1 Lecture 22: Optical Properties Reflectivity (R)= !"#$"%&"’ !"#!$%"& Transmitivity (T)= !"#$%&’!!() !"#!$%"& if the medium is transparent: T+R=1
1
Lecture22:OpticalProperties
Reflectivity(R)= !"#$"%&"'!"#!$%"&
Transmitivity(T)=!"#$%&'!!()
!"#!$%"&
ifthemediumistransparent:T+R=1
3
v:speedoflightinmediumc:speedoflightinvacuumAbsorption:𝛼(absorptioncoeff.)
𝐼 X = I!e!!!W:characteristicpenetrationdepthdistanceatwhich
theintensityoflightwavesreducesto!!(37%)ofits𝐼!
R,T,𝛼,n,W→opticalconstantsofmaterialsDispersion:Dependenceofrefractiveindex,reflectivity,…onthewavelength
5
ClassicalTheory:Drudetheory&Lorentztheory• Drude:freeelectrons;canbeacceleratedbyanexternalelectricfield.
‐Explainsthedependenceofopticalconstantsonwavelength,butcouldn’texplainabsorptionband.
• Lorentz:Boundelectrons;electronsareboundtothenucleusbyspringwhichdeterminesthenaturalfrequency
𝐹 = −𝑘𝑥 𝜔! =km
harmonc oscillatorApplyingelectricfield,displacesthepositivecharge(createselectricdipole) →forcedvibrationThedipoleisassumedtobeharmonicoscillator
6
‐ Anoscillatorabsorbsamaximumamountofenergywhenoscillationnearsitsresonancefrequency.
Theclassicaltheory(Drudle&Lorentz)canexplainthedependenceofRonthewavelengthformetalsandsemiconductorsandinsulatorsasfollows:
Inmetals:Lowfrequencyregion:R=0,𝜔! = 0FreeelectronscanexplaintheIRHighfrequencyregion:Electrons(harmonicoscillators)canexplainabsorptionbands.Insulators&semiconductors:Canbeexplainedbyharmonicoscillators(boundelectrons)DrawbackofClassicalTheories:Theyfailedhowtoexplainthatelectronsshouldbeconsideredsometimes(atlowfrequencies)freeandconsideredothertimes(athighfrequencies)bound.
7
QuantumMechanics:Lightabsorption:‐Intrabandtransitions‐Interbandtransitions ‐direct
‐indirectInterbandTransitions:Electronstransitionbetweendifferentenergybands
Directinterbandtransition:Verticaltransition,looktotheelectronbanddiagraminthefigure.k(ofelectron)remainsconstant.
Note:themomentumofphoton(intheopticalreigon)ismuchsmallerthanforelectron.Looktothefigure:Therearemanypossibletransitions
8
‐opticalspectraofmetalsaredominatedbydirectinterbandtransitionsIndirectbandtransitions:Involvesphononorlatticevibrations.Looktotheenergyband,theexcessmomentumisabsorbedfromortransformedtothelatticethroughphononexchangeIndirectinterbandtransitionsmaybeneglectedformetals.(lessthandirecttransitionby2or3ordersofmagnitude)However,theyareimportantinsemiconductors
Note:Direct&indirectbandtransitionsarepossiblefrombandtomuchhigherbandthantheabove(i.e.skippingbandsispossible)Intrabandtransitions:Electronsareexcitedtohigherenergystateswithinthesameband(impliesphonon).
9
Effectofelectron‐electroninteractionisignored(independentelectronapproximation–reasonablyvalid)–effectofelectron‐ioninignored(independent‐electronapproximation;completelyinvalid).
‘plasmafrequencyMean free timebetweencollisions is τ (probabilityofcollisionperunit time is1/t;probability of having a collision in infinitesimal timeinterval dt is dt/t) ‐τ isindependentofelectron’spositionorvelocity(agoodassumption).
10
Onlyobservedinmetals,becausetheyhaveunfilledenergybands.Theydon’toccurinintrinsicsemiconductorsorinsulatorsIntrabandtransitionsareonlyinmetalsandsometimeshighlydopedsemiconductors.Inlowfrequency,theydominate,notquantitized.Interbandtransitions:Occurinmetals,insulators,semiconductorsYoucanseethatintrabandtransitionscorrespondtofreeelectronsinclassicaltheoryandinterbandtransitionscorrespondtoboundelectrons Note:Freeatomshavesharpabsorptionandemissionlines,butsolidshavebroadabsorptionandemissionbandsduetobandformation