
9.2 Properties and specifications of photodetectors [7,8]
•
elel el•
ph ph ph
number Z of electrons in circuit Z ZEQE = = =
number Z of incident photons Z Z
(9.1)phI / q=
©JHW 262
Iph = photogenerated detector current,Φ0 = incident optical power.
( )
0Φ / hν
EQE < 1 due to optical losses: Reflection losses at surface, incomplete absorption
(transmission losses) electrical losses: incomplete collection (recombination in bulk or at
surfaces).
αExternal quantum efficiency EQE:
ηcoll = (electrical) collection efficiency,rΦ = reflectance at surface,
α = absorption coefficient of detector material,
(9.2a)(1 ) [1 ( )]coll effEQE r exp wΦη α= − − −
αΦ
©JHW 263
weff = effective thickness of photodetector.
Internal quantum efficiency IQE:
(9.2b)IQE = EQE /(1 )rΦ−since EQE = IQE (1- rΦ)
= ratio of photogenerated detector currentand incident optical power
number of photons per time produces photocurrent (if ll h t t d)
ph·Z phphI q Z
•=
9.2.2 Responsivity, spectral response ℜ [A/W]
(if all photons converted),
optical power (Watts) at energy hν results in
electric current Iph = qΦ0 / hν (if all photons converted),ph0Φ h Zν
•=
with quantum efficiency EQE:
Iph = EQE q = EQE qΦ0 / hν = ℜΦ0phZ•
(9.3)
©JHW 264
p ( )
[ℜ] = A/W
(9.4)[ ]µm
1.24ph
0
qEQE EQE
Φ h
Ι λν
ℜ = = =
ideal spectral response ℜ: linear function of λ
for semiconductor detector with gap Eg and α = 0 for hν < Eg,
ideal quantum efficiency EQE = 1, independent of λ
For quantum detectors:
EQE and ℜ = 0 for hν < Eg or λ > λg = hc/Eg
from Eg = hνg and c = νλ it follows: λg = hc/Eg and
(9.5)[ ] [ ]1 24
µmeVg
g
.
Eλ =
For thermal detectors:
©JHW 265
For thermal detectors:
Quantum efficiency EQE: from Eq. (9.4) it follows:1 24
[µm]
.EQE
λ= ℜ
0
0
output signalconst.;
Φ= ℜ =
ℜm
ℜ
ℜ0Thermal
Quantum detector1
EQE
λ
0 detector
λgλ g
λ
©JHW 266
Fig. 9.1: Wavelength dependence of external quantum efficiency andspectral response ℜ of quantum detectorsand thermal detectors.
9.2.3 Detection sensitivity [9,10,11]
Detectors: characterized by a minimum detectable signal.Physical reason: Noise. Types of noise relevant in detectors:
within detector itself:within detector itself:- Quantum noise of incident photon flux- shot noise of current (dark current + photocurrent)
in resistor of detector circuit:- Thermal (Johnson or Nyquist) noise.
©JHW 267
Minimum-detectable signal: defined as mean signalyielding signal-to-noise ratio (SNR) of
SNR = 1.
9.2.4 Response time, frequency response [12,13]
In general: detectors ≙ low-pass filters:
(9.7)1
2 2
( ) =
1 (2 )
0
r
f
fτ + π
RR
f = frequency, ℜ0 = responsivity at 0 Hz, τr = response time.
Response time τr represents several physical delay effectslike transit times caused by different times for
drift within field regions
©JHW 268
g
diffusion outside of field regions (see below).
In addition to physical effects:
RC time constant of photodetector circuit;
electronic response time limit;
upper cut-off frequency fc:
CD = capacitance of detector (pn-junction)
RL = load resistor of detector
(9.8)1
2π cD L
fC R
=
©JHW 269
9.3 Photoconductors [14,15]
Two main types:
a) b)
hν-
hνEC
E
EFhν
E
EC
ED
EA
localized
-
hν
+
-
©JHW 270
Fig. 9.2: a) intrinsic and b) extrinsic photoconductors.
+EVEV+
Extrinsic type: for IR-detection; IR-detection with intrinsic detectors:
low Eg high dark currents low detection sensitivity.
