Table Of ContentMauro Nisoli
Semiconductor Photonics
Principles and applications
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Contents
Preface
Band structure of semiconductors
1.1 Crystals. latticesandcells
1.1.1 TheWigner-Seltzcell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thereciprocal lattice ~16-
1.3 Electrons in a periodic crystal 10
1.4 Theconceptoteitective mass 13
1.5 Energy bands 14
1.5.1 Electronsand holesinasemiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Calculationoithebandstructure 15
1.6.1 Tight-bindingmethod . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.2 Crystalwithone-atombasisandsingleatomicorbital 18
1.6.3 Linearlattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.4 Slmplecubiclattice . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.5 BandstructureofsemiconductorscalculatedbyTBM 22
1.7 Thek - p method 25
1.8 Bandstructuresofafewsemiconductors 28
1.8.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8.2 GalliumArsenlde . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8.3 Gallium Nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - - . . . . - - i i ' - - - - 30
Electrons in semiconductors
2.1 introduction 33
5.7 Competition between radiativeand nonradiative recombination
QLlCJl‘iii.ll*.“ ‘1‘.’e-ll=1
6.1 introduction
6.2 Electronic states
6.2.1 Electronicstatesinthevalence band . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Density ofstates
6.4 Electron density
6.5 Transition selection rules
6.5.1 lntersubbandtransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Absorption and gain in a quantum well
6.7 lntersubband absorption
6.8 Strained quantum wells
6.9 Transparency densityanddifferentialgain
6.9.1 Differential gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Excitons
6.10.1 Absorptionspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10.2 Excltonsin quantumwells . . .
Light i:r‘r\-iitin:.; -T.=iP»<:iei~'
7.1 Basic concepts
7.2 Double heterostructure LEDs
7.3 Carrierleakage over barrier
7.4 External efficiencyofa LED
7.5 Emission pattern of a LED
7.6 Luminousefficiency
7.7 Blue Light Emitting Diodes
Se-:"Yiici'_»n<tu<;i;_\' L~;‘?sK.=-.'$
8.1 introduction
8.2 Rateequationsandthresholdconditionsfor laseraction
8.3 Temperature dependence
8.4 Output power
8.5 Quantum well lasers
8.6 Laserstructures "
8.7 Spectral and spatial characteristicsofdiode laseremission
Disti'iti-liteu t"E*F3diDCJ(t14. Lasers
9.1 Basic concepts
9.2 Coupled-modetheory
9.2.1 Thresholdconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 DFB laserwith uniform grating
9.3.1 Approxlmated evaluation ofoscillatingfrequenciesandthreshold gain
9.4 DFB laserwith ,\/4-shiftedgrating
9.5 Distributed Bragg Reflector (DBR) laser
Vertical Cavity Surlace~~Emitting Lasers
10.1 Basic structure
10.2 Threshold conditions
10.3 Distributed Bragg reflectorsforVCSELs
10.4 Threshold conditionsandcurrentconfinement
10.5 Advantagesandapplications
Quantum cascade lasers
11.1 Quantum cascade lasers
11.2 Gain coefficient
11.3 Rateequationsand threshold conditions
11.4 Outputpower, slope- and wall-plug efflciency
11.4.1 Wall-plug efflciency . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Numerical example
11.6 Applications
Fundamental constants
index
Preface
The termphotonics was introduced in the late sixties by PierreAigrain, who gave
the following definition: "Photonics is the science ofthe harnessing oflight. Pho-
tonics encompasses the generationoflight, thedetectionoflight, themanagement
oflight through guidance, manipulation and amplification and, most importantly,
its utilisation for the benefit ofmankind." In the last decades impressive progress
in the field ofphotonics hasbeen achieved, thanks to remarkable advances in the
understandingofthe physical processes at the heart oflight-matter interaction in
photonic applications and tothe introduction ofcrucial technological innovations.
Photonics is anextremelylarge field, as clearlypointedoutbythe above definition,
since it refers to all types oftechnological device and process, where photons are
involved.
Thisbookdoes notaim toanalyseallaspectsofphotonics: afewexcellenttextbooks
alreadyexist,whichpresentseveraltopicsrelevantforphotonicapplications. Theaim
ofthisbookistointroduce and explain importantphysicalprocesses attheheartof
the opticalpropertiesofsemiconductordevices, suchaslightemittingdiodes (LEDs)
andsemiconductorlasers. Itissuitableforahalf-semester (oraone-semester) course
inPhotonicsorOptoelectronicsatthegraduatelevelinengineeringphysics, electrical
engineeringormaterialscience. ltoriginated froma graduatecourse (PhotonicsI)
which I amteachingatthe PolitecnicodiMi1ano since 2006. The concepts ofsolid
statephysicsandquantum mechanicswhichare required tounderstand thesubjects
discussed in thisbookare addressed in the introductorychapters. Itis assumed that
the reader hashadcourses onelementaryquantum mechanics, solid-state physics,
andelectromagnetictheoryatthe undergraduatelevel.
The book presents a selection oftopics, which I consider essential to understand
the operation ofsemiconductordevices. Itoffers a relativelyadvanced analysisof
the photo-physics ofsemiconductors, trying toavoid the use ofexceedinglycomplex
formalisms. Particularattentionwas devoted toofferaclearphysicalinterpretation
ofallthe obtained results. Variousworked examples are added throughoutall the
chapters to illustrate the application ofthe various formulas: the solved exercises
are evidenced bythe coloured boxes in the text. The numerical examples are also
importantsince theyallowthe reader to have a direct feeling ofthe orderofmagni-
tude ofthe parameters used in the formulas discussed in the text. The greyboxes
containconcise discussions ofsupplementarytopics ormore advanced derivations of
particularresults reported in the maintext, which maynotbe easilyderived bythe
reader.
