EE 232 Lightwave DevicesLecture 2: Basic Semiconductor Physics dOti lPand Optical ProcessesInstructor: Ming C. WuUniversity of California, BerkeleyElectrical Engineering and Computer Sciences DeptElectrical Engineering and Computer Sciences Dept.EE232 Lecture 2-1©2008. University of CaliforniaOptical Properties of SemiconductorsIntraband Transition(Free-Carrier Absorption)ECAbsorption Emission(FreeCarrier Absorption)EVInterbandTransitionEAImpurity-to-BandTransition• Optical transitionsTransitionTransition– Absorption: exciting an electron to a higher energy level by absorbing a photon–Emission: electron relaxing to a lower energy state by EE232 Lecture 2-2©2008. University of Californiaggyyemitting a photonBand-to-Band Transition• Since most electrons and holes are near the band-edges, the photon energy of band-to-band (or interband) transition is approximately equal to the bandgap energy:• The optical wavelength of band-to-band transition ghv E=can be approximated by1.24λ≈ : wavelength in mgEλλμ≈EE232 Lecture 2-3©2008. University of California: energy bandgap in eVgEEnergy Band Diagram in Real Space and k-SpaceE1Effective Mass ApproximationEECB*212eC eEEmv=+22*2eCkEEm=+ECk2em*eekmv=Momentum:VBxkEV*212hV hEE mv=−22*2hVkEE=−xReal Space K-Space2hmEE232 Lecture 2-4©2008. University of CaliforniaBand-to-Band TransitionCBhνCBhνCBhνhνVBhνVBhνVBhνhνAbsorptionSpontaneous StimulatedAbsorptionpEmission EmissionPhotodetectors;Solar CellsLEDOptical Amplifiers;Semiconductor LasersEE232 Lecture 2-5©2008. University of CaliforniaConservation of Energy and Momentum• Conditions for optical absorption and emission:CBEE–Conservation of energyCBhνE2νhEE =−12– Conservation of momentumVBhνE121hkkkν−=VB1kk1k221212,~2hkkakνππ12()()2~~0.5 ~1hkanm mνπλλμ<<Optical transitions are “vertical” linesEE232 Lecture 2-6©2008. University of California21kk⇒=LatticeConstantDirect vs Indirect BandgapsCBCBPhhνhνXPhononVBVB• Direct bandgap materials– CB minimum and VB maximum occur at the • Indirect bandgap materials– CB minimum and VB maximum occur at same k– Examples• GaAs, InP, InGaAsPdifferent k– Example• Si, GeEE232 Lecture 2-7©2008. University of California• (AlxGa1-x)As, x < 0.45 • (AlxGa1-x)As, x > 0.45– Not “optically active”Absorption Coefficient• Light intensity decays exponentially in semiconductor:• Direct bandgap semiconductor has a sharp absorption edgexeIxIα−=0)(has a sharp absorption edge• Si absorbs photons with hv > Eg= 1.1 eV, but the gabsorption coefficient is small– Sufficient for CCD•At higher energy ( 3eV)•At higher energy (~ 3 eV), absorption coefficient of Si becomes large again, due to directbandgaptransition toEE232 Lecture 2-8©2008. University of Californiadirect bandgaptransition to higher CBReview of Semiconductor PhysicsElectron and hole concentrations:() ()nfEEdEρ∞=∫() ()() ()CVneEEphnfEEdEpfEEdEρρ==∫∫Fermi-Dirac distributions:1()nfE−∞=⎛⎞()1exp1()nnBfEFkTfE⎛⎞−+⎜⎟⎝⎠=()1exp: electron quasi-Fermi levelppBnfEFEkTF=−⎛⎞+⎜⎟⎝⎠EE232 Lecture 2-9©2008. University of Californiaq : hole quasi-Fermi lnpF evelElectron/Hole Density of States (1)• Electron wave with wavevector kik re⋅• Periodic boundary conditionszLLe() () ()xyzik r L x ik r L y ik r L zik ree e e⋅+ ⋅+ ⋅+⋅===   • An electron state is defined byxLyLee e e===()222kkk lπππ⎛⎞↑↓ ↑↓⎜⎟kkΔzk• Number of electron states between k and ()222,, , ,xyzxyzkkkmnlLLLπππ↑↓=↑↓⎜⎟⎜⎟⎝⎠kkk+Δzk + Δk in k-space per unit volume22224()222kkdk kdk k dkVπρπππ⋅==ykEE232 Lecture 2-10©2008. University of California2()222kxyzVLLLρππππxkSpin Up and DownElectron/Hole Density of States (2)• Number of electron states between E and E + ΔE per unit volume22 2**2CeekkE E dE dkmm=+ ⇒=()**22222()eCeemEEmkdk dE E dEρππ−==3/2*2221()2eeCmEEEρπ⎛⎞=−⎜⎟⎝⎠• Likewise, hole density of states3/2*21()hmEEE⎝⎠⎛⎞⎜⎟EE232 Lecture 2-11©2008. University of California22()2hhVEEEρπ=−⎜⎟⎝⎠Electron and Hole Concentrations2*22211() ()21expCCene CEEnmnfEEdE EEdEEFρπ∞∞⎛⎞== −⎜⎟⎛⎞−⎝⎠+⎜⎟∫∫1/21expCCBnCCkTFEnNFkT+⎜⎟⎝⎠⎛⎞−=⋅⎜⎟⎝⎠1/23/2*2222CBeBCkTmkTNππ⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠()Fermi-Dirac Integral1jx∞∫2VpEFNFπ⎝⎠−⎛⎞⎜⎟()01 (1)1Gamma FunctionjxxFdxjeηη−=Γ+ +∫1/23/2*p=2VpVBhBNFkTmkTNπ⋅⎜⎟⎝⎠⎛⎞=⎜⎟3()22πΓ=EE232 Lecture 2-12©2008. University of California2222VNπ=⎜⎟⎝⎠Approximation of Electron/Hole Concentration()1/23e when 11 4(1)1when 1jjxxFdxjeηηηηηη∞−⎧⎪=⎨⎛⎞Γ+ +⎪⎜⎟∫∼0(1)1when 13When (Boltzmann approximation)jeFEηπΓ+ +⎪⎜⎟⎝⎠⎩When (Boltzmann approximation)CnBnCEFkTCFEnNe−−≈⋅3/2When (Degenerate)4nCnCFEFE⎛⎞−⎜⎟CnNBoltzmann ApproxDegenerate43nCBCkTnNπ⎜⎟⎝⎠≈⋅Quasi-FermiLevel Cross Band EdgeEE232 Lecture 2-13©2008. University of