material properties and characterization optical …¤ckel/mnsm_matprop.pdfphotonics represents the...

Master of Science in “Micro and Nanosystems”

Material Properties and Characterization

Optical Processes in Semiconductors I - IIITutor:

Prof. Dr. Heinz Jäckel, Electronics Laboratory, High Speed Electronics and Photonics Group,

Department of Information Technology and Electrical Engineering, [email protected], www.ife.ee.ethz.ch, Tel.: +41 44 632 27 57

course dates: 20.11.07, 27.11.07, 04.12.07 CAB H57

18/11/2007

Material Properties and Characterization: Optical Processes in Semiconductors Electronics Laboratory, ETH-Z 01.12.2006


Preface:Photonics represents the synergy between optics and electronics and forms the base of well established, but still rapidly expanding fields such as optical communication, micro-optics, optical storage, display and image technology, optical sensors and actuators etc. Essential for the fast advancement of the field is the tight interplay between optical materials, devices and systems. Optical materials and optical processes in these materials have been enabling for the revolutionary developments and will continue to do so in the current progress of the field toward nano-photonics.Although Photonics covers many fields from basic physics, quantum mechanics, classical optics, electronics, device concepts and realizations, materials science, fabrication technologies etc., advancements of very different optical materials systems have been key issues for success and commercial implementations. Due to their outstanding role in combining electronics and photonics and miniaturizing photonics down to the limits of the optical wavelength or the DeBroglie-wavelength, semiconductors have been highly successful as optical materials and also paved the way towards the emerging Nano-Photonics. In view of these developments and the limited available space, this course on “Optical Processes in Semiconductors” focuses on the description of optical processes in semiconductors. Never the less, many of the presented concepts are also directly applicable to other important materials systems such as insulators, metals, organics etc. Due to their importance we will also cover, but with less emphasis, metals and insulators. The selection of the course topics is driven pragmatically by the practical device realizations (to be presented in the MNS-Master course on “Devices and Systems“ in the 8th semester) for photonic applications and systems.

The course is based to a large extent on the excellent textbook by M. Fox, “Optical Processes of Solids” (see literature reference). The broad scope of photonics from Quantum Mechanics, Solid State Physics, Classical Optics and Mechanics makes strong demands on the required working knowledge of the students. Quantum Mechanics will only be used where classical concepts simply fail, but otherwise the course will follow a quasi-classical approach. Forced by the limited amount of lecture hours extended, proofs are deferred to appendices or to the many excellent and more detailed textbooks. Working knowledge is expected (or should be “refreshed” from previous undergrad-courses) in: Electro-Magnetic Field Theory, Classical Optics, basic Semiconductor Theory.

Summer 2006 H. Jäckel0-2


0-3

Literature:Primary Literature:M. Fox, Optical Properties of Solids, Oxford University Press, 2001The course is based on this book, which is the main source of reference. The topics are well developed for their concepts, without too much mathematical overhead.

J. H. Simmons, K.S. Potter, Optical Materials, Academic Press, 2000Well written textbook, which goes more into the formal details.

Secondary Literature:B. E. A. Saleh, M. C.Teich, Fundamentals of Photonics, John Wiley, 1991 Excellent, easy to read introduction, which covers a much broader range of topics in photonics beyond semiconductor materials, with a strong focus also on devices.

L. Coldren, S. Corzine, Diode Lasers and Photonic Integrated Circuits, John Wiley, 1995As the title indicates this is a solid and detailed introduction semiconductor lasers.

A. Yariv: Quantum Electronics, Wiley, 1967R. Loudon: The Quantum Theory of Light, Oxford Science Publications, 2000J. I. Pankove, Optical Processes in Semiconductors, Prentice Hall, 1971 Easy to read introduction covering in a more descriptive way the material of the course. Despite its age most relevant concepts are covered with the exception of quantum-confinement in semiconductors.

D. Marcuse, Principles of Quantum Electronics, Academic Press, 1980

Material and derivations on optical transitions in Chap.4,5 are treated in more detail in the course:Prof. H. Jäckel “Optoelectronics and Optical Communication” (6.sem.) at D-ITET

Script-download on: www.ife.ee.ethz.ch/education/lectures


0-4

Structure of the Course “Optical Processes in Semiconductors”:

How does light interact with solids, in particular semiconductors?Electro-magnetic (EM-) waves interact by their electrical vector field E(r,t) and magnetic vector field H(r,t) with the electrical charges, electrons (holes) or the charged crystal lattice of the solid (phonons).The interaction with matter modifies the properties (amplitude, frequency, wavelength, propagation velocity, propagation direction, polarization, etc.) of the lightwave. Photonic devices control this interaction for a functional goal.

Course Plan: IntroductionApplicationsState-of-the-Art

Classical Light WavesPropagation of classical EM-wavesDipole interaction

Electronic Physics of MetalsCharge motion in quasi-free state

Nonlinear OpticsCharge motion in nonlinear potentialsNonlinear bound dipolesNonlinearities in quantized systems

Quantum Size Effects Charge motion in the periodic potential of semiconductors with linited degrees of freedom

Linear OpticsElectronic Physics of SemiconductorsCharge motion in the periodic potential of semiconductors,quantized systems and bound states

self-study

Maxwell, EM-waves repeat yourself

optional, self-study


0-5

Content:1. Introduction 1-11.1 Motivation 1-21.1.1 Advantage of optics 1-21.1.2 Driving forces present / future 1-31.1.3 Why Semiconductors as Materials for Photonics / Optoelectronics 1-31.2 Photonics Applications 1-41.3 Optical Effects and Photonic Devices / Applications 1-51.4 State-of-the-Art of Photonic Devices / Applications 1-61.4.1 Fiberoptic Communication 1-61.4.2 Tb/s Technology 1-71.4.3 Display and Illumination Technology 1-81.4.4 Image Sensing 1-81.4.5 Optical Data-Storage 1-91.4.6 Optical Pump and Power Sources 1-91.4.7 Optical Integration 1-101.4.8 Nano-Optics 1-111.4.9 Optical measurements 1-121.4.10 Optical Medical Applications 1-12

2. Classical Propagation of Lightwaves 2-12.1 Basic Optical Processes 2-22.1.1 Wave propagation and basic optical processes 2-22.1.1.1 Basic Relations between optical propagation constants 2-32.2 Basic Optical Materials 2-92.2.1 Crystalline insulators and Semiconductors 2-102.2.2 Glasses 2-112.2.3 Metals 2-122.2.4 (Molecular Materials) 2-122.2.5 (Doped Glasses and Insulators) 2-13


0-6

2.3 Microscopic Polarization Models 2-142.3.1 Dipole Oscillators of bound charges 2-142.3.2 Kramers-Kronig Relation 2-192.3.3. Dispersion: phase and group velocity 2-202.3.4 Optical Propagation in anisotropic media: Birefringence 2-212.3.4.1 The technique of the refractive Index-Ellipsoid 2-222.3.5 Lattice Oscillations 2-242.3.6 Free electron Oscillations 2-24Appendix A2.1 EM-field as quantized harmonic oscillator: Photons (optional) 2-26

3. Optical Properties of metals: free electrons 3-1 3.1 Drude-Lorentz-model of reflectivity of metals 3-23.2 Interband absorption effects in metals (opt) 3-53.3 Free carrier effects in doped semiconductors 3-5

4. Optical Properties of Semiconductors: Interband absorption (3) 4-14.1 Basic Interband Transitions in Semiconductors: an overview 4-24.1.1 The Einstein Model of Optical Transitions in a discrete 2-Level System 4-34.1.2 Quantum mechanical optical transition in a 2-level systems 4-84.1.3 Generalization to optical transitions in energy-band systems 4-134.2 Interband Absorption in Semiconductors 4-184.2.1 Band edge absorption and direct band structure for III-V compounds 4-194.2.2 Joint-density of states in SC with parabolic dispersion 4-204.2.3 Absorption spectra of direct SC 4-214.2.4 Frank-Keldysh Effect (optional) 4-224.2.5 Absorption spectra in indirect SC 4-234.3 Static and dynamic absorption measurements 4-244.4 Excitons 4-254.4.1 Free exciton absorption 4-254.4.2 Free excitions in a high electrical field 4- 27


0-7

4.5 Phonons 4-284.5.1 Lattice absorption /reflection processes 4-294.5.2 Inelastic and elastic light scattering by phonons 4-31 Appendix A4.1 Optical Transitions between discrete energy states 4-34Appendix A4.2 Optical Transitions between energy bands 4-37

5. Optical Properties of Semiconductors: Luminescence and stimulated emission 5-1 5.1 Light emission by spontaneous or stimulated emission 5-25.1.1 Carrier decay by spontaneous emission 5-25.1.2 Direct and indirect interband Luminescence 5-35.1.3 Photoluminescence (PL) 5-45.1.3.1 Spontaneous emission (PL) 5-45.2 Stimulated Emission and Optical Gain 5-75.3 Semiconductor Materials 5-95.4 Electroluminescence (electrical carrier injection) 5-11

6. Quantum Size Effects in Semiconductors 6-16.1 The Concept of Quantum Confinement 6-26.2 Quantum confinement in Heterostructure semiconductors 6-36.3 Band engineering: Electronic States in SC-crystals 6-46.1.1 Quantum Well Structures 6-46.3.2 Electronic Levels in QWs 6-56.4 Optical absorption and Emission in quantum-confined SCs 6-76.4.1 Matrix-elements and Selection Rules in QWs 6-76.4.2 Density of States in Quantum-Confined SCs 6-96.4.3 Excitons in quantum-confined structures 6-106.4.4 Quantum confined Stark-Effect 6-106.4.5 Intersubband Transitions (optional) 6-11


0-8

6.5 Fabrication of quantum confined structures (optional) 6-126.5.1 Technological Requirements and Challenges 6-126.5.2 Molecular Beam Epitaxy (MBE) 6-136.5.3 Metal-Organic Vapor Depostion Epitaxy (MOVPE) 6-156.5.4 Nano-meter scale structuring by epixial growth 6-176.5.4.1 Lithography and etching 6-176.5.4.2 Structured Growth 6-176.5.4.3 Self-organized Growth 6-18

7. Nonlinear optical processes (11) 7-17.1 Nonlinear Susceptibility Tensor 7-27.2 Non-resonant Nonlinearity: Anharmonic scalar oscillator 7-47.3 Resonant Nonlinearities 7-77.4 Second-order Nonlinearities 7-87.4.1 Pockel’s or the linear Electro-Optical effect 7-117.4.2 Three wave mixing 7-127.5 Third-order Nonlinearities 7-137.5.1 Optical Kerr Effect and Nonlinear Refractive Index 7-147.5.2 Stimulated four-wave mixing 7-157.5.3 Resonant 3rd-order Nonlinearities in Semiconductors 7-16


0-9

Symbols:A = Einstein coefficient

= AreaA = vector potentialα = absorption coeff.

B = Einstein coefficientB = magnetic flux, Inductionβ= propagation vector

C = expansion coeff. c = vacuum velocity of light

D = electric field displacementvector

d =dipole separation δ = Dirac function

E = electric field vectore = electron chargeei = unity vectorε0 = dieelectric permittivity of

free spaceεr = relative dielectric constantε = dielectric permittivityEQF= quasi Fermi-level

F = forcef = frequency

g = gain

H = Hamiltonian= magnetic field intensity

h = Planck’s constantħ = h/2π reduced Planck

constant

I = Intensity= current

i = current

j = imaginary unitj = current density

k = propagation constant k = Boltzmann constantk = Ks= spring force constant

L = lengthl = length

M = magnetizationm = massm0 = electron massmeff = effective massme = effective electron massmh = effective hole mass

N = particle numbern = particle densityn = refractive index

P = polarization denistyp = polarizationp = momentum

q = elementary charge

R = transition rate (also: G)r = transition probabilityr = spatial coordinate vector

T = absolute temperature t = time

μ0 = magnetic permeability offree space

μr = relative magnetic constantμ = magnetic permeabilityu = Bloch function

v = velocityν = frequencyV = volume

W = optical spectral energy density (also: u, ρ)

ω= radial frequency

x = cartesian coordinate

y = cartesian coordinate z = cartesian coordinate

Ψ= time and spatial dependentSchrödinger wavefunction

ψ= spatially dependentSchrödinger wave function

λ= wavelength

ρ= electronic density of states

σ = conductivity

χ = electric susceptibility

Γ= damping

γ= damping

τ= life time

Δ = Laplace operator = nabla operator

× = vector product• = scalar product∇


1. Introduction

Goals:• what makes semiconductors an important and unique optical material ? • what are the driving applications of semiconductor photonics • state-of-the-art of semiconductor photonics

1-1


1. Introduction1.1 Motivation 1.1.1 Advantages of Optics• extremely large bandwidth (10-50THz@200THz) for digital / analog signal transmission / processing• large spatial modulation density, high parallelism• non-interacting signal transmission, high spatial and spectral density• μm-wavelength and high miniaturization potential• non-contacting and 3-D information storage• light waves with high temporal and spatial coherence

1.1.2 Driving Forces, present / future• fiberoptic communications, optical LANs and computer interconnects• optical data storage• Laser pump sources• display technology, printing and illumination• Image sensing

• monolithic integration• optical routing and switching• nano-photonics, optical signal processing and computing 1-2

Future


1.1.3 Why Semiconductors as materials for Photonics / Optoelectronics ?

• direct current-pumped light generationsimple, compact electrical current pumping for efficient high gain optical amplification

• conceptually “easy” integration with electronics, technology for miniaturization • insulating or conducting material• advanced planar, batch-processing technology• advanced process technology inherited from electronicsbut:• non-silicon very challenging compound-semiconductor (III-V, II-VI, IV-IV) technology• modest production volume• very challenging integration technology and device scaling for IOCs

Important non-semiconductor optical materials:• metals (reflectors, contacts, …) • glasses (coatings, waveguides, …)• (organics)

Optical materials incompatible with semiconductors: • many nonlinear solid state materials• magneto-optical solid state materials

1-3


1-4

1.2 Photonic ApplicationsApplications Summary:semiconductor based


1.3 Optical Effects and Photonic Devices / Applications • Material related optical effects and device implementations

1-5

Harmonic GeneratorsAll optical switchingSelf-phase Modulation / Pulse CompressorsHolographic memories

Optical Nonlinearity(EX -field dependent material properties)

Modulators, DeflectorsAcousto-Optic effect(mechanical stress-field dependent material properties)

Lasers, optical amplifiersStimulated Emission

Modulators, sensorsElectro-Optical effect(E-field dependent material properties)

Faraday rotators, sensors, optical memoriesMagneto-optical effect(H-field dependent material properties)

Thresholdless Lasers, LEDs, photonic crystalsNanoscale PhotonicsQuantum size effects

Application, deviceEffect, material

Photodetectors, PhotoconductorsAbsorption modulatorsSolar Cells

AbsorptionInterband, intersubband, band-to-discrete levels, inter-level

LEDs, DisplaysRecombination, Spontaneous EmissionInterband, intersubband, band-to-discrete levels, inter-level


1.4 State-of-the-Art of Photonic Devices / Applications 1.4.1 Fiberoptic Communication

1-6

1980 1990 2000 2010108 YEAR

TRA

NSM

ISSI

ON

CA

PAC

ITY

(b/s

)

109

1010

1011

1012 1 Terabit/s

100Mb/s

400Mb/s

2.5Gb/s

10Gb/s

40Gb/s

160Gb/s ?

e-mail HTML e-commerceMusic Video

3.6 Tb/s (research, TDM/WDM)

ALL-OPTICAL(TDM/WDM)

ELECTRONICOPTOELECTRONIC

Fiberoptics and optical processing was and still is the most powerful driver of semiconductor based Photonics

- THz Bandwidth- unmatched low dispersion and attenuation

(< 0.5 dB/km) of glass fibers- small size, although micro-minaturization is

a challenge- the electronic bottle-neck is limiting although

200 Gb/s electronics will emerge- 150 million fiber-km deployed

Tb/s hero experiments:25 Tb/s OWDM2.5 Tb/s OTDM


1.4 State-of-the-Art of Photonic Devices / Applications

1.4.2 Tb/s Technology • Experimental OWDM / OTDM-Systems

have broken the 10 Tb/s-barrier• Electronics is moving up towards data rates

of 200-300 Gb/s• Technology is feasible, but challenges still are

- sub-ps pulse generation for OTDM- dispersion compensation- extreme sub-nm wavelength control for

sources and filters for OWDM-systems

300 GHz UTC-photo-diode:

1-7

500 fs InP-based all-optical switch (H. Melchior et al. , ETHZ)


1.4 State-of-the-Art of Photonic Devices / Applications

1.4.3 Display and Illumination Technology • semiconductor based, current pumped

Light Emitting Diodes are very efficient and cover the visible spectrumfrom 380nm – 900nm

• all solid state with unlimited life-time• low cost

1.4.4 Image Sensing • Band-to-band transitions in semiconductors

provide charge images in CCD or diode 2D-arrays in the visible (silicon) and IR (InGaAs)range

• pixel counts up to 10-20 million• huge industrial and consumer potential

1-8


1.4 State-of-the-Art of Photonic Devices / Applications 1.4.5 Optical Data Storage • high density and capacity by contact-less

coherent laser-beam • read-only and erasable 2D-storage media

metallic and amorphous thin-films, magneto-optic optic thin-films

• wavelength scaling towards 300-400nm

• potential for ultra-high capacity 3-D volumedata storage ?

1.4.6 Optical Pump and Power Sources • continuous wave (CW), 500mW power diode laser for fiber laser pumping

• continuous wave (CW), kW- high power diode laser arrays for Solid-State Laser-Pumping

1-9


1.4. State-of-the-Art of Photonic Devices / Applications

1.4.7 Optical Integration Motivation: • conventional optical devices are large (typ. 100mm – cm)

low integration density• weakly guiding optical (Δn<<1) wave

guides have large bend-radii (mm)

Challenges:

• device materials and structures are very heterogeneous

complex fabrication technologydifficult lateral coupling betweenheterogeneous waveguide-sections

• sub-micron control

1-10

residualroughness

1.5μm

SOA WG

M.Smit, COBRA-TU Eindhoven

Optical amplifier / passive waveguide



1.4.8 Nano-Optics

Motivation: • scaling of optics to the dimension of theoptical wavelength- high contrast (large Δn) waveguides- Photonic Crystal WG (optical bandgap)

• Manipulation of spontaneous emission life time

• efficient dispersion manipulation(slow light)

• photon localization to the limit

Challenges:• nm-technology and processing• active photonic devices • interfacing with conventional devices

1-11

Ref

M.K.Chin et al., 1999

5μm

High contrast photon wires Photonic Crystals

Photonic Crystal Diode Laser Photonic Crystal Resonator with Quantum Dot



1.4.9 Optical Measurements • laser ranging and radar• fiber gyroscopes for rotation sensing• optical fiber sensors for temperature, stress, voltage and current• small displacement• flow and velocity measurements• Sub-wavelength Scanning Near-field Microscopy• X-ray laser sources and Nuclear Fusion• fs-second spectroscopy and material characterization

1.4.10 Medical Applications • laser surgery• fiber endoscopy• biostimulation and photoradiation therapy• bio-sensors based on waveguides

1-12


1. Introduction

Summary: • THz bandwidth, ultra-low dispersion and attenuation make fiberoptics the

winning technology

• Semiconductors access the visible (0.3μm) and IR (3μm) - and partially FIR (20μm) –wavelength range

- injection pumping- SC-planar processing and micro-miniaturization- SC-epitaxy

• Photonics Integration makes slow but steady progress and will be mandatory infuture optical signal processing applications

• in measurement technology optics provides the fastest controlled events

• Photonics is an enabling technology in many application fields

• for low cost applications organic photonic materials are highly desirable, but thereare still major unsolved challenges

1-13


2. Classical Propagation of Light Waves

Goals:• Generic light wave propagation in polarizable, lossy and non-isotropic media • Generic description by formal frequency-dependent parameters ε, n, α, … • Charge-motion and resulting polarization by the classical harmonic oscillator model for non-

resonant excitation• Field and potential representation of electro-magnetic fields (no photon concept)

2-1


2. Classical Propagation of Light Waves2.1 Basic Optical Processes (prerequisite from Physics I&II)

Wave properties of photons are described by Maxwell’s equations neglecting the particle properties and the quantisation of the EM-energy.• EM-waves do not interact without the intermediation of matter• the interaction with matter results from the EM-forces (electrical force, magnetic force) of

the light field (E-, H-field) on the charges in matter:- quasi-free mobile charges (metals)- weakly bound, mobile charges in the atomic lattice (SC) - strongly bound charges in atoms

• Charge motion in matter reacts back by its own field on the driving field and modifies it α, n, ε

2.1.1 Wave propagation and basic optical processesHarmonic EM-plane waves are characterized by:- oscillation frequency ω- wavelength λ and a phase velocity vph=ω/k, resp. propagation constant k- amplitude of the field-vectors- polarization, direction of the field vectors

Photonic devices: modifying wave properties by interaction with matter

( ) ( )E r ,t , H r ,t

2-2

( ) ( )0

j t krrE r ,t E e e ωα −−=


The wave propagation in matter is modified (attenuation k, α, wavelength λ, intensity I, polarization, direction) :

in linear materials ω is not modified !