SemiconductorPhotonics isorganizedas follows. Chapter 1 focuses onthe description
ofa fewconcepts ofsolid state physics, which are relevantforthe calculation and
analysis ofthe band structure ofsemiconductors. The Bloch theorem is introduced,
which describes the wavefunction of electrons in periodic structures. The tight-
binding method is considered, with a few simple examples, and the k ~ p method,
which are used tocalculated the band structure ofsemiconductors. Chapter2 deals
with the discussion ofthe main properties ofcharged particles (electrons and holes)
inintrinsicanddopedsemiconductors. The densityofstatesis firstcalculated andthe
essentialconceptsofcarrierstatistics insemiconductorsare discussed. Basicconcepts
ofquantum mechanics are contained inChapter3. In particular, the densitymatrix
formalism is introduced, which is used in the book forthe calculation ofthe optical
susceptibility ofa semiconductor. After a veryshort overview ofessential aspects
ofclassicalelectromagnetic theory, Chapter4 analyses the interaction ofelectrons
with an electromagnetic field. The expressions ofthe interaction Hamiltonian, which
are extensively used throughout the book, are derived in this chapter. Chapters 5
and 6 build on the previous chapters. in particular, Chapter S deals with the opti-
calpropertiesofbulksemiconductors, i.e., semiconductorswithspatial dimensions
muchlargerthan the de Broglie wavelengthofthe electrons involved in the relevant
physical processes. Absorption and gaincoefficients are calculated and the radiative
and non-radiative recombination processes insemiconductors are analysed. Chapter
6 analyses the principles ofthe photo-physics in semiconductorquantumwells, i.e.,
insemiconductorstructureswhere the electrons are confined in one direction bya
potentialwell,with a thickness smallerthan the electron de Brogliewavelength.
In the remaining five chapters the general results obtained in the first part ofthe
bookare applied to the investigation ofthe main optical properties ofsemiconduc-
tor devices: light-emitting diodes and lasers. The general philosophy adopted in
these chapters is the following: the fundamentalphysical processes are investigated,
ratherthan the technological characteristics ofthe devices. Aftera shortand general
analysis ofsemiconductorlasers inChapter8, based on the rate equation approach,
Chapter9 contains a detailed theoretical analysis ofthe Distributed Feedback (DFB)
lasers, based on the use ofthe coupled-modeequations. Byusinga simpleperturba-
tive approach, the threshold laserconditions are obtained. Vertical CavitySurface
EmittingLasers (VCSELs) and Quantum Cascade Lasers are analysed in thefinal two
chapters.
Final note: whyEschefs “Skyand WaterI" on the bookcover? This print suggested
me various connections with photonics in semiconductors. It is light which, by
playingwith shapes, gives rise to an evolutionwhich transforms fishes inwaterinto
birds inthe skyand the transformationis closelyrelated to the reciprocal interaction
among elements. In a similar manner, it is lightwhich, by playingwith electrons,
triggers dynamical processes inside matter, with the generation ofnew properties
and functions, where complex interactions among particles and surrounding envi-
ronment play a crucial role. The analogy can be pushed even more considering
that in Escher’s print the transition from skytowaterstarts and ends with realistic
shapes ofbirds and fishes, respectively, and moves from one real shape to theother
through a sequence ofshapes that, in particularin the central portion ofthe print,
resemble but are not exactly neither birds nor fishes. Forcing a little the analogy,
we can saythat typical photonic processes in semiconductors evolve from a given
physical (real) state to anotherphysical (real) state through a sequence ofquantum
superpositionofstates,whichdonothaveananalogyinthe realmofclassicalphysics.
Milano, 11 November2016 MauroNisoli
l. Band structure of semiconductors
Crystals, latticesand cells
Acrystal is composedbyaperiodic repetitionofidenticalgroupsofatoms: agroup is
calledbasis. The correspondingcrystallattice isobtainedbyreplacingeachgroupof
atoms bya representative point, asshown in Fig. 1.1. Acrystal can be also called
latticewitha basis. Whenthebasis is composedbyasingle atom thecorresponding
lattice is called monoatomic. In aBravais lattice the position Rofall points in the
latticecanbewritten as :
R=T1181 +71,282 +71.383, (1.1)
where a1, a2 and a3 are three non-coplanar translation vectors, called primitive
vectorsandn1, ‘fig andn3 are arbitrary (positive ornegative) integers. This definition
ofBravaislattice implies thatthislatticelooksexactlythe samewhenviewed from
anylattice point. Notonlythe arrangementofpointsbutalsotheorientation must
be exactlythe same from everypoint in a Bravais lattice. Therefore, two points in
o'o'o'o' - - -
O. 0.0.0.0. ' . _ _
o'o'o'o' - - - -
la) rs) (cl
Figure 1.1: (a) Basiscomposedbytwodifferentatoms; (b) bidimensionalcrystaland
(c) correspondinglattice.
the lattice, whose positions vectors are given byr and r’ = r+ R are completely
equivalentenvironmentally. Forexample, the two-dimensional honeycomb lattice