Photoconductive sensors: use change of conductivity
(9 9)σ = q n + q p
with n, p = electron- / hole density,
µn, µp = electron- / hole mobility.
High sensitivity: large changes in conductivity required between
illuminated and dark state („light/dark-ratio“):
(9.9)σ = q n μn + q p µp
©JHW 271
σph = photoconductivity and σd = dark conductivity
(9.10)( )( )
ph n ph pph
d d n d p
q n p
q n p
μ μσσ μ μ
+=
+
Linear device:
G(Φ) = light-induced generation rate of electrons and holes, lif ti f h t t i
(9.11)nph = nd + G(Φ)τ and pph = pd + G(Φ)τ
τ = lifetime of photogenerates carriers,Φ = light intensity.
(9.12)1 1ph n p
d d d n d p
Gn p
σ μ μΔσ τσ σ μ μ
+= + = +
+
©JHW 272
Sensitive detectors require high carrier lifetimes τ.
9.4 Photodiodes [16,17,18]
qVd
Ep n
j1
j3EC
E
jph(n)
hν
jph(p)
j j = majority carrier currents diffusion currents across barrier V
+ +
SCR
j4
j2+ + + + + + + + + + + + +
+
EV
EFEC
EV
EF
++
hν
©JHW 273Fig. 9.3: Operation principle of photodiodes (and solar cells).
j1, j2 = majority carrier currents, diffusion currents across barrier VD(diffusion voltage of junction)
j3, j4 = minority carrier currents, drift currents across the spacecharge region (SCR)
jph(n), jph(p) = photocurrents
Equilibrium condition in the dark: Σ ji = 0 , i = 1, ..., 4
(9.13)1 (ideal diode eqation)d 0
qVJ = J exp -
kT
Illumination: Additional minority carrier currents Jph(n) and Jph(p);increase of majority carriers negligible (low excitation regime).
Photocurrent Jph: in reverse direction;
added to dark current:d phJ = J - J :
©JHW 274
J(V)-characteristic of photodiode and of solar cell
(9.14)10 ph
qVJ = J exp - - J
kT
J
V
Φ = 0
Jph
V
JphJph
Φ = Φ1
photodiode regime solar cell regime
©JHW 275
9.4.1 Detailed look at pn-photodiodesSpecial properties, especially in comparison to pin-photodiodes(see below): see Fig. 9.5:
Fig. 9.4: Current/voltage curve of solar cells and photodiodes.
p n
+- Structure
hν
Space charge density ρ(x)
-
+
x
Space charge density ρ(x)
Electric field E(x)
©JHW 276
( )
x
-
+
-
+
-
+
Electron energy E(x)
EF
EC
EV EC
EV
V
p nSCR
EFhν
hν
and photocarrier collection ( ---- )Photocarrier generation G(x) ( )
G(x)
drift diffusiondiffusion
+ + V
x
©JHW 277
Fig. 9.5: Scheme of pn-photodiode.
hν
drift diffusiondiffusion
x
G(x)
Space charge- and drift region narrow (dimension: < 1 µm);
for efficient light absorption (Φ(abs.) ≈ 0.8 Φ0):thickness of (crystalline) silicon photodiode of ≈ 50 µm required;
Fi 9 h i i !
Discussion:
see Fig. 9.5, photocarrier generation!
Collection losses: increase with distance from SCR;see shaded area below G(x)-curve.
pn-photodiode: Diffusion-controlled detector
Main contribution to photocurrent due tophotocarrier diffusion from outside space charge region.
©JHW 278
pn photodiode: Diffusion controlled detector.
9.4.2 The pin-photodiodeImprovement of carrier collection by increase of width of field-/ drift-region; see Fig. 9.6!
hνp n
+- Structure
i
Space charge density ρ(x)Space charge density ρ(x)
-
+
x
Electric field E(x)
©JHW 279
x
hν-
+
Electron energy E(x)
EF
EC
EV EC
ViEF
-
+-
and photocarrier collection (- - -)
Photocarrier generation G(x) ( )
hν G(x)
+EV
x+
©JHW 280
Fig. 9.6: Scheme of pin-photodiode.
drift diffusiondiffusion
x
Discussion:
Goal: broadening of field region
Possible approach: very low doping of p- and n-region
(dSRC2 ~ 1/(ND , NA)); however: low conductance of p- and n-region,
hi h i l i i !high resistance losses in operation!