Reflection Propagation Transmission

by the processes of:Reflection R /Transmission T: caused by the refraction difference at interfaces Refraction: reduction of wave velocity vph(ω)=c/n(ω) ; n=refractive indexAbsorption: attenuation / amplification of intensity I(z)=I(0)e-αz ; α(ω)=absorption coefficient

Luminescence: undirectional light emission by recombination in excited matter Scattering: light attenuation by redirection of propagation and possible change in ω

(elastic and inelastic scattering)

2.1.1.1 Basic Relations between optical propagation constantsThe wave motion of photons is described by EM-waves obeying Maxwell’s equations. EM-waves are described by 2 coupled vector-fields with 6 components:

2111

nR Tn

−⎛ ⎞= − = ⎜ ⎟+⎝ ⎠

2-3


Not all components are independent. It can be proven that only 2 components are independent.

Electric and magnetic Polarization of isotropic and linear materials:The E-field vector induces a separation of positive and negative charge centers (e) forming atomic dipoles

Polarization density P with a dipole density N:

Analog for magnetic dipoles:

Maxwell’s Equations:

- Field Equations: - Materials equation:

2-4

( ) ( )( ) ( ) ( )( )x y z x y zr ,t r ,tE r ,t E ,E ,E , H r ,t H ,H ,H= =

p( E ) ed( E )= −

( )

0

0 0 01 1r r

definitionP Np( E ) Ned( E ) E ; susceptibility

and dielectric flux

D E P E E ; relative dielectric constant

χε χ χ

ε ε χ ε ε ε χ

= = − = =

= + = + = ⇒ = +

0 0 1M r r MM H and B H M Hχ μ μ μ μ χ= = + = ⇒ = +

2 2 22

2 2 2

0free charg e

conduction displacement with , , ;x y z x y z

D

BE B

t

B j j j Dt

ρ ρ

∂ ∂ ∂ ∂ ∂ ∂∇ = ∇ = Δ = + +

∂ ∂ ∂ ∂ ∂ ∂

∇ ⋅ = =

∇ ⋅ =∂∇ × = −∂

∂ ⎛ ⎞⎛ ⎞∇ × = + = + ⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠

( )0 0

0

1r

r

cond

D E E

B H

j E

ε ε ε χ

μ μ

σ

= = +

=

=

d

-p

dk

E

+

Potential-Representation of EM-fields: (optional, without proof)

Definition of a Vector-Potential and scalar Potential representing the fields E, H:

Maxwell’s equations for and : (ref. J.Singh, p.346-351)

The vector potential leads to a simple relation for the field-energy density, resp. photon density nph of the EM-field and finally also to a simplified Schrödinger-equation for an electron in the EM-field.

Using the definitions of E and B and the Poynting-vector S relates the energy density to A:

( )( ) ( ) ( ){ }

2 202 2

2 0 0 2

2

2 20 0

1 1 1 0for JA ; const

o o o

A A A J A At t t

At

Wave solution for vector potential A r ,t : A r ,t A exp j k r t cc ; k

ρφε φ ε ε

μ μ μρφε

ω εμ ω

= =∇⋅ = = =

⎛ ⎞∂ ∂ ∂∇ ∇ ⋅ + − ∇ + = ⎯⎯⎯⎯⎯⎯→ ∇ − =⎜ ⎟∂ ∂ ∂⎝ ⎠∂ ∇ ⋅ + ∇ = −∂

⎡ ⎤= − + =⎣ ⎦


2-5

( )A r ,t ( )r ,tφ

( )A r ,t ( )r ,tφ

( ) ( )

( )

( )

22 20

0

220 0

0

22 20

4

1 2

2

ph e ph e

T

ph e

ph

S r ,t E H v k A sin k r t k with v c / and k k / k

By time averaging : dt S r v A kT

energy density : S r / v A / c

ω εμ

εμ ω

εω

= × = − = =

− = =

⇒ =

∫

( ) ( )

( ) ( )

0B r ,t A r ,t and gauge ( coulomb ) A

E r ,t A r ,tt

φ

= ∇ × ∇ ⋅ =∂= − − ∇∂

(wave eq. for A)


Remark:In quantum mechanics the vector potential A is often used because it leads to a very simple description of the interaction Hamiltonian between electron and EM-field.

Wave-equations for the E- and H-field for current and charge free media:loss-less medium

Elimination of one field variable from MW-equations by elementary vector-operations yields the wave eq.

Generic harmonic (ejωt) plane wave-solution with propagation constant k(ω):

With the resulting propagation constant k(ω):2-6

0 0 0j E ; ;σ σ ρ= = = =

( )( )

( )

2 22

0 0 2 2 2

1r r

r r

r r

cE E E with vt v t

cand the definition of the : v n realn

ε μ ε με μ

ε μ

∂ ∂∇ = = =∂ ∂

= ⇒ =refractive index n

( ) ( ) ( )( ) ( ) ( ) ( )0 0 0 0j k r t j kz tE r ,t E , e resp. in z direction : E z,t E , eω ω ω− −= − =

( ) ( )2 22 20 02 2

ph

ph ph ph

in the photon - picture : n = local photon density in the optical mode

S r / v A / c n A n /εω ω εω= = → =

( ) zGeneric solution : E z,t ~ f tv

⎛ ⎞⇒ −⎜ ⎟⎝ ⎠

u sin g : jt

ω∂ ≡ ⇒∂

( ) ( ) ( )2k n realv c

π ω ωω ωλ

= = =


Wave-equations for the E- and H-field for lossy (conducting ) medium:Again the elimination of one field variable from MW-eq. by elementary vector-operations yields

Defining the complex index of refraction :

Complex index of refraction and dielectric constant :defining

2-7

0j E ;σ ρ= =

( )( )

( ) ( ) ( )

( )( )

0

22

0 0 2

2 20 0 0

0 0

wave equation with lossesr r

j kz t

r

r r r

E Et

with a damped or growing wave solution of : E z,t E , e

k j complex propagation constant k

Et

ω

ε μ ε μ

σμ μ

σμ

ω ε μ ε μ ω

μ

ω

−

∂∂

∂∇ = +∂

= ⇒

⇒ = +

n

( )k n k complexcω= =

n rε

( )

( )

describes propagation velocity ; wavelength describes dattenuation and gaindefinitionn n j ; n real index of refraction ; extinction coefficient of the field amplitude

relates to :

k n jc

κ κ

ω κ

= − = =

= −

0σ >


Interpretation of : damping of the wave

Introducing in analogy to the relation a complex dielectric constant:

Real- and imaginary parts of are coupled ! (see also Kramers-Kroenig-relation)

Maxwell’s equations do not specify anything about the nature of the polarization and its dynamics in the time- or frequency domain (assume instantaneous response of the dipole).The polarization is given by the microscopic properties of the different optical materials. 2-8

( ) ( )0 0 0

2 4

j n z t j n z tzj kz t c cc

dampingE z,t E e E e E e e

from the intensity - damping factorc

ω ωωω ωκω

κω πκα αλ

⎛ ⎞ ⎛ ⎞− −−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠= = =

⇒ = =

( )

( ) ( )

2

2 2 21 2 1 10

22 02

2

rn j n j n n

nn

κ

κ

ε ε ε κ ε κ ε

εε κ κ

→+

→+ +

⎛ ⎞= = + = − ⇒ = − ⎯⎯⎯→ =⎜ ⎟⎜ ⎟

⎝ ⎠

= − ⎯⎯⎯→ = −

n and ε

k

rn ε=


2.2 Basic optical materials Common optical materials are classified into the categories of:- crystalline insulators and semiconductors (for lasers, optical amplifiers, modulators, detectors, …)- glasses (for fibers and waveguides, coatings)- metals (electrical contacts, mirrors)- doped glasses and insulators (lasers, optical amplifiers)- molecular organic materials

The different material properties originate from the different interactions (polarization, conduction) between light and the motion of free or bound charge components of the material.

Atomic / molecular dipole: (local, bound) Lattice interaction: (quasi-free, distributed)

unit-cell a 2-9

- + - + - +


2.2.1 Crystalline insulators and semiconductorsThe distinction is mainly by the transparency region and the bandgap Eg,which is high in insulators such that their free carrier density is small at RT.Interacting mobile electrons are bound to the atoms of the crystal lattice.

Dominating interactions of the EM-light wave are with -polarization effects of bound electrons- conduction effects of free electrons - lattice vibration effects (phonons)

Typical properties:- low attenuation (α, k, n2 , ε2≈0) in transparency range

(low absorption between 2 strong absorption bands)

- high losses at low (phonon-absorption in FIR-region ~5μm range)and high optical frequencies (band-to-band UV-absorption of bound electrons ~0.3μm)

- depending on lattice symmetry, the properties canbe anisotropic (eg. birefringence)

- semiconductors (SC) have relatively high refractiveindices n~3 decreasing with increasing bandgap Eg

- SC can also have substantial free-carrier absorption at low frequencies

Most optical materials are non-magnetic μr=1. 2-10

gEω <

transparent

transparent

D

ngr

n

D

Dispersionminimum

n, ngr

2.2.2 GlassesGlasses are based on amorphous oxides (eg. SiO2) with similar characteristics as crystalline insulators• ultra pure glasses have low loss (α<1dB/km) and are the material of choice for optical fibers• over the visible and NIR-range the frequency dependence of n(ω) and the resulting pulse

broadening, resp. dispersion ngr(ω) and D(ω) are relatively weak (see chap.2.2.3)• control of the refractive index n by addition of metal-oxides

• Attenuation: (silica) Dispersion:(silica)


2-11

~20 THz ~20 THzOH-absorption


2.2.3 MetalsMetals contain a large number of unbound, free electrons interacting strongly with EM-light waves and screen the metal from the EM-wave in the visible and IR-range.

Typical properties are:- high reflectivity up to the plasma frequency ωp

(in the UV-range)- for ω>ωp the metal becomes UV-transparent- residual interband transitions are present

2.2.4 (Molecular materials) (not course topic, see ref.Fox)

Although organic materials are becoming interesting for many applications due to their properties- good transmission properties in the visible and NIR- strong polarizability, high nonlinearity- low dielectric constant- active lasing transitions

and in particular due to the easy fabrication technology and compatibilities to SC.Major drawback of molecular materials is the limited temperature stability (glass temperatures ~200-3000C)

2-12


2.2.5 (Doped Glasses and crystalline Insulators) (not course topic)

• Glasses as a matrix are often doped by nano-size particles (metals, SC, etc.) and the nano-scale particles show size-dependant scattering and absorption effects.

• Glasses and crystalline insulators are used as transparent host-matrix for doping with optically active metal-ions, which provide efficient lasing transitions.

The interaction of the dopands with the crystal-potential of the host may modify the atomic transition-levels

Example: Ruby as a Cr-doped Al2O3 (first Ruby-laser):

Other examples: Nd-doped Yttrium-Aluminum-Garnet (YAG)

Nd-doped Glass

2-13


2.3 Microscopic Polarization Models • EM-waves exert electrical and magnetic forces on bound or free charges in the crystalline solid.

Electric force:

Magnetic force: (Lorentz-force)

• Moving charges re-radiate EM-fields modifying the original fieldBecause in SC the charge velocities are small the Lorentz-force can be neglected

• Optical phonons also produce quantized oscillating lattice dipoles interacting with the EM-field

2.3.1 Dipole Oscillator of bound chargesMaxwell’s equations contain the polarization P(E), without specifying the microscopic charge motion.

Classically, atoms in a EM-field are considered as harmonic oscillators, where electrons of mass memove in a local “parabolic” potential Veff(r) of the heavy nucleus of mass mN around the equilibrium charge position r0:

repulsive equilibrium attractiveforce force 2-14

EK qE q V= = − ∇ ⋅

( )1HK / c v B= ×

r

Veff

r0

+ _( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

00

0

22

0 0 02

2

12

02

eff effeff eff

r

Seff eff

r

r r r'

parabolic potential

V VV r V r r r r r ...

r ! r

KV r' V r'− =

∂ ∂+ − + − +

∂ ∂

⎯⎯⎯⎯→ = +

2

02 0e S eff S

Equation of motion in the local potential withoutexternal time dependent field :

m r' F V ( r') K r' with equilibrium position r' , r rt r

−∂ ∂= = − = − = =∂ ∂

displacement potential


The atomic charge center separation d=r+-r- creates a dipole p:

The equilibrium restoring force FS is assumed to be linear in (r-r0)=x, therefore the corresponding potential Vs(x) is parabolic with x=0 as equilibrium:

KS=atomic force constant (unknown)

The classical model, which can not represent the energy-quantization of real atoms, is useful for oscillation frequencies ω far away from absorption at the atomic resonance-frequency ω0.All motion parameters KS, me have to be fitted from macroscopic measurements.

Only a quantum-mechanical treatment provides a self-consistent description of the quantized charge motion close to resonance, where energy exchange (absorption) is important.

Classical Equation of motion including damping γ in an electrical field E: (eg. collision energy or radiation losses of the atom)

Newton equation of motion

2-15

( ) ( ) ( ) ( ) 0p E q r r qr E qd E with r+ − − += − = − = − =

( ) ( ) ( ) 22S S S SF x K x V x K / x= − ⇒ =

( )

( ) ( ) { }( ) ( ) ( ){ } ( ){ }

( ) ( )

2

2

0 0

0 0 0

00 02 2

0

e e S X

j tX

j t j t

e S

e

m x m x K x eE tt t

For a harmonic field : E t E cos t E Re e

and a harmonic solution x t X cos t X Re e Re X e

eE / m KStationary Solution : X with resonance frequencymj

ω

ω φ ω

γ

ω

ω φ ω

ω ωω ω γω

−

− + −

∂ ∂+ + = −∂ ∂

= =

= + = =

−= =− −


From the definition of the microscopic polarization p(ω)=-eX(ω) the macroscopic komplex dielectric constant ε(ω) or the refractive index n(ω) can be determined:

We get for the relative dielectric constant

Simplifications close to resonance:

phase lag I phase lead normal I anomalous dispersion

2-16

( ) ( ) ( )( ) ( ) ( )

2

0 0 2 20

2

0 0 0 0 0 0 0 0 0 02 20 0

1

1

resonante

background resonant bg re

NeP Np NeX Em j

Newith D E P P E E E Em j

ω ωω ω γω

ε ε ε χ ω ε ε ε ωε ω ω γω

= = − =− −

= + + = + + =− −

( ) ( )( )

( ) ( )

( )( ) ( )

2

1 22 20 0

2 220

1 2 22 20 0

2

2 2 22 20 0

11

1

r bge

bge

e

Ne jm j

Nem

Nem

ε ω χ ε εε ω ω γω

ω ωε ω χε ω ω γω

γωε ωε ω ω γω

= + + = +− −

−= + +− +

=− +

( ) ( )( ) ( ) ( )

2 20 0 0 0

2

20 0

2

10 1 1r st re

Neand andm

ω ω ω ω ω ω ω ω ω

ε ε χ ε ε χε ω ∞

→ Δ = − → − ≈ Δ

= = + + ∞ = = +

( ) ( )221 2r j n n jkε ω ε ε= + = = −

ε1, ε2: n, κ:


Properties:• Absorption α (imaginary part) is only relevant very close to resonance ω0, in contrast n is

effective over a large frequency range.

• Below resonance ω0 the phase angle of εr is >0 meaning that the polarisation P lags the driving field E and therefore delays its propagation. This leads to a shorter wavelength λ=λ0/n than in vacuum. Above resonance the situation is reversed.

• Below and above resonance the real refractive index n(ω) increases with ω

→ normal dispersion, “blue” light propagates slower than “red”

• In the vicinity (above) of resonance the real refractive index n(ω) decreases with ω

→ anomalous dispersion, “blue” light propagates faster than “red”

Multiple ResonancesAtoms and crystals have a multitude of optical resonances / transitions. If the transitions are inde-pendent the polarization can be described by the superposition (sum) of the individual transitions:

2-17

( ) ( )( )

( ) ( )( )

01 2 2

0

02 2 22

2 14

14

st

st

;ω ω ωε ω ε ε ε λ γω ωω λ

ω γε ω ε εωω λ

∞ ∞

∞

ΔΔ = − − =ΔΔ +

Δ = −ΔΔ +

∼

∼

( ) ( ) ( ) ( )2 2

0 2 2 2 20 0

1 1j jj j r

j j je e, j j , j j

j

N e fNeP N p Em mj j

f oscillator strength

ω ε ωω ω γ ω ω ω γ ω

= = ⇒ = +− − − −

=

∑ ∑ ∑

minimum of vgr


2-18

Electronic interband UV-absorption λ~150nm

SiO2-molecularFIR-absorption λ~9 / 21 μm

( ) ( )

( ) ( )

( )

1

1

ph

gr

nDispersion relation : k

ck cPhase velocity : v

n

kGroup velocity : v

ω ωω

ωω ω

ωω

−

−

=

⎛ ⎞= =⎜ ⎟⎝ ⎠

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

Schematic dependence of α(ω) and n(ω): α(ω) and n(ω) of fused silica (SiO2) glass:


2.3.2 Kramers-Kronig-Relation (proof see lit. A.Yariv)

As the real and imaginary parts of complex refractive index have the same dynamicorigin, it is obvious that the must exist a relation between them.

resp.

Satisfying the condition of causality:

P=principal part of the integral

Consequences:• if the complete frequency dependence of n(ω) or κ(ω) is known, eg. by measurement, the other

one can be calculated in principle.

• practically it is important, that any change in optical gain or loss g(ω), resp. α(ω) results in a change in the refractive index n(ω) and vice-versa.

2-19

( ) ( ) ( ) ( )00 0 02 2

0

eeE / mX X ' jX ''j

ω ω ωω ω γω

−= = +− −

( ) ( )2

1 22 20 0

11re

Ne jm j

ε ω χ ε εε ω ω γω

= + + = +− −

( ) ( ) ( )

( ) ( )

( ) ( )

11

11

n n j

'n P d '

'

n 'P d '

'

ω ω κ ωκ ω

ω ωπ ω ω

ωκ ω ω

π ω ω

+∞

−∞+∞

−∞

= −

= +−

−= −

−

∫

∫

( ) ( )rn ω ε ω=

2.3.3 Dispersion: phase and group velocityFor practical applications the velocities of the carrier wave phase vph(ω) and the envelope- or group vgr(ω) of a pulse-modulated EM-wave are important.

If the dispersion relation k(ω) is known we will obtain: (see exercise)

At λz the positive and negative GVD of “below” and“above” UV- and FIR-resonance cancel → GVD=0.

