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Lecture22:OpticalProperties

Reflectivity(R)= !"#$"%&"'!"#!$%"&

Transmitivity(T)=!"#$%&'!!()

!"#!$%"&

ifthemediumistransparent:T+R=1

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Refraction:refractiveindexn

n =c𝑣

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v:speedoflightinmediumc:speedoflightinvacuumAbsorption:𝛼(absorptioncoeff.)

𝐼 X = I!e!!!W:characteristicpenetrationdepthdistanceatwhich

theintensityoflightwavesreducesto!!(37%)ofits𝐼!

R,T,𝛼,n,W→opticalconstantsofmaterialsDispersion:Dependenceofrefractiveindex,reflectivity,…onthewavelength

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ClassicalTheory:Drudetheory&Lorentztheory• Drude:freeelectrons;canbeacceleratedbyanexternalelectricfield.

‐Explainsthedependenceofopticalconstantsonwavelength,butcouldn’texplainabsorptionband.

• Lorentz:Boundelectrons;electronsareboundtothenucleusbyspringwhichdeterminesthenaturalfrequency

𝐹 = −𝑘𝑥          𝜔! =km

harmonc oscillatorApplyingelectricfield,displacesthepositivecharge(createselectricdipole) →forcedvibrationThedipoleisassumedtobeharmonicoscillator

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‐ Anoscillatorabsorbsamaximumamountofenergywhenoscillationnearsitsresonancefrequency.

Theclassicaltheory(Drudle&Lorentz)canexplainthedependenceofRonthewavelengthformetalsandsemiconductorsandinsulatorsasfollows:

Inmetals:Lowfrequencyregion:R=0,𝜔! = 0FreeelectronscanexplaintheIRHighfrequencyregion:Electrons(harmonicoscillators)canexplainabsorptionbands.Insulators&semiconductors:Canbeexplainedbyharmonicoscillators(boundelectrons)DrawbackofClassicalTheories:Theyfailedhowtoexplainthatelectronsshouldbeconsideredsometimes(atlowfrequencies)freeandconsideredothertimes(athighfrequencies)bound.

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QuantumMechanics:Lightabsorption:‐Intrabandtransitions‐Interbandtransitions ‐direct

‐indirectInterbandTransitions:Electronstransitionbetweendifferentenergybands

Directinterbandtransition:Verticaltransition,looktotheelectronbanddiagraminthefigure.k(ofelectron)remainsconstant.

Note:themomentumofphoton(intheopticalreigon)ismuchsmallerthanforelectron.Looktothefigure:Therearemanypossibletransitions

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‐opticalspectraofmetalsaredominatedbydirectinterbandtransitionsIndirectbandtransitions:Involvesphononorlatticevibrations.Looktotheenergyband,theexcessmomentumisabsorbedfromortransformedtothelatticethroughphononexchangeIndirectinterbandtransitionsmaybeneglectedformetals.(lessthandirecttransitionby2or3ordersofmagnitude)However,theyareimportantinsemiconductors

Note:Direct&indirectbandtransitionsarepossiblefrombandtomuchhigherbandthantheabove(i.e.skippingbandsispossible)Intrabandtransitions:Electronsareexcitedtohigherenergystateswithinthesameband(impliesphonon).

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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).

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Onlyobservedinmetals,becausetheyhaveunfilledenergybands.Theydon’toccurinintrinsicsemiconductorsorinsulatorsIntrabandtransitionsareonlyinmetalsandsometimeshighlydopedsemiconductors.Inlowfrequency,theydominate,notquantitized.Interbandtransitions:Occurinmetals,insulators,semiconductorsYoucanseethatintrabandtransitionscorrespondtofreeelectronsinclassicaltheoryandinterbandtransitionscorrespondtoboundelectrons Note:Freeatomshavesharpabsorptionandemissionlines,butsolidshavebroadabsorptionandemissionbandsduetobandformation

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Plasmafrequency“𝜔!”:Separatesthereflectiveregionfromthetransmissionregion

𝜔! =𝑒!𝑁!ℰ!𝑚!