Good approach:insertion of i-layerwith thickness in the range of 50 µm;
majority of photocarriers collected by drift,
only small contribution by diffusion from p- and n-layer;
©JHW 281
improved collection efficiency;see Fig. 9.6, shaded area below G(x)-curve;
pin-photodiode: Drift-controlled detector.
Time-dependent behavior of pn- and pin-photodiodes:
a) pn-photodiode: controlled by diffusion time:
Diffusion length L of mobile carriers:
τDL =
Diffusion constant D and carrier mobility μ:interrelated according to Einstein-relation:
τDL =D = diffusion constant, τ = carrier lifetime
q
kTD μ =
©JHW 282
It follows:
q
(9.15)qkT
L
D
L
μτ
22
==
τ substituted by diffusion time tdiff and L substituted by diffusion distance
ddiff :
(9.16)2diff
diffT
dt =
mVµ
with VT = temperature voltage = kT/q = 0.025 V.
pn-junction of photodiode assumed in center of structure at ddiff = d/2:
©JHW 283
(9.17)T
diff µV
dt
4
2
=
b) pin-photodiode: controlled by drift time:
Drift velocity vd of charge carriers in a semiconductor:
V
with di = thickness of the i-layer.
(9.18)i
d d
VmmEv ==
©JHW 284
Drift time tdrift:
A li ti f V 25 V t i h t di d d d d
(9.19)2
i idrift
d
d dt = = .
Vμv
Application of e.g. V = 25 V to a pin-photodiode, and di ≈ d:
(9.20)4 4 0 025 1
25 250drift T
diff
t V .=
t V
μμ
⋅= =
Upper cut-off frequency of pin-photodiode
©JHW 285
pin-photodiodes useable for frequencies up to 50 GHz.
pp q y p p
remarkably higher than that of pn-photodiode.
9.4.3 The Schottky photodiode
Schottky photodiodes: even higher cut-off frequencies
than with pin-photodiodes: up to 100 GHz.
©JHW 286
Operation principle:
EVac
Electron energy E(x)
Schottky barrier height
qΦΒ = qΦΑ − χ
qΦΑ = work function of metal
χ = electron affinity of
Vac
ECEF
Eg
qΦΑ
qΦΒ
χ-
hν1
-
+
qΦΒ qΦΑ χ
©JHW 287
Fig. 9.7: Band diagram of Schottky photodiode.
semiconductorEV
Eg
Metal Semiconductor (n-type)x
hν2
+
Characteristic features:
Semiconductor coated with thin semitransparent metal film
Rectifying barrier obtainable with
- metal with high work function ΦA on n-type semiconductor ormetal with high work function ΦA on n type semiconductor or
- metal with low work function on p-type semiconductor.
Absorption in metal film: electron injection across interface,
when hν > qΦΒ:
th h ld l th
©JHW 288
threshold wavelength:
[ ] [ ]1 24
μmeV
(1) (1)th th
B B
hc .; ;
qΦ qΦλ λ= = (9.21)
Absorption of radiation in semiconductor:
1 24h(9.22)[ ] [ ]
1 24μm
eV(2) (2)th th
g g
hc .; ;
E Eλ λ= =
Schottky diodes: feasible on materials available only n-type
or only p-type, e.g. on II-VI compounds.
©JHW 289
9.5 Photodiodes with internal gain: Avalanchephotodiodes (APDs) [19,20,21]
Avalanche photodiode: converts each absorbed photon into a cascadeof moving carrier pairs; basic process: impact ionisation in high-field
region inside reverse biased pn junctionregion inside reverse-biased pn-junction.
++
-
2-
1Ekin > Eg
p nEC
EV
EF
©JHW 290Fig. 9.8: Principle of avalanche multiplication in an APD.
+
--
2
+
1hνg
Eg
E
Threshold energy Ekin,th(n) for impact ionization of electrons:
(9.23)2
( ) 1
eff, p
eff, nkin, th g
eff p
mm
E n Em
+=
Example: meff, p = meff, n : Ekin,th(n) = 3/2Eg
In most semiconductors: m ff > m ff :
1 eff, p
eff, nm+
©JHW 291
In most semiconductors: meff, p > meff, n:
(9.24)3
( )2g kin,th gE E n E≤ ≤
In practice: region for light absorption and region for carrier multiplication are separated in order to
provide sufficiently thick absorber region,
avoid junction breakdown, andj
generate multiplication of only one carrier type (electron or hole).