The classical model describes n(ω), ngr(ω), β2 farfrom ω0,j accurately if fitted with data for Ks, fj, me,j,… 1-42

GVD>0below UV-resonance

GVD<0 λZabove IR-resonance

GVD=0λ~ 1.3mm (glass)


( ) ( ) ( )

( ) ( )

( )

( )( )

0

1

1 1

00

00

ph

gr

gr

nDispersion relation : k k n

ck cPhase velocity : v

n

kk nGroup velocity : v n k

cnc / nn

Defining GVD group velocity dispersionas a measure of pulse envel

ω ωω ω

ωω ω

ωω ω ω

ωω ω

−

− −

= =

⎛ ⎞= =⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞∂∂ ∂= = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞∂= + =⎜ ⎟∂⎝ ⎠

( ) 2

2 20 0

1 2gr

op distortion :

/ v n nGVDc c

ωβω ω ω

∂ ⎛ ⎞∂ ∂= = = +⎜ ⎟∂ ∂ ∂⎝ ⎠

vgr

vph

2-20


2.3.4 Optical propagation in anisotropic media: BirefringenceInitially we made the assumption that the polarization displacement and the dipole moment are parallel to the field vector as expected for an isotropic solid.

In anisotropic solids, eg. caused by the anisotropic crystal structure, the vectors need not to be parallel and the scalar properties of χ become tensors:

susceptibility tensor

Choosing x,y,z to be the principle axis of the crystal the susceptibility tensor χij becomes diagonal:

If the susceptibility becomes a tensor, it is to be expected that the refractive index depends on the propagation direction , resp. propagation vector and the polarization.

The question is: what are the wave solutions of Maxwell’s equation in such an anisotropic media ?

It can be proven from Maxwell’s equation, that:

the propagating wave in an anisotropic medium is the sum of 2 waves a and b propagating at different velocities, resp. with different refractive indices na, nb.

2-21

d ( )p EE

11 12 13

0 0 0 21 22 23

31 32 33

x x

y y

z

ιj

z

P EP E P E P E

P E

χ χ χε χ ε ε χ χ χ

χ χ χ

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟= ⇒ = = =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

χ

( )11

122 0 0

033

0 010 0 1

0 0ij ij ij ij

Definition

'' D E E or E D D

'

χχ ε ε ε χ η ε

εχ

−

⎛ ⎞⎜ ⎟ ⇒ = = + = =⎜ ⎟⎜ ⎟⎝ ⎠

χ =

ks( )n k

2.3.4.1 The technique of the refractive index ellipsoid: (proof see lit. A. Yariv)

The crystal lattices of III-V Semiconductors are often centrosymmetric and isotropic (eg. GaAs, InP).

• semiconductors often show a natural or a field-induced anisotropy (electro-optic (EO) effect)• anisotropy and EO-effect (nii(E)) are important for SC light-modulators and for phase-matching

The propagation of the 2 wave solutions and are described by the refractive index ellipsoid:

For a given propagation direction the fields and are found by:

• intersecting the normal-plane to the propagation vector with the index-ellipsoid

• determines the intersection ellipse. The orthogonal principle axis of the intersection ellipse give

1) direction of and and 2) the axis-length are equal to na and nb

• using the dielectric tensor εij to calculate from the known completes the solutionMaterial Properties and Characterization: Optical Processes in Semiconductors Electronics Laboratory, ETH-Z 01.12.2006

2-22

sDa

Db

Eb

Ea

na , λa

nb , λb

anisotropes Material

1b

6b

6a

1a Β=Δφ

2 2 2

2 2 21xx yy zz

x y zn n n

= + +

b bD , na aD , n

intersection ellipse

kβ = a a aD , E , n b b bD , E , nβ

a bD ; DbD

a bE ; EaD

Assuming the known principle axis εij tensor, we define the corresponding refractive index tensor nij:

n11 , n22 , n33 define the index ellipsoid and the anisotropy of the crystal.

Isotropic: n11=n22=n33 (sphere) Uniaxial: n11=n22≠n33 (uniaxial ellipsoid) ; Biaxial: n11≠ n22≠ n33 (ellipsoid)

Semiconductors are often isotropic. Chap.7 shows that external fields can modify the coefficient nij

Typical Properties:

2

211 11 11

222 22 22

233 33 33

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 0

r

ij ijdefinitionu sin g n

' n' n

' nε

χ εχ ε

χ εε

=

⎛ ⎞+⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟+ = = ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= 1 + χ =

2-23



2.3.5 Lattice oscillationsThe EM-field also interacts with the charged crystal lattice introducing lattice polarization.

Phonons are collective oscillations of the crystal-lattice. The atoms are bound by linear forces into their equilibrium crystal position. Typical III-V-compound semiconductors (eg.GaAs) have different lattice atoms with different masses and charges producing dipole-oscillations.

(details see chap.4.5)

2.3.6 Free electron oscillationsOscillations of free electrons without restoring forces (KS=0) is typical for metals.

(details see chap.3)

2-24


2. Classical Propagation of Light waves

Summary:• Propagation of light waves is characterized by the complex refractive index n(ω), where the

real and imaginary part ,n and κ describe the phase velocity and the attenuation.

• The real and imaginary part of the complex dielectric constant are related by the Kramer-Kronig relation. A change in attenuation/gain results in a change in refractive index.

• Off-resonance polarization of bound electrons in atoms and molecules can be modelled by a classical harmonic mechanical oscillator with a resonance at 1014-1015 Hz

• Free electrons in metals can be described by a dipole oscillator with the selfresonance ω0= 0

• The polarization slows down the light wave as described the refractive index n. The slowdown is caused by phase delays in the multiple scattering process of the dipoles (coherent elastic scattering).

• Losses lead to frequency-dependent attenuation α and extinction κ close to the resonance frequency

• The frequency dependence of the refractive index n is responsible for dispersion effects.The frequency dependence is given by the structure of multiple resonances in the solid.

• Optical anisotropy must be described by tensor-relations leading to orientation-dependent propagation properties and birefringence

• The classical description does not describe the quantization of the field energy leading to the concept of the photon

2-25


Appendix: 2A A 2.1 The EM-field as a quantized harmonic oscillator: (optional, see Lit. P.K. Basu, p.37)

Without going into the details of the proofs we want to demonstrate that the EM-field described by Maxwell’s eq. can be represented by a quantum mechanical harmonic oscillator with equidistant energy levels En.

Classical mechanical harmonic oscillator and its Hamiltonian H(p,q):

Momentum p and coordinate q are the generalised system coordinates.

To connect to the energy representation of the harmonic oscillator we consider the vector-potential representation of the EM-field:

2-26

( ) ( ) ( )

20

2 2 22 2 2 2 2 2 2

0 0 0

0

2 2 2 2Quantum Mechanicsp ,q momentum, positionoperat r

r

oq r ; p i

p kx p kq with x q and the resonance frequency k / mt t

p k p pT ; V q q mq H p,q mq H p,q mqm m m

ω

ω ω ω

∂∂

= =−

∂ ∂+ = + = = =∂ ∂

= = = → = + ⎯⎯⎯⎯⎯⎯⎯→ = +

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

22

2 2

2 22 2

2

22 2

0 02

1

1i

j t jk ri

The vector potential A r ,t obeys the wave equation :

A A separating the solution : A r ,t X r A tc t

cX r A t constA t tX r

A t A e and X r X r k X r X r X ec

ω

ω

ω−

− −

∂∇ = =∂

∂∇ = = = −∂

→ = ∇ = − = − → =


Transforming the vector potential variable A by 2 new general “momentum” und “position” variable P,Q:

Generalized harmonic oscillator equation for the EM-mode k (classic)

Hamilton-function for the harmonic oscillator:

2-27

( ) ( )

( )

( ) ( )

( )

2 2 2 0 1 2

k

k

x y z x y z

j t jk rk

k

j t jk rk k k

k k

Quantization by periodic boundary condition :

A r ,t A r L,t

k n , n , n n , n , n , , ,...L L L

A r ,t A e cc

per definition : E r ,t A r ,t E j A e cc andt

H r ,t

ω

ω

π π π

ω

− +

− +

= + →

⎛ ⎞= = ± ±⎜ ⎟⎝ ⎠

= +

∂= − = = +∂

∑

∑ ∑

( )0 0

20

1

12

2

kj t jk rk k

k k

o oV

k k k k

jA H k A e cc

Field energy : E E E H H dV

Time averaging for the k th mod e : E A A V

ω

μ μ

ε μ

ε ω

− +

∗

= ∇ × = = × +

= ⋅ + ⋅

− = ⋅

∑ ∑

∫

( ) ( )

( )

120

2 2 2

4

12

k k k k k k k

k k k k

A V Q jP e with unity vector e

E P Q

ε ω ω

ω

−= + −

⇒ = +

( ) ( )2 2 2 2 2 2 21 12 2QM : p,q are operatorsClassic : H p q , k / m Quantum Mechanic : H p rω ω ω= + = ⎯⎯⎯⎯⎯⎯⎯→ = +


For the solution of the stationary Schrödinger-equation:we obtain the equidistant energy eigenvalues En of the system (after some lengthy calculation, see. eg. Lit. A.Schiff)

Interpretation:

EM-fields are quantized in a way similar to the quantization of the motion of an bound electron, described by a quantized harmonic oscillators.The evolution of the energy of the light-field can be represented by a quantized harmonic oscillator.

The EM-field is quantized into equidistant energy levels separated by the energy quanta ħω (Photons).A mode in the energy state En contains contains n+1/2 photons.

A mode containing no external field, resp. no external photonsn=0 contains in its ground state still an energy E0=1/2 ħω (zero point energy).

This energy is called the “vacuum-energy” and can be seen as the origin of the spontaneous emission process as a stimulated emission by the vacuum-field.

( ) ( ) 1 0 1 22n n n nH r E r E n n , , , ...ψ ψ ω⎛ ⎞= → = + =⎜ ⎟

⎝ ⎠

2-28


3. Optical Properties of Metals: quasi-free electrons

copper single crystal

Goals: copper single crystal

• Highly mobile electrons in metals show a strong interaction with EM-fields resulting in ahigh reflectivity up to the plasmafrequency

• scattering losses of mobile carriers also introduce substantial losses• reflectivity and losses are important for practical devices

3-1


3-2

3. Optical Properties of Metals: quasi-free electrons Metals and doped semiconductors contain a large number of free, unbound electrons which can interact with light fields and influence the dielectric and absorption properties.Metals play an important role as optical reflectors or electrical device contacts, heavily doped semiconductors are incorporated in many optoelectronic semiconductor devices. Therefore, the frequency dependence of reflectivity and losses are important. In contrast to the bound, polarizable electrons in insulators, we can consider these free-carrier systems as the collective motion of the electrons without a restoring binding force.

3.1 Drude-Lorentz-model of reflectivity of metalsAs the simplest model we describe the motion of the free electron by setting the atomic force constant KS (chap.2.3.1) to zero KS=0, ω>ω0=0 → χ <0. The resonance frequency ω0 becomes zero, ω0=0 and the displacement phasor X0(ω):

The plasma frequency ωp increases with higher carrier density

( ) ( ) ( )

( )

( )

0

2

00 0 02

0 0 0

22 2

2

1

0 0

1 1

e

0 r 0 0

pr p

0 e 0 e e

Assuming a free electron density N, we obtain :

D = X

plasma frequencyj

eE / mX X ; Xj

E E P E eN

e N e N Nwith them j m m

ε ε ε ε ω

ωω γω

ωω γω

ωε ω

ε ω γω ε+

= = ∞ ∞ =+

= + = − ⇒

= − = − =+

∼


3-3

For weakly damped electron motion γ=0 n imaginary n real

In realistic metals there are loss or damping processes, such as the carrier scattering (with scattering times τ), which also causes ohmic losses, described by a finite conductivity σ(τ) (simple to measure).

To relate the damping constant γ to τ, resp. σ(τ), we transform the equation of electron motion:

( ) 2

2

2

1

0 1

0 0 1

0 1 1 1

11

pr

p r r

p r

p r r

: negativ, n imaginary R

: n R

: positive, , n positive, R

nR intensity reflection coefficientn

ωε

ω

ε

ε

ε

ω

ω ω ε

ω ω

ω ω ε

= −

< = < = = =

= = = =

> = > < = = < <

−=+

( )

( )

( )

( )

( )

2

2

0

0

0

2

0

1 1

11

1 11 1

e e X e e

x

e

e

p xm x m x p eE t with the momentum p m v m and /t t t t

for E E cos tev E carrier velocitym j

the velocity v produces a current density j Nev E

e N with the scatteringm j j

γ γ ττ

ωτω

ωτσ ω

τσ ω σ τωτ ωτ

∂ ∂ ∂ ∂+ = + = − = = =∂ ∂ ∂ ∂

=

= −−

= − = ⇒

= = =− −

( )23 1 14 1310 10 10

scattering time limited conductivity

For metals : N cm and s

σ

τ− − −−


3-4

The frequency dependent conductivity σ(ω) determines the relative dielectric constant εr of the metal.

Using the relation from chap.2 for lossy dielectrics:

The table shows that metals are good reflectors up to the UV-region (ωp) where they become transparent.

For realistic modelling we have to use the effective mass of the metal electron.

The reflectivity at IR-frequencies is less than 1 (80-90%) and drops towards the UV-region. Metals are not excellent mirrors at IR-wavelength, making low-loss dielectric Bragg-reflectors necessary.

( ) ( )

20

2

2 2 20

1

1 1 111 1 1 1 11 1p

r0 e 0 e 0 e 0 0/

usingj jj

e N e N e N j jm m m j

σ ω τ

γ τε

ω γω ωτω

σ ωστωε ε ω ε ω ε ω ωτ ε ω

τ =

=+ ++

⎛ ⎞= − = − = − = + = +⎜ ⎟ −⎝ ⎠

( )

( )

22 2 21 2 1 2

2 2 2

1 2 12 2 2 2

22

0 02

2

11 1

2

22 22

r

p p

p

we get after some algebraic manipulation for small << 1/ :

n j n j n and n

;

and the wave intensity attenuation / c

/~

c cThe penetr

ω τε ε ε κ ε κ ε κ

ω τ ω τε ε ε

ω τ ω ω τ

α κω

ω τωω εα σ ωμ ω

= = + = − ⇒ = − = −

= − = >>+ +

=

= = =

0 0

2 2

ation (skin) depth δ of the field into the metal

into metal at low frequencies is : δα σ ωμ

= =


3-5

3.2 Interband absorption effects in metals (optional)The Drude-Lorenz model of free metal electrons is an oversimplification because metals also show a bandstructure which allows direct transitions between the bands (interband transition with k-conservation).

These interband transitions absorb part of the slightly penetrating wave at frequencies ω<ωp and thus reducethe reflectivity R from the high free carrier value below ωp.

Red coloured copper is an example where this interband absorption is very significant (for details see lit. M.Fox).

3.3 Free carrier effects in doped semiconductorsStrongly n- or p-doped semiconductors have high concentrations of quasi-free electrons (n) and holes (p), which cause also plasma effects and free carrier absorption.

Because the carrier concentrations (typ. <1018-1019 cm-3 compared to 1023cm-3 for metals) are low, the plasma frequency ωp is also low and may cause absorption even below the bandgap absorption edge ωg=Eg/ħof the semiconductor.

In addition free carrier losses can be substantial in transparent, low loss semiconductor waveguide structures.

In the following treatment we will incorporate the considerable background polarization of bound electrons in SC by Pbound:

InterbandAbsorption


3-6

Pbound is contained in εopt

The free carrier plasma frequencies in SC are typically in the IR-region (metals deep-UV).

InSb reflectivity:

band-to-band absorption edge: λ~6um

phonon absorption edge: λ~50um

plasma reflectivity edge: 14 – 32um depending on doping

( )

2

2

0 0 0 0 0

22

2

1

11

n,p0 r 0 bound free carreier 0 opt 0 r

eff

n ,pr opt p n,p opt eff ,n ,p

opt eff

D =j

e NE E P P E E E

m

e Nwith e N / m

m j

ε ε ε ε εω γω

ε

ε ε

ε ω εε ω γω

+= + + = − = ⇒

⎛ ⎞= − =⎜ ⎟⎜ ⎟+⎝ ⎠

( )

13

21 2

2 2 2

1 2 12 2 2 2

101

11 1

p

r

p popt opt opt

p p

we get

in SC s and is low, therefore for IR - frequencies we have :from n j

;

As a result of the typical SC parameters we g

τ ωωτ

ε ε ε

ω τ ω τε ε ε ε ε ε

ω τ ω ω τ

−

>>= = +

⎛ ⎞= − ≈ = <<⎜ ⎟⎜ ⎟+ +⎝ ⎠

∼

( ) ( ) ( )2 2opt

et free carrier losses

in the IR - region : n ; / nε κ ω ε ω→ ≅ ≅

( )1p opt opt/ω ω ε ε= −

free carrier absorptionin bandgap


3-7

Using the same procedure to determine the absorption α as in the case for metals we get now:

Band-diagram representation of free electron absorption in a n-doped SC:

The optical field can only induce direct (constant k) verticaltransitions from filled electron states to empty states above EF.

In addition a scattering event (eg. phonons) with the timeconstant τ is necessary to conserve the momentum Δk.

The free carrier absorption for holes is similar, except that the hole band structure (eg. 3-bands, lh, hh, so)allows strong direct intervalence band absoption processe, not requiring a scattering event.

2 2

2 20

1opt p n,pfree carrier

eff ,n ,p

N enc m nc

ε ωα

ω τ ε τ ω= =


3-8

3. Optical properties of metals: quasi-free electrons

Summary:• metals show a non-resonant (ω0=0) dielectric behaviour leading to a strong dielectric

response below the plasma-frequency ωp. The classical dipole model including the carrier scattering time as damping mechanism is a1.order description (Drude-Lorentz-model)

• metals change from reflective behaviour to transparency above the plasma frequency

• free carrier absorption losses caused by carrier scattering is substantial in metalsand highly doped semiconductors.Typically absorption losses increase with carrier concentration, resp. doping in semiconductors.

• The plasma frequency for metals is in the UV-region and for doped semiconductors in theIR-region.

• Coherent and quantized, collective oscillations of the electron plasma are possible andcalled plasmons.


4. Optical processes in Semiconductors:Interband absorption

Goals:• Semiconductors show strong optical transitions and absorption in the visible and

near infrared region (NIR-region) • The band structure and the dominant band-to-band transitions require a quantum

mechanical description• Excitonic effects and the interaction with phonons have to be considered 4-1


4. Optical processes in Semiconductors: Interband absorption

The classical continuous oscillator model fails to explain the energy-level quantization of the electron system observed in atoms and SC.

But it also fails to explain the energy quantization of the EM-field: the photon.

Atoms with localized discrete states show sharp absorption lines. Only at this resonance energy can be exchanged between the light wave and the atom. Off resonance, the classical model is a quite valid description.

In the resonant interaction, the field-energy is converted into atomic excitation-energy and vice-versa by an exchange of energy quanta.

At least the motion of electron-system requires a quantum-mechanical treatment for arealistic materials model.

4.1 Basic Interband Transitions in SCs: an overviewMajor optical processes in a simple 2-level (≡“2-band”) system

• absorption• spontaneous emission• stimulated emission• (non-radiative recombination) 4-2

(non-radiative)


These transitions only occur if they are compatible with QM-selection rules for the transition.

Energy E and momentum k conservation hold for the photon-electron interaction:

i= initial state ; f= final state ; ωif= transition frequency

Absorption: Annihilation of a photon and creation of an electron-hole (e-h) pair in SCSpontaneous emission: Creation of a photon and annihilation of an e-h-pair in SCStimulated emission: Induced creation of a photon with the same frequency ω, phase φ and

propagation vector k and annihilation of an e-h-pair in SC

As phonons (in the solid) can generate dipole-waves they can also interact with photons and electrons in a 3-particle process → indirect transitions in SC.

4.1.1 Einstein Model of Optical Transitions in a discrete 2-Level SystemFor isolated atomic 2-level systems with an- atom density N=N1+N2, Ni= density of atoms in state i- quantized discrete energy levels E2 and E1, interacting with

- an optical broad-band field at the transition frequency ω=(E2-E1)/ħ and an energy density ρ(ω),Einstein postulated for the electrons the followingtransition probabilities rkl, resp. transitions Rkl rates between the energy states 1↔ 2: 4-3

f i if

f i

Energy conservation : E E

Momentum conservation : k k k

ω= +

= +


Definitions of theTransition probability i→f:

Transition rate i→f: Ni=atom density in the initial state i

The transition rate R is the number of particles per volume making a transition from i → f per unit time

N1(2)= density of atoms in state 1(2) p1=probability of electron occupation of level 1 , E1p2=probability of electron occupation of level 2 , E2

with p1+p2=1

Einstein’s equations: (postulates)

Remark: energy levels are discrete, whereas the optical field is describes by a

spectral energy density ρ(E):

Assuming a statistical distribution function fi(Ei) of the occupation of the energy states E1, E2 by electrons and considering that for isolated atoms the occupation of a state means that the other state is necessarily empty: 4-4

fif

pr

t∂

=∂

fif i if i

pR N r N

t∂

= =∂

( )

( )

12 2 12 12 1 1 12 21

21 1 21 21 2 2 21 21

21 1 12 12 2 2 21

1 2

2 1

2 1 ,spont ,spont ,spont

Absorption : r p R r N N B Et

Simulated Emission : r p R r N N B Et

Spontaneous Emission : r p R r N N At

ρ

ρ

∂→ = = =∂∂→ = = =∂

∂→ ≡ = =∂

( )2

EMWEV E

ρ ∂=∂ ∂


Einstein’s Transition Rates with occupation probabilities:

Relation between Einstein’s A and B-coefficient:Spontaneous emission is a necessity for a system to reach again thermal equilibrium after switching off the interacting light field.

From the thermodynamic equilibrium (sum of up and down rates =0) at temperature T between the blackbody-radiation ρbb(E21) and a 2-level system obeying a Boltzmann occupancy distribution we obtain (proof see exercise II , see also lit. Yariv, Jäckel, Coldren)

Rate-equations (particle conservation in an energy level):We consider the N states at E2 and E1 occupied by N2 and N1 electrons as “reservoirs” for electrons storing electrons. Because the particles have to be conserved, we have the continuity equation of the particle-flux in an d out of a reservoir: 4-5

( ) ( ) ( ) ( ) ( )

( ) ( )

12 21

3 3 3 321 21

21 21 21 21 21 213 3 3

21 21 21 21

8 8mode

modeBB

spectral mode density

phaverage number ofphotons per mode

B B B

n E n hvA E B E E E B E B vh c c

from Blackbody Radiation spectral energy density :

E E n E

π πρ

ρ ρ

= =

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠−

= ( ) ( )3 2

1 2121 21 3 3

81 modei

otonenergy

n Ewith : n exp E / kT ; Eh c

πρ−= − =⎡ ⎤⎣ ⎦

( ) ( )( ) ( ) ( ) ( )( )

( )

12 12 21 1

21 21 21 2 21 21 1

21 21 2

1

,spont

Absorption : R B E Nf E

Simulated Emission : R B E Nf E B E N f E

Spontaneous Emission : R A Nf E

ρ

ρ ρ

=

= = −

=


The change of particle in a level per unit time = difference between the inflow – outflow of particles

Rate-equation (particle conservation):

If ; N1>N2 the absorption transition dominates, the material absorbs light(thermodynamic equilibrium)

If ; N1=N2 no light is absorbed (net) and the material is transparent, resp. the amount of absorption and emission is equal

If ; N1<N2 stimulated emission dominates, the material amplifies light(population inversion)

For simplicity we assumed:

Conclusions:• in thermal equilibrium N1>>N2~0 absorption (mostly up-ward transitions) is dominant

• if N1≅N2=N/2 the material is transparent (~equal up- and down-ward transitions)• under so called inversion N1<N2 stimulated emission or amplification dominates ~(N2-N1)

• stimulated emission is proportional to the field intensity I(ω) at resonance ω=ω21= (Ε2−Ε1)/ħ4-6

( ) ( ) ( )( )2 2 12 21 21 21 2 1 2

1 1 2

pump ,spont pumpLevel E : N R R R R R B E N N N At

Level E : N Nt t

ρ∂ = + − − = − − −∂∂ ∂= −∂ ∂

12 21R R=

12 21R R>

12 21R R<

12 21 21,spontR ;R R>


Pumping:In thermodynamic equilibrium N1>>N2 and the material is absorbing e-reservoirand does not amplify (R12>R21) or generate light.

For population inversion N2>N1 and gain practical opto-electronic devices need an external pump mechanism R12,pumptransferring electrons from level 1 to 2

The rate-equations are modified by the pump rate to:

If no external field is present (no absorption) and the system is stationary:

Pumping makes a carrier inversion and gain possible !

4-7

( )( )2 12 12 21 21 12 21 2 1 2,pump ,spont ,pumpN R R R R R B E N N N At

ρ∂ = + − − = − − −∂

( )21

12 2 2 12

0 0

2,pump ,pump

E ;t

R N A N R / A N /

ρ ∂= =∂

= → = >

N2 , E2

N1, E1

Rpump R12 R21 R21,spont

ρ(E)


4.1.2 Quantum mechanical optical transitions in a 2-Level System(only description of solution concept, for details see appendix A4.1)

Calculation of A and B requires the quantum-mechanical analysis of the electron motion in the• time-independent atomic electrostatic potential V(r) and • the superposed time-variant vector potential of the EM-field A(t), represented by the simplifiedoptical electric potential Vopt(t).

As shown in appendix A.4.1 the time dependent occupation probabilities

p1(t)=IC1(t)I2, p2(t)= IC2(t) I2 of the energy levels 1 and 2 change

under the influence of a quasi-monochromatic, x-polarized optical field E(t) at location x:

• E(t) generates a time-dependent optical potential Vopt(x,t) at the position x (separating atom-electron):

• The motion of the bound electron of the atom under theinfluence of both potential fields is described by the

time-dependent Schrödinger-equation:

System with discrete Energy levels E1, E2 and discrete optical field energy E=ħω 4-8

( ) ( ) ( )2 j t j tx xE t E cos t E / e eω ωω −= = +

( ) ( ) ( ) ( )0

2x

j t j topt x x xV x,t E t dx' xE cos t xE / e eω ωω −= − = − = − +∫

( ) ( ) ( ) ( ) ( ) ( ) ( )2

0 02op ptt opH r H r ,t r ,t eV r Ve r ,t j r ,tm t

r ,t⎡ ⎤ ∂⎡ ⎤+ Ψ = + + Ψ = − Ψ⎢ ⎥⎣ ⎦ ∂⎣ ⎦

xEx

-eVopt(x,t)

V0(x)

Vtot(x,t)


• The optical field changes the occupation-probability of the 2 energy states, introducing an atomic polarization and transitions between the states at resonance (ħω= ħω21=E2-E1).

• The solution of the time-dependent Schrödinger-equation can be approximated by using the stationary solutions for the 2 energy levels ψ1(r,t) and ψ2(r,t)weak perturbation approximation:

From the probability interpretation of the normalized wavefunction ψi, resp. ψiψi* we obtain the occupation probabilities pi of the energy-levels:

A lengthy calculation gives for small optical perturbations (see. Lit. Jäckel, R. Loudon):

4-9

) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

) ( ) ( )( ) ( ) ( ) ( ) ( )

1

2

1 1 1

2 2 2

1 1 2 2

1 1 2 2

1 0 0

2 0

opt i

j E / t

j E / t

i

Stationary solutions : E t ;V C const.

r ,t r f t r e

r ,t r f t r e

r ,t C r ,t C r ,t

Time dependent solution : E t C t

r ,t C t r ,t C t r ,t

ψ ψ

ψ ψ

= = → =

Ψ = =

Ψ = =

Ψ = Ψ + Ψ

− ≠ →

Ψ = Ψ + Ψ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 21 1 1 1 2 2 2 2 1 2 1* *p t C t C t C t ; p t C t C t C t with p t p t= = = = + =

( ) ( )

( ) ( ) ( )( )

1 2

222 12

2 2 2 0 1 2 12212

0 1 0 0 0 1

21 14

p ; p for t the electron is in the lower state

sin t /p t C t xeE with dipole matrix element : x

ω ωψ ψ ψ ψ

ω ω

→ = = =

−⎡ ⎤⎣ ⎦= = << −−⎡ ⎤⎣ ⎦

Absorption 1 2 :

~1/t0

~1/4 t02

area ~t0

ωkm-ω

In the same way we obtain for stimulated emission:

Fermi-Golden Rule:

In real systems with finite interaction duration t=t0 or atomic level-broadening due to atomic collisions

1. the optical wave is not single frequency ω but has a frequency uncertainty (spectral width) Δωopt~1/t0→ energy density of the field2. the final state has a certain energy spread ΔE=ħΔω due to the same reason or due to

perturbations processes → line width function3. the final (and/or) initial states are a part of a state-continuum, with the energy state density ρ(ω)

Therefore we get the averaged transition probability by a weighted integration of the transition probabilities over the uncertainty range Δω of the process:


4-10

( ) ( )( ) ( ) ( ) ( )

1 2

22 2 2 2 212 1 1 0 2 1 21 1

0 0 0 1 0 2

4

: p ; p for t the electron is in the upper state

at resonance : p t C t e E / x t r p t / t ~ tω ω ψ ψ

→ = = =

= = = = ∂ ∂

Stimulated Emission 2 1

( ) ( ) ( ) ( )22 2 2 2 212 2 2 0 2 1 12 24at resonance : p t C t e E / x t r p t / t ~ tω ω ψ ψ= = = = ∂ ∂

( ) ( ) ( )( )

( )

222 12

2 2 2 0 1 2212

2

2 0 1 122

214

14 2t

sin t /p t C t eE x

eE x t

ω ωψ ψ

ω ωπψ ψ δ ω ω→∞

−⎡ ⎤⎣ ⎦= =−⎡ ⎤⎣ ⎦

⎯⎯⎯→ ≅ −

( ) ( ) ( ) ( ) ( )2 22

2 2 2 1 12208

opt

opt optep t C ',t ' d ' x ' ' td '

ω ω

πω ρ ω ω ψ ψ ρ ω δ ω ω ωεΔ

= = −∫ ∫


This leads immediately to the Einstein coefficient B, the matrix element M and also to A:

Transition Selection rules:Obviously a transition is quantum mechanically possible or allowed only if the matrix element M≠0. M=0 represent a forbidden transition.

Interpretation:Both states have either even or odd-symmetry

Only transitions between initial and final states ofdifferent symmetry are allowed (because x has odd-symmetry).

In the detuned interaction we only have oscillatory and small changes of the occupation probability, resulting in a weak polarization of the atom, resp. ground state. No real absorption of quanta takes place.

We have neglected in this treatment the Lorentz-force and only considered the electrical dipole as an approximation. For a detailed discussion using the vector potential A of the optical field see app.A4.1.

4-11

( ) ( )

( ) ( )2 2

2 2

12 12 2

2 2 312 21 2 1 1 22 2

0

20

0

1 124

4 4

opt

e eB B x M with the dipole matrix - element : M r x r

x

d

er

rπ πψ ψ ψ

πω ψ ψ ρ ωε

ψε ε

∗=

=

= = − = ∫

E

E2

E1

ψ2

ψ1

xx

Observe: ρopt(ω) is a frequency-density !


4.1.3 Generalization to optical transitions in an energy-band System(Description of solution concept, details see appendix A4.2)

Electrons move quasi-freely through the periodic crystal lattice and form quasi-continuous energy bands of extended, delocalized states (the energy quantization in the makroscopic crystal is very dense).Optical transition between the extended (V=crystal volume) energy states in the conduction (C) and valence (V)-bands are described by similar matrix-elements as atomic transitions.

The number of possible transitions per energy-interval dE is described by the density of transition pairs , respectively the joint density of state function ρj,12(E). ρj,12(E) can be expressed by the density of state functions ρ1(E) and ρ2(E) of the initial and final bands:

discrete ↔ continuum continuum ↔ continuum transition

density of state ρ2(E2)

4-12

( )2

ii i

NDef .: EE V

ρ ∂=∂ ∂

Remark:In SCs the final state is not necessarily empty as in isolated atoms.

description by occupation probability functions f(E)

( )2 2E dEρ

( )1 1E dEρ

( )12 2 1j , E ,E dEρ

Conduction band: E2

Conduction band: E1

EC(k)

EV(k)

monochromatic field


For the excitation with an ideal monochromatic optical field (ω) the transition rates are proportional to the joint density of state functions (appendix A.4.2 , Fermi Golden Rule!):

Summary: ρj,12(ωħ) is the joint density of final states

• The energy dependence of ρj,12(E=ħω) determines together with the occupation probability f(E) thespectral distribution of absorption and emission.

• In SC-crystals the atomic states start forming bands from the bonding and anti-bonding orbitals.

• The lattice-periodic (a) wave functions ψ(r,t) are extended and delocalized (quasi-free electrons).• Bands have a finite energy-width and are separated by energy gaps Eg.

• The dispersion relation Ef(k) connects the level energy Ef and the corresponding momentum k. 4-13

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

212 21 21 2 1 1 2

221 21 21 2 1 2 1

221 21 21 2 1 2 1

2 1

2 1

2

vc j ,cv v c

vc j ,cv c v

,net vc j ,cv c v

R M E ,E f E f E ;

R M E ,E f E f E

R M E ,E f E f E

with the joint density of state function expressed by the density of state func

πω ω ρ

πω ω ρ

πω ω ρ

= −

= −

= −

( ) ( ) ( ) ( ) ( )( )2 1 2 1 2 1 4 2 2j ,cv c v c v

tions of the bands :

E ,E E E / E E see chap. . .ρ ρ ρ ρ ρ= +

( ) ( ) ( )212

2i f j ,R Mπω ω ρ ω→ =

( ) ( )21 2 1

0 2 1

vc opt opteM E HmeE r

ψ ω ψ

ψ ψ

=

→ ≅

Matrix-Element:


Density of state functions for bulk SC-crystals:From basic semiconductor physics: • electrons and holes can be considered as free quantized particles moving in a macroscopicpotential-boxV(r)=constant over the size of the crystal V=L3.

• in the effective mass me,(h) approximation the dispersion relation of the quasi-free particle is:

• the energy states in the band obey the time-independent Schrödinger-equation with energyEigenvalues Ei , the momentum ki and the Eigenfunction-function ψi(r,t) (in 1D):

The solutions are standing plane-waves inthe volume L3 in the x-, y- and z-direction.

( )( )

( ) ( )2 2

02c ,v c ,v x y z

e h

E k k E with k n ,n ,n km

= + = Δ

4-14

Dispersions-Diagram E(k):

Δk=2π/L

Ec(k)

Conduction-Band

Valence-Band

Ev(k)

Ψc,i , Ei

Ψc,i+1 , Ei+1

EC

EV

k0, k1, k2, k3, ...

kki

( ) ( ) ( )

( ) ( )( ) ( )

2

22

2

2

crytal i i i

i i crytal i

p eV r r E rm

r E eV r rm

ψ ψ

ψ ψ

⎡ ⎤+ = →⎢ ⎥

⎣ ⎦− ∇ = −

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 yx jk yjk xx x y y

x y z

x u x e ; x u x e ; .....L L

with r x y z

ψ ψ

ψ ψ ψ ψ

= =

=


The standing waves in the volume V=L3 fulfil the boundary conditions (at infinite potential walls):

Ψ(0,y,z,t)= Ψ(x,0,z,t)= Ψ(x,y,0,t)= Ψ(L,y,z,t)= Ψ(x,L,z,t)= Ψ(x,y,L,t)= 0

The momentum vector ki must therefore be quantized:

And a corresponding energy eigen value: E(kx,ky,kz)

In momentum- or k-space each energy state E(k), resp. (nx, ny,nz) is represented by the k-vector and an associated incremental volume dk=dkxdkydkz.

Density of energy states in the bands for 3-D bulk crystals

these energy densities are used to calculate the joint energy pair density !

the extension to 2-, 1- and 0-dimensional systems is topic of an exercise !

Remark:The counting of 3-D standing matter waves is identical to counting EM-field modes of the blackbody radiation. 4-15

( ) ( ) ( )( )2 3

2 eff ,n pCn p V

mE E Eρ

π= −

Solutions in k-space:

E~k2

~ΔE

2 2 0 1 2i i ik n n k with k and i x, y,z and n , , ,...L Lπ π= = Δ Δ = = =

dk

k


Dipole-matrix element for extended stated in SCs:

We showed as a fundamental result:

Mif(ω) is the matrix element of the transition i→f, describing the strength of the transition.

Matrix-element between extended states in SCs:The wave functions and the optical field are extended, not localized waves.

From elementary perturbation theory, assuming a harmonic, resonant perturbation at ω=ω21=ω2−ω1:

Using a simplified electrical potential Vopt(r,t) of the optical field in the time dependent Schrödinger-

equation of the classical photon field acting on a local dipole , resp. electron:

4-16

( ) ( ) 3f i f i

bra ket abreviationr H ' r d r H 'ψ ψ ψ ψ∗

−=∫

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( )

0 0 1 2

1 i

jk r jk r jx jx jxopt

jk ri i i

p er'H ' r pE r pE e er ' E e u sin g : e / e e

Static solutions : energy states i is a plane waves modulated by lattice - periodic Blochfunctions u r :

r / V u r e normalizedψ ψ ψ

+ − ⋅ + − ⋅ + − + −

⋅

=−= − = − = = +

=

( ) ( )

( ) ( )( ) ( )

( ) ( )

30

1

1 1

1 2 3

f

f i

i

jk rf f f f

jk r jk r jk rf i f i

i x x y y z z i j

r / V u r e normalized

eH ' u r e r ' E e u r e d rV

lattice periodicity of the Blochfunction : u r ' n a n a n a u r ' ; n , , ,....

ψ ψ ψ

ψ ψ

⋅

− ⋅ + − ⋅ ⋅∗

=

= =

→ =

− + + + = =

∫

( ) ( )22i f if j ,ifR Mπ ω ρ ω→ =

( )i rψ ( ) 0j t kr

i iE r ,t E e ω= ∓

( )E r ,t p er '= −

x

a

r’

a

r’

a

r’

a

r’


4-17

Due to the periodicity of the Blochfunction we represent the integration over the total crystal volume

V , by a summation of integrals over the unit cells , :

The momentum conservation enforces that all contribution from each unit cell is in phase, otherwise each unit cell would have a different phase factor and the integral would average to M=0.

Conservation of momentum:

Interband-transition are only efficient between states with thesame momentum k for band-to-band transition ρ(E) means the joint density of statesfunction under k-conservation.

• Direct transitions are “vertical” transitions in the dispersion diagram.

( )( )

( )( ) ( ) ( ) ( )( )30 0

1

2 2

f i

unit cell

f i i

jr k k kf i f i f i

i , r ia r ' V

using the momentum conservation assumption

k k k k / photon momentum k ~ / a carrier momentum

e eH ' u r r ' E u r e d r u ia r ' r ' E u iaV V

π λ π

ψ ψ−

− + +∗ ∗

= +

= ± = <<

= = + +∑∫ ∫ ( )

( ) ( )

( )( ) ( ) ( )( ) ( )0 0unit cell unit cellunit cell unit cell

i i

f i unit cell f i f iN V / V unit cellV V

r ' dr '

and the lattice periodicity of Blochfunktion : u r u r a a lattic vector

e eH ' N u r ' r ' E u r ' dr ' u r ' r ' E u r ' dr 'V V

ψ ψ− −− −

∗ ∗−

= −

− = + =

= =∫ ∫

2 2f i f i

f i f i

k k k with k / and k ,k / a a lattice period

k k ,k k k vertical transition in k space

π λ π= ± = < =

<< → = −

( )x y za a ,a ,a=V

...dV∫ 3

unit cell unit cell

unit cellN V

...da with N V / a− −

− =∑ ∫

kfEf

Eiki

f

i


4.2 Interband Absorption in SemiconductorsInterband transitions in SC are classified into direct and indirect transitions.