Fig. 9.9 shows an APD structure.
©JHW 292
p p p n
Structure
++ - +(π)
hν
-
x
EB
Electric field E(x)
Space charge density
--
+
(t)ρ
xhν
- -- -
- --
--
+ +
++
Electron energy E(x)
E
EE
V E
C
F
V
C
multipli-cationregion
©JHW 293
Fig. 9.9: Scheme of reach-through p+-p-p-n+ avalanche photodiode(RAPD).
++
EEF
V
x
„Reach-through“: applied reverse bias must be high enough fordepletion layer (SCR) to reach through p and πregion into p+ contact layer.
Field strength: adjusted by doping:
threshold field strength EB for carrier multiplicationconfined to the desired multiplication region.
For visible light detection: silicon most common material.
Compound semiconductors, III-V- and II-VI-compounds, also in use:
f h t i t d t f i d ti
9.6 Materials and detector configurations
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for photoresistors due to ease of economic production(II-VI-compounds);
for optical communication purposes via glass fibers (λ = 1.3 - 1.6 µm);multinary III-V-compounds required.
Insulator
Electodeshν
hν
Semiconductor
p i n
hν
a) Photoresistor
Semi-conductor
b) pin-photodiode
n-AlGaN
p-AlGaN
Au MaskSemi-transparent-Ni
p-GaN
AlGaNMQWs
©JHW 295Fig. 9.10: Examples of structures of photodetectors.
Metal
c) Schottky-diode
Au
n-SiC substrate
Ti
d) Reach-through avalanche photodiode (RAPD)
9.7 Internet Links
1. Formation of a PN Junction Diode (Applet): http://www.acsu.buffalo.edu/~wie/applet/pnformation/pnformation.html
2. Formation of a PIN Junction (Applet): http://jas2.eng.buffalo.edu/applets/education/pin/pin2/index.html
3. PN-Junction Simulation (Applet):http://fiselect2.fceia.unr.edu.ar/fisica4/simbuffalo/education/pin/pin/index.html
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9.8 Literature
1. G. Winstel und C. Weyrich, Optoelektronik II (Springer-Verlag, Berlin, 1986).
2. R. Paul, Optoelektronische Halbleiterbauelemente (Teubner, Stuttgart, 1992), pages 213-277.g , ), p g
3. H. G. Unger, Optische Nachrichtentechnik, Teil II (Hüthig Buch Verlag, Heidelberg, 1992), pages 451-500.
4. M. Fukuda, Optical Semiconductor Devices (Wiley, New York, 1999), pages 211-264.
5. Pages 19-23 in Ref. [1]. 6. Pages 488 – 489 in Ref. [3].7. Pages 35-54 in Ref. [1].8 Pages 217 226 in Ref [2]
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8. Pages 217-226 in Ref. [2].9. H. G. Wagemann and H. Schmidt, Grundlagen der
optoelektronischen Halbleiter-bauelemente (Teubner, Stuttgart, 1998), pages 152-153.
10. D. A. Ross, Optoelectronic Devices and Optical Imaging Techniques (The Macmillan Press, London and Basingstoke, 1979), pages 48-67.
11. D. Jansen, Optoelektronik (Vieweg, Braunschweig, 1993), pages 127-146.
12. Pages 37-38 in Ref. [10].
13. Pages 234-239 in Ref. [4].
14 Pages 119-138 in Ref [9]14. Pages 119 138 in Ref. [9].
15. Pages 20-30 in Ref. [10].
16. Pages 141-165 in Ref. [9].
17. Pages 55-88 in Ref. [1].
18. Pages 211-243 in Ref. [1].
19. W. Heywang, H. W. Pötzl, Bänderstruktur und Stromtransport(Springer-Verlag, Berlin, Heidelberg, New York, 1976), pages 260 264
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260-264.
20. Pages 243-264 in Ref. [4].
21. Pages 99-132 in Ref. [1]