Indirect SC (eg. Si, Ge, AlGaAs at high Al-content):minimum of the conduction and the maximum of the valence band have different k-vectors

Direct SC (eg. GaAs, InP, GaN, and many III-V-compounds, but also diamond):minimum of the conduction and the maximum of the valence band have the same k-vector k~0

(“vertical” transitions)

Optical processes are much more efficient for direct transitions because no 3rd particle (phonon assited) is required for momentum conservation.

4-18


4.2.1 Band edge absorption and direct band structure for III-V-compounds:

Depending on crystal structure and crystal atomic composition the conduction and valence bands may contain several bands with different effective masses and density of states eg. GaAs.

Multiple bands will influence the joint density of states if the transitions are allowed.

Typical band structures: Generic III-V-band structures:GaAs. InP: simplified 4-band model:

Transitions from the heavy hole (hh), light hole (lh) and spin-orbit splitting (so) bands to the conduction band are allowed.

The bands can be approximated by a parabolic dispersion relation E~k2 for k~0. 4-19


4-20

ΔN-states in Δk

ΔN-states in Δk

Δk

ΔE1

ΔE2

ΔΕ=ΔE21

δk (equidistant k-quantization)

ΔE1 und in ΔE2 we have ΔN=Δk/δk states

4.2.2 Joint density of states in SC with parabolic dispersion:Due to the momentum conservation only states in the conduction- and valence band having the same k-vector can interact. For a given k the corresponding energy-levels EC(k) and Ev(k) in the bands, as well as the transition frequency ωcv(k)=(EC(k) - Ev(k))/ħ are defined, resp. related by the dispersion relations:

For a hh → c or a lh → c-transition at the a common k-vector we have:

Using and the joint density of state function ρj,CV(k) in k-space for bulk material:

( )ch kω

( ) ( )2 2 2 2 2 2 1 1 1

2 2 2cv g ge h e h

k k kE k k E E withm m m m

ωμ μ∗ ∗ ∗ ∗= = + + = + = +

( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )

2

2

3 21 2

2 2

22

2

1 22

0

j ,cv j ,cv j ,cv

j ,cv j ,cv

//

j ,cv g g

j ,cv g

kE dE k dk with k

E k / dE / dk

:

E E for E

and E for E

ρ ρ ρπ

ρ ρ

μρ ω ωπ

ρ ω

= =

→ =

⇒

⎛ ⎞= − >⎜ ⎟⎝ ⎠= <

joint density of states

( ) ( ) ( ) ( )2 2 2 2 2 2 2 2

2 2 2 2C g hh lh soe hh lh so

k k k kC band : E k E ; V band : E k ; E k ; E km m m m∗ ∗ ∗ ∗− = + − = − = − = −Δ −


4.2.3 Absorption spectra of direct SCDue to Pauli’s exclusion principle a transition from an initial state to a final state is only possible if the final state is empty (unoccupied).

Therefore, the occupation probability functions fC(E) and fV(E) have to be known. In SC under non-equilibrium one assumes f to be quasi-Fermi-functions.

In thermal equilibrium and for relatively large bandgaps Eg>>kT or T→ 0 the levels in the conduction band are mostly empty (fC(EC)~0, fV(EV)~1)

all up-ward (absorption) transitions for ħω>Eg are possiblethere are no down-ward transitions.

Assuming: fC=0, fV=1

Because the absorption α(E=ħω) is related to Rvc(E) and Rcv(E) by:

α(ħω) ~ Rvc(E)fv- Rcv(E)fc= Rvc(E) ~ ρcv(E)

Strong absorption occurs for:

- low effective mass

- large matrix element

- high optical frequencies

( ) ( )( ) ( )

1 2

0

/

g g

g

~ E for E

for E transparency

α ω ω ω

α ω ω

− >

= <

gE Eα −∼

bandgap

4-21

Eg E

−α

transparency absorption


4.2.4 Frank-Keldysh Effect (optional)

The Frank-Keldysh effect describes the change of the absorption edge (bandgap lowering), resp.the absorption in a bulk-SC due to a strong external electrical E-field

e-h tunnelling into the bandgap:

absorption control by electrical field• Apparent reduction of the bandgap Eg(E)

• Oscillatory modulation of the absorption spectrum for ω>Eg Experimental absorption modulation:

Schematic band-bending and carrier-tunnelling into the bandgap:

a) small b) large spatial probability overlap of e-h pairTunnelling makes transition to states with energy separation <Eg possible !

The change in absorption together with the Kramers-Kronig-relation causes a change in the refractive index Δn(E)

index controlled by an electrical field.4-22


4.2.5 Absorption spectra of indirect SCTransitions in indirect SC (eg. Si, Ge) require the simultaneous interaction of a photon and a phonon (phonon emission or absorption with momentum q and frequency Ω), because the photon momentum ħk is too small to conserve the momentum.

Energy and Momentum Conservation in photon-phonon-electron interaction:

The probability of this 3-particle interaction, resp. the transitionrate Rcv and therefore the absorption α is much smaller than indirect SCs.

Eg,Si Eg,GaAs4-23

f i

f i

Energy : E E

Momentum : k k q

ω= + ± Ω

= ±


4.3 Static and Dynamic Absorption MeasurementsStatic transmission measurements: ps-Pump-Probe transmission measurement:

The probe pulse samples the sample transmissionat a delay τd after the exciting pump-pulse

Δtpump<<τ

Spectrometer acts as a narrow band

filter4-24

t

intensity

Luminescence λ21

τ

Pump signal λpump

τd


4.4 ExcitonsExcitons are electron-hole pairs bound by Coulomb-attraction forces and are usually generated by band-to-band absorption processes and modify the absorption edge.

Similar to H-atoms they form quantized 2-particle systems with discrete binding energy levels E(n).

Depending on their size (∅=ax) the neutral excitons can move freely through the crystal

Because of their large diameter, freeexcitions can be considered ase-h “hydrogen” atoms in a dielectric with the high dielectric constant εr of the SC.→ large ∅ ; small E(n)

free excition (∅>>a) tightly bound excition (∅~a)(typ. Semiconductors) (typ. Oxides)

Since their binding energies E(n) (~10meV) are comparable to the thermal energy kT, excitonsin bulk crystals are easily ionized by phonon collisions and are not stable (they separate into regular e-h pairs).

4.4.1 Free exciton absorptionThe binding energy of excitons for the relative motion can be estimated by the model of a hydrogen atom in a high εr dielectric and with reduced electron (m0) and hole masses. 4-25

εr


The nth energy level E(n) with respect to the ionized state becomes:

After a photon absorption the photoelectron forms with the valance-band hole directly an exciton.The absorption energy of the exciton is therefore is Eg’(n)=Eg-E(n). Excitons form narrow absorption bands below the interband absorption edge.

Remark: for electron and holes to form an exciton, their group velocities vg,e=vg,h must be the equal !4-26

( )

( )

( )

2 20

2 20

2 20 0

1 0

13 61 1 1

10

4 5 3

r

H X

H

eff ,e eff ,h

n H X H

H

r

with the Rydberg constant :

with the Bohr radius of the hydrogen atom :

R RE nm n n

R . eV for hydrogen and

reduced effective massm m

Exciton radius :mr n a n a ~ a

a / m e .

μ

μ

μπε

ε

ε

= − = − <

=

= +

= = >>

= = ⋅ 1110 m )−

V(r)

E2E1


therefore excitons are mostly formed at the band edge only.

4.4.2 Free exciton in high electrical fieldsAn external E-field (ε) tends to separate the bound electron and hole charges and brakes (ionize) the exciton. If the external potential difference across the exciton ΔV=Ernbecomes comparable to the binding energy E(n) ~ ΔV=Ernionization occurs:

Application: the field-modulation of the narrow exciton absorption line is used in intensity electro-absorption modulators (EA).

4.4.3 Exciton nonlinearity in intense optical fieldsIf the density of excitons becomes very high (eg. by light absorption)then excitions overlap and the number of exciton-collisionsincreases until the excitons dissociate

→ Nonlinear, intensity-dependent bleaching of the excitionabsorption

Application: Exciton bleaching is used eg. in all-optical switches.

( ) ( )1 1

0 0

g ,e n g ,h h

g ,e g ,h

v E k v E kk k

because in the Brioullin zone center k / k v v

∂ ∂= = =∂ ∂

= → ∂ ∂ = → =

( ) ( ) 2

2 21 0

1 11 Hmax

r H

E RE nr m a

με

= ≈ =

4-27


4.5 PhononsPhonons are quantized oscillation modes of the crystal lattice (lattice positions in the solid).The oscillating lattice atoms may form eg. oscillating dipoles which can interact with photons. The phonon oscillation frequencies are quantized and typically in the optical-to-FIR-range

(1012-1013Hz whereas electronic dipole oscillations are in the visible-to-UV-range 1014-1015Hz ).

From basic physics it is known that phonons are characterized by their momentum p, wave vector q, frequency Ω and : • a dispersion relation Ω(q), resp. E(q)= ħΩ(q) → acoustic (A) and optical (0) phonons• a transverse (T) or longitudinal (L) polarization Phonons have a momentum p=ħq<h/a, which is often much larger than the photon momentum p=h/λ.Photon and phonon can interact under energy and momentum conservation by mutual destruction or creation processes (phonon emission and absorption).

Phonon Spring-model:

Quantum mechanical selection rules determine possible interactions (eg. coupling TO-phonon–T-photon)4-28

~1-10 Thz

No dipoles

With dipoles(strong charge separation)+

++

_ _ _

a


• Photons and phonons couple through EM-field forces, mechanical charge polarization isnecessary (ionic crystals (eg. III-Vs), not in covalent crystals (eg.Si)).

• The optical phonons usually show a weak dispersion.

4.5.1 Lattice absorption/reflection processes (first order processes)FIR-photons (THz ≡ 10-30meV) may be destroyed by the creation (emission) of a phonon which is a first order process and has a high efficiency.Not many practical devices exist and the process is used more for material characterization.

Classical Oscillator Model for Phonons:Optical phonons can be considered as polarization waves where the positive and negative lattice atoms are polarized from their equilibrium positions (x=0) by the EM-wave of the photons. The restoring force of the dipole is described by the “spring constant” K. Assumption: the opt. frequency ω is close to Ω (the resonance frequency of the phonon mode) and q≈k ≈ 0:

4-29

( ) ( )

( ) ( )

2

2

2

2

1 1and using :

positve ions : equation of motion

m x m x K x x qE tt t

negative ions :

m x m x K x x qE tt t

1 and x x xm m

γ

γ

μ

+ + + + + −

− − − − − +

+ −+ −

∂ ∂+ + − =∂ ∂

∂ ∂+ + − = −∂ ∂

= + = −

optical wave

phonon wave

+

_


For a harmonic excitation of the damped dipole equation we obtain for the dielectric constant of the lattice and the electronic polarization (χ):

Knowing εr(ω) we can calculate as for conventional dipoles in the THz-range:

phonon resonance εr<1 4-30

( ) ( )2 2

22 2 TO

TO

subtraction of the oscillator equations :K q qx x x E t x x x E t

t t t t

K / resonance frequency

γ γμ μ μ

μ

∂ ∂ ∂ ∂+ + = ⇒ + + Ω =∂ ∂ ∂ ∂

Ω =

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

0

2 2

22 20 0

2

2 2

11 0 1 1r st r rTOTO

TOr st

TO

E t E cos t

Nq Nqwith andj

j

ω

ε ω χ ε ε χ ε ε χε μ ε μω γω

ε ω ε ε εω γω

∞

∞ ∞

=

= + + = = + + = ∞ = +ΩΩ − −

Ω⇒ = + −

Ω − −

n, n, , resp. Rκ α

n=imag.


4.5.2 Inelastic/ elastic light scattering by phonons (second order processes)

Light Scattering: light can interact with excitations of the propagation medium (phonons, polarons, excitons, etc.) and change the initial photon frequency and momentum (ω1, k1) to (ω2, k2). Besides elastic Rayleigh and Mie-scattering where ω1=ω2 , inelastic scattering ω1≠ω2 is important.

Light scattering processes involving 2 photons (ω1, k1, ω2, k2) and one phonon (Ω,q) (or other particles, excitations of the solid) are categorized by the nature of the energy exchange:

Raman-scattering from an optical phonon

Brillouin-scattering from an acoustic phonon

scattering with phonon emission → Stokes scattering ω2= ω1-Ω

scattering with phonon absorption → Anti-Stokes scattering ω2= ω1+Ω

Conservation rules of the 3-particle interaction: energy (frequency up- or down shift): ω2= ω1±Ω momentum:

Although Raman- and Brillouin-scattering are weak processes because they involve 3 particles to interact simultaneously, there are a few important device applications in SC. The resonant interactions with phonons can also enhance nonlinearities substantially.Anti-Stokes processes depend strongly on temperature, because they need exciting phonons (phonon absorption). Meaning the temperature T must be high for a strong effect. 4-31

1 2k k q= ±


The interaction of photons and phonons can be described by the oscillator model of the phonon.

Phonons have a limited lifetime τ (damping γ) of ~1-10ps due to strong phonon-phonon scattering and due to nonlinear interactions between phonons.

Remark: Phonons interact also with the charge carriers in SC and are responsible for indirect optical transitions, ohmic resistance, interband-thermalization, etc. ….

4-32


4-33

4. Optical processes in SCs: Interband Absorption

Summary:• Interband absorption occurs when electrons make a dipole transition between a lower

and a higher energy state. The transition corresponds to a destruction of a photon in the optical field and the creation of an excited state in atoms or an electron-hole pair in SCs.

• Energy levels and energy bands can only be described quantum-mechanically

• Interband transitions are only possible for photon energies larger than the bandgap – below bandgap the SC is transparent.

• The transition rate of absorption is proportional to the dipole matrix element,the joint-density of pair-states and the field intensity. The transition rate is also a function of the occupation probabilities of the initial and final state in the SC, described by the quasi-Fermi-distribution functions.

• The energy and momentum has to be conserved in direct transitions. Because the photon momentum is very small, the momentum of the electron and hole are virtually identical.

• Excitons are bound (by Coulomb-attraction) electron-hole pairs, which are only stable at RT in quantum wells (chap.6). Strong optical exciton transitions close to the bandgap are possible.

• Excitons can also be ionized by strong electric fields.

• Phonons are quantized lattice oscillations. The classical description is a harmonic polarization oscillator with resonance frequency in the range of 1-10 THz.


4-34

Appendix: 4AA 4.1 Optical transitions between discrete energy states

( a short course in perturbation theory, which is not complete)

A time-dependent resonant optical field induces transitions of an electron between the stationary energy states in a stationary potential V .

Classical motion of an electron in a time-dependent EM-field and a static potential:The EM-field is described by its vector potential . The static potential is .The classical Hamiltonian H of the charge motion is (for derivation, see Lit.: D. Marcuse) proposed as:

It can then be proven that this Hamiltonian reproduces Newton’s law of motion and therefore is correct:

Transition to Quantum Mechanics: ( change of classical variables → QM-operators)

For no EM-field present A=0 we describe the static motion by:

( )A r ,t ( )V r

( )212

H p eA eV ; e particle chargem

= − + =

1m v e E v H using the definitions : E V A and H At t

μμ

∂ ∂⎡ ⎤= + × = −∇ ⋅ − = ∇ ×⎣ ⎦∂ ∂

( )( )( ) ( ){ } ( )

2

2 20 0

12

1 12 2opt

H p eA eVm

H p eA t eV H H t j with H p eVm t m

= − +

∂⎧ ⎫Ψ = − + Ψ = + Ψ = Ψ = +⎨ ⎬ ∂⎩ ⎭

( ) ( ) ( )

( )

2

2

12

12

kEj t

k k k

H p eV j using the separation for r and t : r ,t r u tm t

p eV E and u t em

ψ

ψ ψ−

∂⎧ ⎫Ψ = + Ψ = Ψ Ψ =⎨ ⎬ ∂⎩ ⎭

⎧ ⎫+ = =⎨ ⎬⎩ ⎭

( )21 2 2E E /ω ω= = −


4-35

We assume for simplicity a 2-level system with 2 stationary Eigenfunctions ψi and 2 Energy-Eigenvalues Ei:

The general state of the system is described by a linear superposition of the 2 solutions:

Simplification of the Hamiltonian for the Dipole-Approximation:

From quantum mechanical commutation: (without proof and assuming A2→0)

Perturbation-approximation for transition probabilities :We consider the perturbation by the EM-field as a temporal change in the occupation probabilities and

of the energy states E1 and E2. We assume a solution-”Ansatz” with time-dependent coefficients C1(t) and C2(t) of the form

( ) ( ) ( ) ( )1 2

1 1 2 2

E Ej t j tr ,t r e and r ,t r eψ ψ

− −Ψ = Ψ =

( ) ( ) ( ) ( ) ( )1 2 2

1 1 1 2 2 1 1 2 2

E Ej t j t

ir ,t C r ,t C r ,t C r e C r e with C probability of state iψ ψ− −

Ψ = Ψ + Ψ = + =

21C

22C

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2

1 1 2 2 1 2 21

E Ej t j tr ,t C t r ,t C t r ,t C t r e C t r eψ ψ

− −Ψ = Ψ + Ψ = +

( ) ( )( ) ( ) ( ) ( )

2 22 20

0 0 0 0

1 12 2 2 2

opt

opt

j kr t j t j t j tr

e e AH H p eA eV p eV pA Apm m m m

for an optical plane wawe : A A e A e ; E A j A e E et

ω ω ω ωλ ω− − − −

>>

⎛ ⎞⎧ ⎫ ⎛ ⎞+ = − + = + + + +⎨ ⎬ ⎜ ⎟⎜ ⎟⎩ ⎭ ⎝ ⎠ ⎝ ⎠

∂= ⎯⎯⎯→ = − = =∂

( ) ( )2 2

012 2 1 2 0 1 2 1

0 21 2 1 0 2 1

2 2optA small

opt

QMarg uments

e e A eH pA Ap p Am m m

eAeM H pA pm m

jeA r eE r dipole approximation

ψ ψ ψ ψ ψ ψ

ω ψ ψ ψ ψ−

⎛ ⎞= + +⎜ ⎟⎝ ⎠

= = =

⎯⎯⎯⎯→ =


4-36

The goal is to calculate C1(t) and C2(t) as a function of the classical applied EM-field:

( ) ( ) ( )0 00 2 j t j tE r ,t E cos t E / e eω ωω −= = = +

Inserting the assumed time-dependent solution into the time dependent Schrödinger-equation in the dipole-approximation with Vopt= -Er:

For the process of absorption (the electron is initially at t=0 in the E1-state and makes a transition to the state E2) after some lengthy, but simple mathematics the solution for C2(t) becomes: C1(0)=1 and C2(0)=0 and the weak perturbation: C2(t)=ε<<1

Non-resonant perturbation: ω≠ ω21=(E2-E1)/ħ→ oscillatory behaviour of IC2(t)I2

Resonant perturbation: ω=ω21

→ Transition IC2(t)I2 ~ t2

transition probability

Applying Fermi’s Golden Rule by weighting the spectral components (chap. 4.1.3) we obtain for the matrix element M12 between discrete energy states:

( ) ( ) ( ) ( ) ( ) ( )( )

2

0

1 2 1 1 2 2

2

0 1

ijE / topt with solution

f t

p eV r eV r ,t r ,t j r ,t r ,t r em t

gives using the wavefuction orthonormality : and the normalization :

ψ

ψ ψ ψ ψ ψ ψ

−

≠

⎡ ⎤ ∂+ + Ψ = − Ψ ⎯⎯⎯⎯→ Ψ =⎢ ⎥ ∂⎣ ⎦

= = =

( ) 212 2p C t ~ t not constant

t∂=∂

( ) ( )2 2

12 12 2 1 12204 opt

er xπω ψ ψ ρ ωε

=( )20

0

2optE .... dρ ω ω

ε→ →∫

( ) ( ) ( )( )

222 2 12

2 1 2 0 1 2212

2114

sin t /C t C t eE

ω ωψ ψ

ω ω

−⎡ ⎤⎣ ⎦= − ==−⎡ ⎤⎣ ⎦


4-37

A 4.2 Optical transitions between energy bands (discrete – continuum)As a prototype system we consider the transitions between a discrete level 0 and a state-continuum ΣS

Assumptions:1) The time-dependent solution of the perturbed Schrödinger-equation can be represented

by a time-dependent superposition of the stationary solutions of the Schrödinger-equation:

2) The perturbation Hamiltonian H’(t) is a harmonic (optical field ω) and so weakthat the occupation probabilities of the levels pi(t)=ICi(t)I2 do not change much over an oscillation period T=2π/ω of the optical field:

3) We consider the transitions from a single state 0 into a continuum ofstates s described by a state density function ρ(E)

4) The stationary solutions are normalized and orthogonal (without proof):

5) we consider the process of absorption 0→ ΣS with the initial conditions C0(0)=1, CS(0)=01) we insert eq.2 in eq.1, 2) we left multiply eq.1 by or and make use of the orthonormality eq.3

( ) ( ) 3f f i i f i f i

abreviationDefinition : Dirac Bra Ket notation r ,t ; r ,t ; H ' d r H 'ψ ψ ψ ψ ψ ψ ψ ψ∗ ∗− − ≡ ≡ =∫

1 0 4f i if ii ff ifì f

with ; eq.ψ ψ≠

= Δ Δ = Δ = Δ =

( )i/ t C t ω∂ ∂ <<

( ) ( ) ( ) ( ) ( )0 0

0

0 0 0 0 0

2

0 3

s ss

sin gle statecontinuum S

S S S S S

r ,t C t r ,t C t r ,t eq.

with H E and H E eq.ψ ψ ψ ψ ω ψ

Ψ = Ψ + Ψ

∑= = = =

∑

0Ψ SΨ

( ) ( ){ }0 01 Si H r eq. for obtaining differential equationH ' r s for C and C,t :t

∂ Ψ = + Ψ∂

discrete state 0

Quasi-continuum ΣS

ρ(ħωs)dE

ħωsmonocromaticoptical field

0ψ

Sψ


4-38

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

0 0 0 0 0 0

0 0

5

6

s

S S'

j tS S

S

j tS S S S' S

S'

i C t H C t H ' t C t e eq.t

i C t H ' t C t H ' t C t e eq.t

ω

ω ω

ψ ψ ψ ψ

ψ ψ ψ ψ

−

− −

∂ = +∂∂ = +∂

∑

∑

For optical fields with frequency ω0 and the vector potential A(t) the perturbation hamiltonian H’(t) is harmonic:

Inserting eq.7 into eq.5 and 6 using the assumption 2), meaning that we can neglect all sum-frequency terms in a following time-integration, we only keep the slow difference-frequency (ωS-ω0) terms in eq.5, 6:

Next, we integrate eq.9 over time t:

( ) ( ) ( )0 0 2 7j t j toH ' t H ' e e H ' cos t eq.ω ω ω−= + =

( ) ( ) ( )

( ) ( ) ( )

00 0

0 0

8

9

s

S S'

j tS S

S

j tS S

i C t H ' C t e eq.t

i C t H ' C t e eq.t

ω ω

ω ω

ψ ψ

ψ ψ

− −

+ −

∂ =∂∂ =∂

∑

( ) ( ) ( )

( ) ( ) ( )( )

( )( ) ( )

0

0 00

2

0 0 00

2

0

10

1 11

S S'

s

tj t'

S S

tj t t'

SS

S S SS

S

S

jC t H ' C t' e dt' eq.

inserting eq.10 into eq.8 :

C t H ' C t' e dt' eq

dense quasi continuum with an ener

.t

making the tr gy state densitansition to a :

H ' '

y E

H

ω ω

ω ω

ψ ψ

ψ ψ

ψ ω

ρ ω

ψ ψ ω ψ

+ −

− − −

= −

∂ = −∂

− =

→

∫

∑ ∫

∑ ( )

( ) ( ) ( ) ( ) ( )( )0

2

0

2

0 0 00

1s

S

S S

tj t t'

S S S S

E d

C t H ' C t' e dt'dt

ω ω

ω

ρ ω

ψ ω ψ ρ ω ω− − −∂ = −∂

∫

∫ ∫

Differential-equation for C0


4-39

We solve the above equation for the following conditions of the matrix-density product:

( ) ( ) ( ) ( )( )

( )( )

( )( ) ( )

( ) ( )0

10

2

0 0 0 00 0

1

S

s

s

S

S

t tj t t'

S S S S

Inverse Fourier Transform: F f t 'f t

int egrating df

C t C t' H ' e d dt f t t'' C t' dt't

ω ω ω

ω ω

ωω

ψ ω ψ ρ ω ω

−−

− − −

= −

∂ = − = −−∂ ∫ ∫ ∫

) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

0

2

0

2 2

0 0 0 0 00

2 20 0 0

1

1 2

2

S S S

S S S

t

S S S S S S

S S S

a H ' constant typical for SC

f t t' H ' t t'

C t H ' C t' t t' dt' H ' C tt

Solution : C t exp H ' t C t exp

ψ ω ψ ρ ω

ψ ω ψ ρ ω πδ

π πψ ω ψ ρ ω δ ψ ω ψ ρ ω

π ψ ω ψ ρ ω

− ≈

→ − ≈ − −

∂ = − − = −∂

⎛ ⎞= − → = −⎜ ⎟⎝ ⎠

∫

Matrix Element - Density Product :

( ) ( )

( ) ( ) ( )

2

0

220 00

0

2

2

S S S

S S SSt small ~

H ' t

r C t H 't

π ψ ω ψ ρ ω

π ψ ω ψ ρ ω→→

⎛ ⎞⎜ ⎟⎝ ⎠

∂= = −∑ ∂

) ( ) ( ) ( )1 1 0S S fb typical for ideal atomic levels fρ ω δ ω ω≈ − − →Matrix Element - Density Product :

( ) ( ) ( ) ( ) ( )( )

( )( )( ) ( )

1

0

2

0 0 00

2

0 01 0

1

1

s f

f

tj t t'

S S S f S

t

S SF

resonanceC t C t' H ' e d dt'

t

H ' C t' dt'

ω ω

δ ω

ωωψ ω ψ δ ω ω ω

ψ ω ψ−

− − −

=

=

∂ = − −∂

= −

∫ ∫

∫


4-40

Occur only in systems without scattering and other broadening processes (coherent states)

Conclusions:• continuum transitions are typical for semiconductor and can be considered as a system withextremely short carrier life-time (broadening processes), no coherent Rabi-oscillations can evolve.

• transitions between ideal states with infinitely long scattering life-times allow the transition oscillations to develop.

( ) ( )

( ) ( )

0 0

2 20 0

1

1 0

f S

f S

Solution : C t cos H ' t

C t cos H ' t Rabi Oscillation between f

ψ ω ψ

ψ ω ψ

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞→ = − −⎜ ⎟⎝ ⎠

)

( )( )( )

( )

( ) ( )

( )

2

0 2

2 20 0

2

0

1 1

1 2

0

14 1

14

S S f

S f p

p S S f

p S S f

c :

H ' K

typical for collision broadened atomic levels fwithout proof :

K H ' C t decaying exponential

K H '

ψ ω ψ ρω ω τ

τ ψ ω ψ ρ

τ ψ ω ψ ρ

≈+ −

− − →

⎧ ⎫ << → =⎨ ⎬⎩ ⎭⎧ ⎫⎨⎩

Matrix Element - Density Product is a Lorentzian

( ) 201 C t decaying periodic Rabi oscillation>> → = −⎬

⎭


4-41

Conclusions:

• The extension to continuum-to-continuum transitions ΣS’ → ΣS, resp. valence-conduction-bandtransitions is straight forward be replacing the state density function ρ by the joint density of statefunction ρjs’s.


5-1

5. Optical Processes in Semiconductors:Luminescence and Stimulated Emission

Goals:• Investigate the inverse process of absorption • stimulated emission is the key to optical gain


5-2

5. Optical Processes in Semiconductors:Luminescence and Stimulated Emission

5.1 Light emission by spontaneous or stimulated emission:Light emission in SC results from a de-excitation, radiative transitions of conduction band electrons to hole states by a spontaneous (or stimulated if an optical field is present) emission of a photon. Luminescence requires a non-equilibrium situation of the carrier-distribution, created by carrier excitation from pumping (injection): - optical pumping: photoluminescence- electrical pumping: electroluminescence

(e-/h-injection in pn-junction diodes)

Electrons can also make down-ward transitions to the valence-band due to parasitic non-radiative transitions(phonon emissions, traps, …)

5.1.1 Carrier decay by spontaneous emissionExcess-carriers in the conduction “band” (upper level) decay according to the carrier-rate equations:

τspont=1/A is defined as the spontaneous emission life-time (large B results in a short lifetime τspont )

( )( ) ( ) ( ) ( )22 21 2 1 2 2 2 0

00 spont

pumpspont,no or small

external field

t /NN B E N N N A R N A Solution : N t N et ρ

τρτ=

∂ = − − − + = − = − =∂


2 22 2

2 2

1 1

1 1 1 11

spont NR spont NR

Rspont spont NR spont NR

total

ideal total

N NN Nt

luminescence efficiency : N / N /t t /

τ τ τ τ

ητ τ τ τ τ

⎛ ⎞∂ = − − = − +⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎛ ⎞∂ ∂ = + =⎜ ⎟⎜ ⎟∂ ∂ +⎝ ⎠

5-3

If there are non-radiative de-excitations mechanisms (τNR) present, the rate-equation is modified to:

Materials with high luminescence efficiency require long τNR

5.1.2 Direct and Indirect Interband LuminescenceMaterials which are strong absorbers (large B) have also short carrier lifetimes (A~B, see chap.4.1.1).

Direct transitions are efficient for light generation and have

- large matrix elements M, strong dipoles and/or

- high carrier densities ρ(E)

The bandgap of the material Eg=ħωg defines the wavelength-ranges:

AlGaAs-, InGaAsP-, GAlN-compounds (see chap.5.3)

Indirect transitions are relative inefficient because

Phonons for momentum conservation are required.

Materials:

Si, Ge

5.1.3 Photoluminescence (PL, measurement of τspont)

5.1.3.1 Spontaneous Emission (PL)

In SCs the Excitation-Relaxation process by carrier-injection and photon-emission is a 2-step process:

1) Fast carrier thermalization (~ps) in the C- and V-bands by phonon emission and carrier-scattering→ establish Quasi-Fermi-distributions f(EF,T),

with EQF,n(p)(t), Tn(p)(t)

1) Electron and hole thermalization:(after short optical excitation)

n

p

ρC EQF,n

EQF,pρV

~kT


5-4

( ) ( ) ( )

( )

( ) ( ) ( ) ( )( )0 0

1 1

1

QF ,n pc v

n p

C C QF ,n n V V QF ,p p

E Ef E / exp

kT

carrier concentrations : n E f E ,T dE ; p E f E ,T dEρ ρ∞ ∞

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟= + →

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= = −∫ ∫

2) Photon-emission by e-h-pair annihilation (spontaneous or stimulated emission)Spontaneous emission:

Stimulated emission:

For optical gain g>1: Rstim,net>0


5-5

( )

( ) ( ) ( ) ( )( ) ( ) ( )( ){ }( ) ( ) ( ) ( ){ }

21

2 1

21 21 12

221 2 1 2 1 1 2

221 2 1

2 1 1

2

stim,net

vc j ,cv C V V C

vc j ,cv C V

Stimulated emission : R E R R

M E ,E f E f E f E

f E f E

f E

M E ,E

π ω ρ

π ω ρ

= − =

= − − =

−

−

=

( ) ( ){ }2 1 0C Vf E f E− >

( ) ( ) ( ) ( ) ( ) ( ) ( )2 221 12 2 11 2 1 21 2 12

1 0

12 2stim,net vc j ,cv vc j ,cvV C

~ ~

absorption : R E R M E , f ME EE Ef E ,π πω ρ ω ρ→ →

⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟− −⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠

≈ ≈⎩ ⎭

− =

( ) ( )( ) ( ) ( ) ( ) ( )( )

20

0

2 2221 2 1 21 0 2 1 2 1

0

3 321

21 3 3

221 21 21 21 21 2 1 2 1

21

2

2

2

8

2 1 1mode

ph

cv cv ph

mode

,spont vc j ,cv C Vph

spont

E s

eWith E E E using M eE r M s r

n ESpontaneous emission : using A= E B Bh c

R E E E M E ,E f E f EE s

R R

ωε

ω ψ ψ ω ψ ψε

πρ

πρ ω ρ

=

= − = ≅ → ≅

⎛ ⎞=⎜ ⎟

⎝ ⎠

= −

= ( ) ( )( ) ( ) ( ) ( )( )( )

2 2

1 21 21 21 21 2 1 2 1 2 1 210

2 2 1mode,spont j ,cv C VeE dE E E r E ,E f E f E dE

enters the carrier rate - equation

πρ ψ ψ ρε

= −∫ ∫


5-6

The gain spectrum g(ω) and the spontaneous emission Rspont(ω) With the relation between stimulated emission and gain we get (see chap.5.2 for details):g(ω)=R21,net(ω)/(vgrsph) ; sph=photon density

The occupied electron and holes-states (pairs) are clustered within ~ΔEQF+kT after after the thermalization time (~few ps) at the C,(V)-band minima, (maxima).

With the joint density of states for bulk-SC and the quasi-Fermi-functions g(ω) becomes:

Low carrier density: ΔEQF<Eg (absorption) High carrier density: ΔEQF >Eg (gain)

N2~0 << N1~N N2>N/2

Bernard-Duraffourg Gain condition: EQF,n –EQF,p> E21>Eg

For gain the quasi-Fermi-level separation must be larger than the transition energy.In pn-junction carrier injection the external diode voltage VD~ EQF,n –EQF,p

The absorption spectrum in SC (ΔEQF<0 ) is therefore much large than the gain spectrum !

( ) ( ) ( ) ( ) ( ) ( ) ( )( )3 2 21 2 221

21 21 21 21 212 2 2 10

41 22 QF ,C Q

//

j F,cv g j ,cr

,Vvg

eE E g f EE M E Ev

f Eωμρ ω ρπ π ε

⎛ ⎞= − → = −⎜ ⎟⎝ ⎠

transparency absorption

Eg E21

g −αΔω<0

QF ,n QF ,pE EωΔ ≈ −g −α

transparency gain absorption

E21Eg

1

-1

( ) ( )( )2 1QF ,C QF ,Vf E f E−

-1


5-7

5.2 Stimulated Emission and Optical GainStimulated Emission is the key process for optical gain g(E) and for optical oscillators and amplifiers.For optical gain g>1 we require population inversion N2>N1 or Rstim,net>0.

To keep the mathematics simple we consider the

Einstein formulation for gain of a 2-level system:

A light beam with a spectral energy density ρ(ω,z) and intensity I(ω,x)= ρ(ω,x)vgr=ħωsph(x) vgr travels through a discrete 2-level (E2, N2, E1, N1) gain medium. In steady state the beam changes it intensity I(ω,z) along z according:

Beam energy density is changed by absorption / emission,which is scattered out of the beam by spont. emission:

Stationary Rate-equation (Rpump=0)

Conceptional inconsistency of Einstein-model with discrete energy levels:

the attenuated/amplified beam is spectrally broad ρ(ω), but the interaction with the discrete 2-levelsystem is monochromatic at exactly ω21.to resolve the contradiction we assume that the energy levels are broadened by scattering processes (collisions, etc.). The probability S(ω) that the 2-level system (ω21) interacts at ωis described by the normalized lineshape function S(ω-ω21).

( ) ( )0 x II ,x I , e Ix x

α ρω ω α αρ− ∂ ∂= → = − ⇒ = −∂ ∂

( ) ( ) ( ) ( )21 21 2 1 21,spont ,netR x R x N N B ,xρ ω= − = − −

S(ω−ω0)

Δω

(ω−ω0)

∼1/Δω( )21 1

Normalization :

S dω ω ω− =∫

x x+Δx Δx=vgrΔt

A

I(ω,x) I(ω,x+Δx)=I(x)+ΔI

xE1

E2

spontaneous emission, R21,spont

stimulated emission, R21,net


The change of optical energy of the wave result from net stimulated emission. The change of the energy density Δρ(ω,x), resp. intensity ΔI(ω,x) in the energy interval dħω during the transition of the volume AΔx in the time of flight Δt=Δx/vgr is :

Optical gain by electrical or optical pumping of active materials is a key achievement for photonics.

Obviously, insulating active materials have to be pumped optically, where as doped semiconductors can form pn-junctions for direct electrical pumping by e-h-injection.

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 21

21 21

21 21 2 1

2

2 211

1

,net ph g

gr ph gr

,net

gr

,net g rrgr

Iwith I v s v and Ix

I x x t R Sx

and t x / v ; for gain : g

R S / v B N N Sv

with N N N

using the stationar

g N ,N , R S / s v

y

ρ ω α

ρω ω αρ ω ω ω ω ω

αωω ω ω ω ωω ωωρ

∂= = = − →∂

∂∂Δ = Δ ∂ = − ∂ Δ = Δ − ∂∂

Δ = Δ = − →

= − = = − −

=

−

+

( )( )

( )

( ) ( )( ) ( )( )

21 2 1 2

2 2

21 2 1

21

0

0

2

2

2 pump

pump ,net spont pump

pump pump

,net pump

pumpgr gr

rate equations for level 2 to determine n R :

R R R R B N N An

for small small signal gainR AN N R / A N

R B N N B R / A N

B Bg S R / A N Nv v

ρ

ρ

ρ ρ

ω ωω ω ω

= − − = − − −

→ →≅ − → = <

= − = −

= − − < ( )S linear gainω

( )gr

B NSv

ω ω−

5-8

g

Rpump

gain

lossRpump,trans=AN/2

0AN

( )gr

B nSv

ω ω+ ( )gr

B NSv

ω ω


5.3 Semiconductor Materials Composition control for a,n, ΔE, Eg:

Requirements on semiconductor systems:

- direct transitions / suitable bandgap energy Eg

- efficient and high p- and n-doping- heterojunctions need composition control for

lattice matching

- composition control for atomic abrupt interfacesatomic thin layer growth

- composition control for refractive index nand heterojunction barrier hight ΔE

- high quality ohmic contacts

- high quality, large single crystal wafers

5-9


5-10

IV-SC (Si, Ge) are useful as passive or absorbing materials but not for light generation.

Limitations of III-V, II-VI-compounds:

- demanding growth (molecular beam epitaxy MBE, metal organic chemical vapour depositionMOCVD) processes for composition controlling for a, n, Eg, doping, ….

- no natural high quality oxide - difficult ion-implantation or diffusion for doping - challenging dry etching processes- moderate wafer size and quality and high costs- limited wafer materials(GaAs, InP, GaP, …)

Wavelength range and Material systems:

5.4 Electroluminescence (electrical carrier injection)

Electroluminescence (excitation by carrier injection) is of enormous practical importance for optoelectronic devices.

The most widely used technique is building pn-heterojunction diodesfor carrier confinement (lattice matching, substrates).Under forward biased current-flow carrier non-equilibrium results from

- hole injection from the p into the n-region

- electron injection from the n into the p-region

generating excess-minority carriers in the bands

non-equilibrium: np>npo , pn>pno EFQ,p< EFQ,n

Separation of quasi-Fermi-levels: EQF,n-EQF,p>Eg optical gain g

np=npo , pn=pno ; EFQ,p= EFQ,n=EF nppn>>ni2 ; EFQ,p> EFQ,n


5-11

EQFnEQFp

EQF,n

EQF,p

eUD

EQF,p

EQF,n


5-12

5. Optical processes in SC: Luminescence & Stimulated Emission Summary:• Luminescence describes the spontaneous emission of photons from excited states by a

dipole transition from the higher to the lower state.

Spontaneous transitions can be considered as stimulated transitions induced by the vacuum field fluctuations.

• The spontaneous emission rate is proportional to the matrix element, the state-pair density and the level occupation, but does not depend on the photon field !

• Strong absorption (direct materials) is coupled to short spontaneous carrier life times

• Stimulated emission is the inverse process of absorption but becomes only effective if the material is inverted in carrier population. Stimulated emission is the most important non-equilibrium process for optical gain and lasing.

• Stimulated photons are spatially and temporally coherent to the stimulating photons • Spontaneous and stimulated emission typically occurs slightly above the bandgap energy

Eg and is controlled by the material and its composition

• The stimulated transition rate has the same dependencies as the absorption rate, but the occupation of the levels is reversed

• Electron and hole injection across pn-junctions is the most important mechanism for carrier inversion and optical gain in SC

• Photo- and electro-luminescence are the base of important pumping mechanisms in solid-state and SC light emitters.


6. Quantum Size Effects in Semiconductors

Stack of quantum dots (QD) Random self-organized QD

ultra-thin InGaAs/AlAsSb Quantum Wells

Goals:• use of quantum-size effects do enhance the energy state density ρ and matrix-elements M • quantum size effects increase optical gain• quantum size effects can help confine carriers spatially to the optical field

6-1


6-2

6. Quantum Size Effects in Semiconductors

6.1 The Concept of Quantum ConfinementInterband transitions Rkl, gain g, absorption α, carrier lifetime τ depend on 1) the matrix-element M and

2) the joint density of states ρj,kl(E).

Mkl and ρj,kl depend on the band-structure, resp. wave functions of the SC.

The band structure (meff, Eg, electron affinity χ, etc.) can be modified by composition andgeometry changes on an atomic resp. nanometer-scale

Band-structure engineering towards artificial quasi-atoms

Size-dependence of optical properties:

Confining an electron (hole) spatially in a range Δx, comparable to the DeBroglie-wavelength λDB=hpof the matter wave, makes the particle momentum p, resp. the energy E in the x-direction uncertainby Δpx, resp. ΔEx

For Econfinement to be observable: (avoid thermal “smearing”)

2 2

22 2x

x confinmentpHeisenberg uncertainty : p x E E ~m m x

ΔΔ Δ → Δ = =Δ

∼

2

2 5 32 002confinment x ,thermal

e,eff

E ~ E kT x ~ nm@T/x

Km

≥ ⇒ Δ ==Δ


6.2 Quantum Confinement in Heterostructure SemiconductorsConcept:Confining the electron motion is possible by building spatial potential barriers ΔE into the SC. Changing the material composition modifies the bandgap Eg and the electron affinity χ , resp. the band-offsets ΔEC, ΔEV.

The potential confinement can be in 1, 2 or 3 degrees of freedom of motion:

confinement: 1-D Quantum Well 2-D Quantum Wire 3-D Quantum dot/box

Realization: simple, film-growth difficult nano-scale structuring or self-organization

Dimensional control on an atomic monolayer-level (<1nm) is required !

dot: low Eg,dot

barrier: high Eg,barrier

6-3


6.3 Band engineering: Electronic states in SC-crystals

6.3.1 Quantum Well StructuresLattice matched films with atomic abrupt interfaces and nm-thickness can be grown by - Molecular Beam Epitaxy (MBE) and

-Metal-Organic Vapor Phase Epitaxy (MOCVD)

(see Chap. 6.5)

Because the bandgaps Eg and electron affinities χ of AlxGa1-xAs and GaAs are different, band-offsets ΔEC and ΔEV in the C- and V-band appear forming the necessary potential barriers.

g ,AlGaAs g ,GaAs C V g ,AlGaAs g ,GaAsE E E E E E> → Δ + Δ = −6-4

GaAs/AlGaAs-System:

Δx

ΔEC

ΔEV


6-5

6.3.2 Electronic Levels in QWsThe electron in the QW can move quasi-freely in the x,y-directions but is confined in the z-direction.

The potential that the electron sees is the superposition of the lattice potential V0(r) and super-posed potential VQW(z) caused by the QW confinement structure due to the composition change:

To find the total wave function we have to solve the Schrödinger-equation in the effective mass approximation:

Localized Wavefunction in QW:

In unconfined bulk crystals the solutions are only extended plane Bloch-waves:

( ) ( ) ( )lattice confinementV r V r V z= + ( )r ,tΨ

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

22

2 2

22

2

2

2

e

j E / t

e

depending on t onlydepending on r only

e

V r jm t

assuming a solution : r ,t r w t

jV r w E w t Cem w t

and r V r r E rm

ψ

ψψ

ψ ψ ψ

−

∂− ∇ Ψ + Ψ = Ψ∂

Ψ = →

∇ ∂− + = = → =∂

− ∇ + =

( ) ( )1 1jk rk k k k

cristal

r r e with normalizedV

ψ μ ψ ψ= =

z

y x

The macroscopic Well-, Wire- or Dot-potential VQC(r) confines the wave function by reducing the freedom of motion in 1, 2 or 3 directions.

It can be shown that a localized solution of the Schrödinger-equation has the approximate form:

The unperturbed Bloch-function uk(r) of the bulk is a solution of

For QWs (no wave propagation in z-direction, standing wave) we can separate the Schrödinger-equation into the (x,y)- and z-dependent terms, resp. into a planar Bloch-function and the Envelope-function:


6-6

( ) ( ) ( ) ( ) ( ) ( )2

2

2 k lattice k ke

r V r r E k r with the Eigenvalue E km

μ μ μ− ∇ + =

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

2 2 22

22

2 2

2 2

2

2

XY XY XYxy k lattice k XY k XY XYe e

z n confinement n n ne

n XY n XY n XYe

r V r r E k r with E k km m

F z V z F z E F z with E confinement energy of level nm

E k E E k E km

μ μ μ⎧− ∇ + = =⎪⎪⎨⎪− ∇ + = =⎪⎩

⇒ = + = +

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

1

1

1

xy xy

k

jk rk k

cristal

jk rk k xy xy

cristal

k

with F r envelope function and lattice periodic Blochfunction

Bulk crystal : r r eV

Quantum Well : r r e with k , r in plane ( x, y )vectorsA

Quantum Wire :

F z

μ

ψ μ

ψ μ

ψ

= − = −

=

= = −

k kψ r = F r μ r

( ) ( ) ( )

( ) ( ) ( )

1

1

x xjk rk x x

cristal

k kdot

r r e with k , r in wire ( x )vectorsL

Quantum Dot : r rV

F y,z

F x,y,z

μ

ψ μ

= = −

=


6-7

Electronic Levels in the QW-Potential VQW: (Exercise III)For an QW of infinite and finite depth in the C-band we find the Eigenfunction and Energy-Eigenvalues

infinite QW-depth: (V0=ΔEC= ∞) finite QW depth V0: (V0=ΔEC)

The solution procedure for Q-wires and Q-dots is just the same but applied to 2- or 3D quantization.

( ) ( )( ) ( )

( )

22

2 22

2

02

12

2 1 2 32 2

z n n confinement ne

n

n

ne

n n n

From the solution of

F z E V z F zm

as energy eigenvalues E and Eigenfunctions F (z)

E n andm d

n nF z sin k z with k ; n , , ...d

π

π π

∇ + − =

=

⎛ ⎞= + = =⎜ ⎟⎝ ⎠

nE no analytic solution

6.4 Optical absorption and emission in Quantum-confined SCs

6.4.1 Matrix-element and Selection Rules in QWsSimilar to the wave functions of the QW the matrix-element M can also be split into a z-dependent and a x-y-dependent part by direct application of the definition of M.

Observe that the optical wavelength λ (μm) is much larger than the unit-cell resp. the dipole charge separation distance r’<<λ.

ΔEC

( )rψ


We expect that M, resp. MDP depends on the polarization of the EM-field (we assumed in-plane polarization):

Making use of the wave functions for an interband transitions v: n → c: n’

From the matrix-element Mn.n’ we obtain for QWs with infinite potential barriers (sinusoidal solutions for Fn(z)): Δn=n’-n=0allowed: 1’→1 , 2’→2 , etc. forbidden: eg. 1’ →2

(odd-even transitions are not allowed)

( ) ( )( )

( )

( ) ( )( )

22 22

0 0

3

1

2 2

xDP

if f i j ,cv f i x j ,cvE , ,

in plane polarization

*DP f i f i

j k ro o

EM

~

eTransition rate i f : R erE x E

M x r x r d r

having made use for the EM - wave of : E r E e E

π πω ψ ψ ρ ω ψ ψ ρ ω

ψ ψ ψ ψ−=

→ = − =

= =

= ≈

∫

f i f i f ix y zψ ψ ψ ψ ψ ψ= ≠

6-8

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) 2

1 1 0

1

xy xy xy xyik r ik ' ri V h,n f C e,n' xy

xy xy

DP CV n,n'

CV C V C V

n,n'

r z e and r z e with r x,y,A A

k k '

M M M / A assuming sepatation of z and x,y in the integration and k - conservation

M x r x r d r dipole momentA

M

ψ μ ϕ ψ μ ϕ

μ μ μ μ∗

= = =

⇒ =

=

= = −

=

∫( ) ( )e,n' h,n e,n' h,nz z dz overlap int egralϕ ϕ ϕ ϕ∗= −∫


6-9

6.4.2 Density of States in Quantum-Confined SCs (Exercise III)Theoretical stair-case QW-absorption spectrum:

Experimental QW-absorption spectrum:

Conclusions:- quantum-confinement increases the density of state ρ(e) at the absorption edge,resp. absorption and gain

- quantum confinement improves the matrix-element M

major device improvements

Density of statesNiD(E)=ρ(E)


6.4.3 Excitons in quantum confined structuresIt can be shown that the quantum confinement of electrons and holes of excitons in QWs increasesthe binding-energy En(n) >>kT through the reduced spatial separation.

Excitons in QW are stable and observable at RT.

6.4.4 Quantum Confined Stark-EffectIn QWs an external electrical field Ez changes the shapeof the well-potential and thus the energy-levels, leadingto a field-dependent shift of the absorption-edge Eg(Ez), resp. bandgap.

Electro-optical Absorption-Modulator

d=QW thickness

6 2 2 4

2

2243

Zg

e E m* dEπ

⎛ ⎞Δ = − ⎜ ⎟⎝ ⎠

6-10

Ez=0 Ez ≠0


6-11

6.4.5 Intersubband Transitions

Quantum-confined SC-structures generate Sub-bands within their conduction- and valence bands.

So called intersubband-transitions between the different subbands within the same band are possible.

These intersubband-transitions are interesting for device applications because of their short carrier life-times in the ps-range.

Intersubband transition in k-space:

(doping)

Conduction band:


6-12

6.5 Fabrication of quantum confined structures: (optional, 8.Sem)

The epitaxial growth of thin (~5nm) and thick (~2μm) defect-free mono-crystalline single and multiple (5-100) layers at high growth-rates over large areas is essential for opto-electronic devices.Typical growth methods:- liquid phase epitaxy LPE (growth from supersaturated metal-liquid, eg. Ga)- Metal Organic Chemical Vapor Deposition MOCVD (growth from gas-phase by gas-decomposition)- Molecular Beam Epitaxy MBE (growth in UHV-vacuum from molecular beams of the crystal components)

6.5.1 Technological Requirements and Challenges

Challenges of III-V and II-VI-compound SCs:• small wafer size, 3-4“, expensive and brittle• doping by ion-implantation and diffusion is difficult (only growth)• no stable native oxide• structuring by dry and wet-chemical processes• difficult regrowth and local growth techniques

• thin and reproducible layers of a few monolayer thickness• atomically flat, resp. abrupt interfaces• perfect lattice matching, resp. composition control• high growth rates (several μm/h if possible) • low defect density• ultra-low and very high doping, abrupt doping and composition gradients

• highly complex layer stacks


6.5.2 Molecular Beam Epitaxy (MBE)Molecular Beam Epitaxy deposits compound crystal films on heated substrates by molecular beams of the different atomic species from so called effusion cells (heated ovens) under ultra-high vacuum conditions (prevent contamination). Thermal beams of elementary molecules (e.g. AlGaAs: Ga-, As-, Al-beams, InP: In-, P-beams) provide the material to the surface for crystal growth. The molecules stick to the surface and then diffuse into the crystal lattice positions.

Atomic layers grow layer by layer with a growth rate of typically ~0.1-0.5 μm/h.

Growth processes: Schematic MBE-UHV-Chamber with Effusion Cells (for elemental beam evapoartion eg. Ga, As, Al-beams)

6-13

1) Absorption from molecular beam (sticking coefficient)2) Surface diffusion3) Desorption4) Incorporation in growth front

1)

3) 4)3)


Effusion Cell: MBE-UHV-Chamber with Effusion Cells, wafer transfer,(heater and ceramic crucibel)

MBE in FIRST-Lab ETHZ:

6-14

2) wafer loading3) wafer transport6) system control7) wafer rotation10) liquid nitrogen-cooled vacuum chamber 11) wafer holder and heater13) beam flux monitor14) beam shutters16) wafer temperature measurement

In the Metal Organic Chemical Vapour Deposition–Epitaxy (pyrolytic decomposition from the gas-phase) a hydrogen carrier gas stream transports as an inert gas ithe highly diluted precursors of metal-organic compounds (metal-methyls, metal-ethyls, etc.) for the group III-elements (Al, Ga, In) and metal-hydrides for the group-V-elements (P,As).

The wafer substrates (GaAs, InP) are heated up to temperatures of 400-800oC, so that the metal-organics decompose by complicated reaction-chains releasing the metal atom.The free metal atoms are absorbed on the wafer and diffuse on the surface to the corresponding lattice-sites of the growing crystal. Growth rate and growth temperature are relatively high, typ. ~1-2 μm/h.

Schematic Growth process: Schematic MOCVD-growth Reactor:

gas flow

substrate, growth front


6.5.3 Metal-Organic Chemical Vapor Deposition (MOCVD)

6-15

pyrolyseof metalorganics


6-16

QuartzReactor

Gasinlet

RF- or optical heating of graphite susceptor holding the wafer

Schematic MOCVD process and MOCVD-Reactor at FIRST, ETHZReaction Scheme for GaAs-Growth:

Gas flow

substrate

substrate


6-17

6.5.4 Nano-meter scale structuring by growthIn order to come above the thermal energy KT the confinement energy, the confinement dimension

must be less than ~10nm @RT for quantum dots.

6.5.4.1 Lithography and etchingObjects with diameters of a few nm are can be exposed by electron beam- or AFM-lithography, but

transferring these dimensions into original quantum-wells by dry- or wet-etching require a difficult compromise between structural etch-damage and structural accuracy.

6.5.4.2 Structured growthConventional MOCVD or MBE growth produces a homogeneous film on the wafer. In general a lateral

variation of film thickness or composition is not possible.

Lateral growth conditions can be influenced during or after growth by the following mechanism:

• Local growth by oxide or nitride growth masksMBE: polycrystalline growth on mask, MOCVD: no growth

• Structured (geometrical topography) substrate topology - lateral variations of layer thickness and composition (mainly for QWs)

- different growth rates depending on crystal facet

• Planar overgrowth over etched structures (critical for etch damage by small dry-etched features)

• Local QW-disordering by mechanical, thermal and optical stress


6-18

6.5.4.3 Self-organized growthTo avoid structural damage or very advanced lithography Different Growth Mechanisms:spontaneous self-organized growth of Q-dots due to layer-by-layer island growthSurface-strain induced nucleation is possible.

Control of dot-size, dot-location and dot-density isdifficult and dependent on the material system.

InGaAs-Q-dots:∅=20nm h=8nm


6-19

6. Quantum Size Effects in SCs:

Summary:• Quantum confinement occurs if the structural dimension are in the order of the DeBroglie

wavelength (SC: typ 10nm) and the confinement energy En is larger than the thermal energy kT

• The confinement of carrier motion can be in 1 – 3 dimensions: Q-wells, Q-wires, Q-dots

the systems exhibit 2 → 0 dimensional physics of electron motion.

• Quantum confinement in SC is realized by potential barriers (due to changes in composition and bandgap) in the conduction and valence bands of the SC

• Confinement energies can be calculated from the envelope function quantization of the Bloch matter wave caused be the macroscopic potential modulation of the confinement structure

• Quantum confinement strongly modifies the density of state functions and to a smaller extent also the matrix elements of dipole transitions. They improve the optical gain in SCs considerably and are fundamental for modern optoelectronic devices .

• Quantum confinement increases the transition energy and also the binding energy of excitons (excitons at RT in QW).

• Fabrication of quantum-confined structures requires an advanced nm-scale fabrication technology with a dimensional control of less than 1nm !

• Quantum confinement enables new quantum effects, which are partially very useful for practical optoelectronic devices (QW-, QD-lasers)


7. Nonlinear Optics

Goals:• In semiconductors nonlinear harmonic generation and up-conversion is of little importance,

but resonant nonlinearities are important in non-linear and ultrafast switching devices • The electro-optic effect, the electrical and optical Kerr-effect are fundamental in many devices• Future optical computing concepts depend to a large extent on optical nonlinearities

7-1

ω

2ωnonlinear X-tal


7. Nonlinear OpticsIntroducing the harmonic oscillator as a model for polarization in atoms, molecules or crystals, we have assumed strict linearity. Linearity is guaranteed by the assumed ideal parabolic potentials of the harmonic oscillator. Wave in linear media do not interact!

Interaction between light waves is only possible with nonlinearities of the medium.

In intense light fields from lasers the optical field-strength becomes comparable to the inner-atomic field (actually still typically ~106 smaller) and the nonlinearity of the potential becomes effective.

This results in nonlinear P(E)-relations and a nonlinear susceptibility χ.

Motivations: Similar to electronic nonlinearities, optical non-resonant dipole-nonlinearities are often used to generate optical harmonic or sum- / difference frequencies at 2ωopt, 3ωopt …. or ωopt1± ωopt2. Nonlinearities for harmonic frequency generation are not important for SC-based devices, but there is increasing interest for resonant nonlinearities (with carrier generation) for ultrafast switching devices.

7.1 Nonlinear Susceptibility TensorGenerally, nonlinearities need at least a quantum mechanical description of the medium.

The starting point for a classical description of optical non-resonant nonlinearities is the EM wave-equation including the nonlinear polarization term P(E):

7-2( )

2 22

0 0 0 02 2E E Et t t

P Eμ σ μ ε μ∂ ∂ ∂∇ − − =∂ ∂ ∂


In general the polarisation of a medium is the summation of the individual dipoles:

the polarization p(E) or/and the dipole-density N(E) can be a nonlinear function of the optical field E.N can be changed if photons of the exciting field are absorbed by resonant transitions (typical for SC for excitation above the bandgap) Possible is also the simultaneous nonlinear interactions with other particles interacting with the field, eg. phonons, polarons, excitons, … .

Methods of solution: Born-Approximation and coupled waves

For non-resonant interactionsLinear: (general tensor relation) Nonlinear: (Taylor-expansion, for simplicity scalar)

7-3

0 1ij r ,ij ijP E ;ε χ ε χ= = + ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

1 2 3

1 2 32 30 0

n th

nonlinear

P P P P .... ; P n order polarization

P E E E ... E Eε χ χ χ ε χ

= + + + = −

= + + + =

( ) ( ) ( )P E N E p E=

The nonlinear wave-equation can be solved by an iterativeprocess, where the previous solution Ei is used for the approxi-mation of the nonlinear term for the solution Ei+1:

First order Born-approximation is just the first iteration

( ) ( ) ( )( )( )

( ) ( )LIN 0 0 0

2 2 22

1 0 0 1 0 02 2 2

2 2 2 22 2

1 1 02 2

0 LIN 0 NL 0

P E E

N20

1 00

L1

E Et t t

P E P E P E

1 nE 1 Ec ct t t

PE E E

ε χ

μ ε μ μ

χ μ

=

∂ ∂ ∂∇ − = =∂ ∂ ∂

∂ ∂ ∂∇ − + = ∇ − =

+

∂ ∂ ∂

⎯⎯⎯⎯⎯⎯⎯→


Generalizing (Taylor expansion) for the anisotropic case (E and P are not co-linear):

Due to the crystal-symmetry, a large number of the susceptibility tensor components become 0 or are identical with other elements. In practice, the number of tensor elements is much smaller.

7.2 Non-Resonant Nonlinearity: Anharmonic scalar OscillatorAssuming a non-resonant excitation ω≠ω0 with virtually no energy absorption by the atomic oscillator the modified classical oscillator model is still useful to explain basic effects.

Modifying the parabolic potential VS(x) to include weak nonlinear terms:

For 2nd-order effects: (non-centrosymmetric force) 7-4

( ) ( ) ( ) ( )1 2 3 2

2 3

nonlinear

Intensity

nd order NL rd order NL

E E E ...χ χ χ χ

− −

= + + +

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 10

2 2 20

3 3 30

9

2 27

3 81

i ij j ijj x ,y ,z

ndi i , jk j k ijk

j x ,y ,z k x ,y ,z

rdi i , jkl j k l ijkl

j x ,y ,z k x ,y ,z l x ,y ,z

P E linear ; i x, y,z terms

P E E order ; i x, y,z terms

P E E E order ; i x, y,z terms

ε χ χ

ε χ χ

ε χ χ

=

= =

= = =

= = → −

= − = → −

= − = → −

∑

∑ ∑

∑ ∑ ∑

( ) ( ) ( ) 3 43

2 2 2 240 0 3 4

1 13 4

12S S S ee eC m x C m xF x V x V x m .....x with : C x C x

xω ω+∂= − ⇒ = + >> >>

∂+

( ) ( ) 20

23S e eSF x V x m x C m x

xω∂= − = − −

∂


Schematic representation of quasi-static nonlinear polarization P(E):Linear, small signal Non-linear

Anharmonic 1D 2nd-order oscillator under single frequency excitation: Harmonic Generation

Inserting the “Ansatz x(t)” (see Lit. Fox) gives for X1(ω) and X2(2ω) as well for χ(1) (ω) and χ(2) (2ω) :

Primary wave at ω generates polarizations at ω and 2ω , respectively to new waves

DC(ω=0), SHG(2ω) ~ nonlinearity C3. SHG becomes large if ω → ω0 (resonance enhancement) 7-5

( ) ( ) ( ) ( )( ) ( ) ( )( )

23

220 02

21 2

1 2

2

2 2 1 2 2

j t j te e e X X

j t j

e

t

Newton : m x m x m x eE t with E t E / e et t

Ansatz for solution with frequencies , : x t / X e X

C m

e c.c.with X X

x ω ω

ω ω

γ ω ω

ω ω ω ω

−∂ ∂+ + + = − = +∂ ∂

= + +>>

( )( )

2 20 3

0

e e X

X

Static polarization :t

m x C m x eE quadratic for x

x E

ω

∂ = ⇒∂

+ = −

⇒

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

21 1

02 20 0

222 1 1 20

02 3 3

1

2 2 2

xe

ex

Ne P ,t E fundamental wavem jm P ,t E second harmonic generation ( G )C SHN e

χ ω ω ε χ ωε ω ω γω

εχ ω χ ω χ ω ω ε χ ω

= → =− +

= → =

non-sinusoidalwith harmonics


Anharmonic 1D, 2nd-order Oscillator-Equation for Frequency Mixingextending to 2 frequency excitation ω1, ω2 → expected mixed frequencies 2ω1, 2ω2, ω1±ω2

Calculation of the susceptibilitiesof ω=0, ω1 ±ω2

2ω1 , 2ω2

(without proofLit.: Simmons p.332)

7-6

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

1 1 2 2

220 01 022

1 2

1 1 11 2

12 20

23

23

2 2

11

2

j t j t j t j te e

e

e e X X

ie i i

m x m x m x eE t with E t E / e e E / e et t

Ansatz for solution for anharmonic perturbation : x t x t x t ....

linear solution : x t x x cc.ex E

m j

C m x

C m x

ω ω ω ωγ ω

ω ω

ωω ω γω

− −∂ ∂+ + + = − = + + +∂ ∂

= + +

= + +

→ =− −

)( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 2 2

0

22 2 22

0 01 022

2 2 2 2 2 21 2 1 2 1

21 13 1 1

2

2

2 2

1

2 2 0

ij ti

j t j t j te e e

j te

e

m x m x m x E / e e E / e et t

solution Ansatz : x t x x

. pertutbation

C m

x

x x

x x

ω

ω ω ω ωγ ω

ω ω ω ω ω

ω

ω

ω

+

− −∂ ∂+ + + = + + +∂ ∂

= + + + +

+

+ −

( ) ( ) ( ) ( )( )

( ) ( )( )( ) ( )

( )

( )( ) ( )( ) ( ) ( ) ( )

1 232 3

1 2 1 22 22 2 2 2 20 0 1 1 0 2 2 0 1 2 1 2 1

232 23

22 2 2 2 20 0 0

32 3

2 2 2 2 2 20 0 0 1 1 0 2 2

1 1

24

1 1 10

i

j t

e

j t

i ie i i i i

e

C Ne e E Em j j j j

C Ne e Em j j

C Nem j j

ω ω

ω

χ ω ωε ω ω γω ω ω γω ω ω ω γ ω ω γω

χ ωε ω ω γω ω ω γω

χε ω ω ω γω ω ω γω

±

± =− + − − ± − ± +

=− − − −

⎡ ⎤⎢ ⎥= +

− − − −⎢ ⎥⎣ ⎦

∓


7.3 Resonant Nonlinearities (carrier generating transitions)Resonant (absorbing) nonlinearities are important in semiconductor based nonlinear optical devices such as ultrafast all-optical switches.

• The frequency ω of the optical field coincides with the transitions- or resonance frequency ω21.

• Photons are absorbed by “stimulated” absorption and the carrieroccupation distribution is changed by carrier transitions.

Example: Absorption-Saturation in a 2-level system creates 3rd-order nonlinearity χ(3)

Concept: Generated carriers in the C-band block the transition by band-filling (all states filled)

t=0 ; α>0 t>0 ; α~0

The resonant nonlinearity has the sameeffect as a 3rd order non-resonant non-linearity, but is ~105 stronger.

glass: n2=10-16-10-7cm2/W

semiconductor: n2= 10-10-10-2cm2/W

These effects play an important role in SC-based all-optical switching devices, where “light controls light” very efficiently at low optical power and over short device lengths (mm versus km in fibers). 7-7

( )( ) 2

nonlin

nonlin

I Kramers Kronig

n n I n I

α α= → −

Δ = Δ ≈

ρ

E

ρ

E

bandfilling absorptionbleaching

ω ω

7.4 Second-order Nonlinearities (C2≠0, C3≠0, C4= …=0)2nd-order effects (C3) occur for crystal potentials with a cubic nonlinearity (C3, resp. χ(2)) → the crystal can not be centrosymmetric.

2nd-order nonlinearities generate frequency “pairs”, resp. 3rd waves with ω3= ωi ±ωj ; i, j = ± 1, ± 2 and ωi= - ω−i .

The generated resulting 3rd wave at ω3 can interact again with the original waves i and j and exchange energy (coupled mode theory).

Major 2nd-order nonlinear effects:

Classical frequency-mixing effects result from the quadratic dependence of the polarisation:


7-8

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 2 2 2

2 23 0 3 1 2 1 1 2 2

2 23 0 3 1 2 1 1 1 2 2 2

20 3 1 2 1 1 2 2 1 2 1 2

2 2 2 2 21 2 1 2 1 2

2

0 2 2

E t E cos t E cos t

eg. P E E

P ,t E cos t E cos t

E E / cos t cos t

generalized frequency combinations : ; ; ; ;

ω ω ω ω

ω ε χ ω ω ω ω ω

ω ε χ ω ω ω ω ω ω ω

ε χ ω ω ω ω ω ω ω ω ω

χ χ ω χ ω χ ω ω χ ω ω

= + →

= = +

= = +

= = + + + −

→ + −

} spont. parametric fluorescence

}parametric oscillation


( ) ( ) ( ) ( )( )

( ) ( )

1 2 1 1

3 3

2 2 2 2 j t k k r22 1 21 1 0 NL 0 0 0 02 2 2

0

j t k r1 1

nE E P E E E ec t t t

solution" wave Ansatz" : E t ,r E e

ω ω

ω

μ μ χ ± − ±

−

∂ ∂ ∂∇ − = =∂ ∂ ∂

− =

3 1 2ω ω ω= ±

3 1 20

i ii i

nk k k with k ec

ω= ± =

Remark:Photon splitting, if permitted by matching conditions is not trivial.It is a quantum mechanical effect with the interaction of vacuum field fluctuations.

Some terms are less important in SC, because the sum-frequencies are often absorbed in the SC and more efficient insulator materials with larger bandgaps exist for non-integrated solutions.

Photon Picture of nonlinear 2nd order interaction: (Feynman-graphs)

photon combination photon splitting

The solution of Maxwell’s wave equation with a nonlinear 2nd order polarization wave, caused by 2 input waves of frequency ω1 and ω2 and momentum k1 and k2 is:

The classical nonlinearity have to conserve energies of the waves. Quantized waves have to conserve particle aspects with respect to energy and momentum:

Energy conservation:

Momentum conservation and phase matching:

The photon momentum is . Momentum conservation is a concept of the photon.2ep k s /π λ= = 7-9


7-10

Phase-Matching Condition:

If the momentum condition is not fulfilled: then the generation efficiency decreases by destructive interference over the interaction length L.

As a consequence the nonlinearity would not be present on a macroscopic scale.

Consider 2 primary wave generating a NL polarization wave:

The phase difference Δφ between between primary and NL-waves is zero independent of position:

NL-waves interfere constructively

Phase matching or phase coherence can be enhanced by suitable choice of the refractive indices andPropagation directions.

1 2 3k k k kΔ = ± −( ) ( )22 2 2eff ~ sin kL / / kL /η Δ Δ

( )( )( ) ( ) ( )

1 1

2 2

1 12 13 3

1 1

2 2

3 3 1 2

3 2 1 3 1 1

j t jk r

j t jk r

j k k rj tj t jk r

E ,t A e e

E ,t A e e

E ,t A e e ~ A A e e

if then if : k k k

ω

ω

ω ωω

ω

ω

ω

ω ω ω

−

−

− ++−

=

=

=

= + = +

( ) ( )3 1 1 0r k k k rφΔ = − + =

r

k1k2

k3


7.4.1 Pockel’s or linear Electro-Optical (EO) EffectThe linear EO-effect is a DC(RF)-field-induced nonlinearity (2nd order) in non-centrosymmetric crystals (eg. Si, Ge, …) and is important in III-V-SC-based phase and intensity modulators.As the light frequency is not changed there is no reabsorption or phase-matching problem. Of course the EM-field has to be phase-matched to the optical field.

Electrical Modulation of refractive index ~E2 ; EO-effect

In the anisotropic case the 2nd-order susceptibility-tensor χ(2)ijk needs 18 elements to be specified.

Due to interchangeability of field order and crystal-symmetries, we can simplify χ(2)ijk to djk:

7-11

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )1 2

2

2 22 1 2 0 1 1 1 2

1 22

22

1

0 0

1

2

r

E t

; Polarization P E cos t E t

because n ~ E t

a RF - modulation of the refractive index n results

E tn n n

nχ χ

ω ω ω ε χ ω ω

ε χ χ

χ+ >>

→ >> ≠ ⇒ =

= + +

⎯⎯⎯⎯⎯⎯→ + Δ = +

Phase-matching: (see M&NM H. Jäckel “Devices and Systems” in the 8th sem.)

Nonlinear materials are often incorporated in waveguides for guiding and confining the lightwave, whereas the RF-signal is guided by a metallic transmission line. The induced 2nd-order susceptibility generates an optical wave, that has to be synchronized to the exciting RF-wave.

For synchronisation (phase matching) the optical and RF-wave must have the same phase velocities.


( ) ( )2 21 3 1 3 2 3 1 2 3

in : out :, ,

χ χω ω ω ω ω ω ω ω ω⎯⎯→ ± = ⎯⎯→ = ∓

7.4.2 General Three Wave Mixing (3 interacting photons)2nd-order nonlinearities in general mix 2 different frequencies ω1 and ω2 and create a 3rd frequency ω3(parametric interaction) which is assumed to be phase-matched to ω3= ω2+ ω1.

The output field can, but must not necessarily be identical to ω3.

The generated nonlinear wave can interact with the generating waves.

Applications:

Up (down)-converter:

Parametric amplifier:

ω3 is the strong pump, ω1 the signal and ω2 theauxiliary Idler signal

Phase-matching conditions must be such that the idler exists

Parametric oscillator:

Parametric amplifier with resonator for ω1 (and also for ω2)

7-12


7.5 Third-order Nonlinearities (C2 ≠0 , C4≠0, C3=C5= …=0)3rd-order nonlinearites occur for crystal potentials with a bi-quadratic nonlinearity (C4, resp. χ(3) ≠0).The crystal must be centrosymmetric (eg. glasses, III-V-compound SC, …). χ(2)

ijk =0.

Major 3rd ordernonlinear effects:

Four-wave frequency-mixing effects result from the cubic dependence of the polarisation:

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 2 2 2 3 3 3

3 34 0 4 1 2 3 1 1 2 2 3 3 4 1 2 3

3 3 3 3 31 2 30 3 2i i i j

E t E cos t E cos t E cos t

P E E E with

generalized frequency combinations for 3 waves :

; ; ; ;

ω ω ω ω ω ω

ω ε χ ω ω ω ω ω ω ω ω ω ω ω

χ χ ω χ ω χ ω ω χ ω ω ω

= + + →

= = + + = ± ± ±

→

± ± ± ±

7-13


7.5.1 Four-wave mixing (4 interacting photons)If 3 waves ω1, ω2, ω3 interact with a χ(3)-nonlinearity then for energy exchange, a 4th wave ω4 isrequired fulfilling the frequency- and phase-matching conditions:

Feynman diagrams for 3rd-order NL:

Stimulated Raman Scattering:ω1=ω, ω2= - ω ; ω3=ωS → ω4= ωS

If ω - ωS=Ωphonon ,

then the interacting waves are coupled to the phonon resonances and can be parametrically enhanced such that Raman-Gain at ωS pumped by ω may result.

Raman-Amplification in fibers have become interesting due to their broad gain in comparison to Er-doped fibers. 7-14

4 1 2 3 i iwithω ω ω ω ω ω± ± ± −= + + = −

4 1 2 3k k k k= ± ±

Four wave mixing optical Kerr effect

3rd harmonic generation THG Raman effect


7.5.2 Optical Kerr Effect and nonlinear refractive index (non-resonant)We consider the input-wave of frequency ω creating a 3rd-order nonlinearity with

ω1=ω2=ω ; ω3=-ω → ω4=ω(No phase matching is required because all 4 wave have the same frequency)

3rd-order non-resonant NL produce refractive index changes depending in the intensity Ias shown also 2nd order NL produce index changes, but proportional to ~√I

DC Kerr Effect as a special case of the optical Kerr-effect:ω1=ω2=0 ; ω3=ω

Δn ~ E(0)2

The DC Kerr effect can also be considered as a χ(3) process 7-15

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 1 2

1 3 2

0

3 22

0 0 20 0 0 0

3

3

0

2 2

0 1

112

12 2

NLr

NLr r r r

r rr

0

because E E

E with and E

From n n n n

En n n I ; I E light intensityn n c n c

n = n n I ; n nonlinear refractive index

χ ε χ

ε ε ε ε χ ε χ εεε ε ε εε

ε εχχ

χ= ⇒ = + +

= + Δ = + Δ = <<Δ= ⇒ = + Δ ≅ + = + Δ

= + = + = =

+ =


7.5.3 Resonant 3rd-order Nonlinearities in SemiconductorsAbsorption/Gain-saturation processes in SC lead to an associated intensity I dependent change in the refractive index Δn(I), which is equivalent to a χ(3)-nonlinearity.

In a 2-level system the static absorption α decreases with increasing field E, resp. intensity I~E2

saturating the resonant transition (bleaching, induced transparency)

However, the resonant effect is orders of magnitude larger than the corresponding non-resonant optical Kerr-effect in glasses.

The temporal dynamics of the intensity dependent refractive index is governed by the fast and slow carrier dynamics effects.

The quasi-optical Kerr-effect in SC is used technically for all-optical switching processes.

7-16

( )

( ) ( )

0

0

11

SS

S

I ; I saturation intensityI / I

with Kramers - Kronig : n ~ I I / I

αα

α α

= =+

Δ Δ −


7-17

7. Nonlinear Optics Summary:• Nonlinear effects at high light intensities change the susceptibility and all properties

depending on χ(ω,E) as a nonlinear function of the electrical field E

• Non-resonant nonlinearities can be described by nonlinear susceptibility-tensors, strongly influenced by the crystal symmetries of the material

• The most important nonlinearities are 2nd and 3rd order depending on the asymmetry of the microscopic crystal potential. In centro-symmetric crystals, only 3rd order (and higher) are different from zero

• Non-resonant nonlinearities can be described classically by anharmonic oscillators

• Resonant nonlinearities are often based on changes of the dipole or carrier densitiesand must be described quantum-mechanically. They are mostly much stronger than non-resonant nonlinearities, but due to the long excited carrier life-times the response time can slow (ps compared to fs)

• Nonlinearities based on coupling the different excitations of the solid like eg. phononsare possible

• 3- and 4-wave mixing occurs in materials with 2nd- and 3rd-order nonlinearities coupling 2 and 3 photon fields together if energy and phase matching conditions are fulfilled

• Nonlinear effects (eg. optical Kerr-effect, self-phase modulation, multiple wave mixing etc.) in SC are interesting for ultrafast all-optical switching elements with response times in the 100fs-region


Information Electronics Laboratory, IfE at D-ITETHigh Speed Electronics and Photonics Group: Research

InP-based Tb/s Photonicssub-ps mode-locked diode laserssub-ps all-optical switches

InP Photonic Crystals for ultradense Optical ICs

Hole depth 2420nm

1μm

2μm

Hole depth3530nm

Holes depth: →3.5um @ 250nm Ø

65-80nm CMOS for mm-wave RF and 40 Gb/s Electronics(ETH-IBM CASE Collaboration)

0.1-60 GHz TWA in 80nm CMOS

7-18

↑200mV /div,2.5ps /div →

↑200mV /div,2.5ps /div →

+200 Gb/s ICs with InGaAsPHeterojunction Bipolartransistors(in-house technology)

80 Gb/s MUX with driver


7-19


7-19

Corrections: (final 19.01.2007)

Major corrections: Minor corrections:p.2-4p.2-6p.4-5p.4-6p.4-11p.4-12p.4-15p.4-16p.4-20p.4-26p.5-5p.5-6p.5-7p.5-8

p.6-8p.

material properties and characterization optical …¤ckel/mnsm_matprop.pdfphotonics represents the...

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