6.973 Semiconductor OptoelectronicsLecture 2: Electronic States in AlxGa1-xAsRajeev J. RamElectrical EngineeringMassachusetts Institute of TechnologyConcepts to be reviewed:•Bandstructure•Bloch functionsIntroduction•Background: p-n junctions•Photodetectors•Modulators•Optical amplifiers•Semiconductor lasers•Heterostructure materials•DFB and VCSEL resonators•Modulation•SystemsSyllabusBasic conceptsAdvancedconceptsGaAs• first created by Goldschmidt, reported in 1929• first report as a semiconductors (Welker @ Siemens) 1952• first semiconductor laser (Hall @ GE, Redhiker @ MITLL) 1962• first double heterostructure laser & CW operation (Ioffe) 1970• first CW room-temperature laser 1975• $2 billion industry for wireless & fiber-optics 2000http://www.ieee.org/organizations/pubs/newsletters/leos/feb03/diode.htmlInP850 nm980 nm1300 nm1550 nmInGaAsN(Sb)Heterostructure MaterialsBandstructureBandstructureof of GaAsGaAs(k)Electronic Energy LevelsElectronic Energy Levels(hyperphysics.phy-astr.gsu.edu)not for periodic b.c.Tighter confinementNodes in wavefunctionHigher EnergyHigher EnergySchrodinger’s Equation…… suggests that the energyof an electron is related to the curvatureof the wavefunctionH2remember H2?Hydrogen MoleculeHydrogen MoleculeHHLow energy High energy11--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basisAdding atoms…• reduces curvature of lowest energy state (incrementally)• increases number of states (nodes)• beyond ~10 atoms the bandwidth does not change with crystal sizeDecreasing distance between atoms (lattice constant)…• increases bandwidthkk= = ππ/ / aakk= 0= 0kk≠≠00Approximate Approximate WavefunctionWavefunctionfor 1for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basisk is a convenient way to enumerate the different energy levels (count the nodes)Bloch Functions:Energy Band for 1Energy Band for 1--D LatticeD LatticeSingle orbital, single atom basisSingle orbital, single atom basislowest energy (fewest nodes)highest energy (most nodes)• Number of states in band = number of atoms• Number of electrons to fillband = number of atoms x 2 (spin)Energy Band for 1Energy Band for 1--D LatticeD LatticeTwo orbital, single atom basisTwo orbital, single atom basismost nodes‘p band’‘s band’Two bands form…one from each atomic orbitalWhat happens as we push atoms together ?BandstructureBandstructureof of GaAsGaAsGa: [Ar]3d10 4s24p13 e- per Gallium atom5 e- per Arsenic atomFCC lattice with 1 Ga & 1 As atom per lattice sitetotal: 8 electrons at each site8 bandsAs: [Ar] 3d104s24p3BandstructureBandstructureof of GaAsGaAs(k)s-orbital likep-orbital likeLowest conduction band3 highest valence bandsEffective MassEffective MassNear a minima, the band is parapolic: where…is the effective mass & contains information on the distribution of energy levelsGaAsGaAsEffective MassEffective Mass0.48 eV∆ = 0.29 eVBZBZMultiplicity of valleys…Effective masses…BandstructureBandstructureof of GaAsGaAs(k)s-orbital likep-orbital likeLowest conduction band3 highest valence bandsaaalightmassheavymassheavymassLighter effective massLarger overlap between orbitalsbasisorbitalpxpypzxValence Valence BandstructureBandstructureBandstructureBandstructureof of GaAsGaAs(k)s like -orbitalp-like orbitalWhat is this split-off band ?heavy hole charge distribution light hole charge distributionSpinSpin--orbit Coupling orbit Coupling WavefunctionsWavefunctionsAngular momentum for quantum state with l =2:2l=z24ºm = 1m = 0m = −155ºm = 2m = −2Orbital Angular MomentumOrbital Angular Momentum-q+Zq-q+ZqL, BlSpinSpin--Orbit CouplingOrbit CouplingThe effective current from the motion of a nucleus in a circular orbit……generates an effective magnetic field…2P1S2P3/22P1/2LBSLBS-q+ZqSSpin up:High EnergyµsL, Bl-q+ZqSSpin down:Low EnergyµsL, BlSpinSpin--Orbit SplittingOrbit SplittingJ = L + S = 3/2J = L + S = 1/2-q+ZqSSpin up:High EnergyµsL, Bl-q+ZqSSpin down:Low EnergyµsL, BlSpinSpin--Orbit Splitting in HydrogenOrbit Splitting in HydrogenExample: l = 1, s = ½Quantum NumbersVectorsj = 3/2 j = 1/2Angular Momentum Addition RulesAngular Momentum Addition Rulesheavy hole charge distribution light hole charge distributionSpinSpin--orbit Coupling orbit Coupling WavefunctionsWavefunctionsheavy mass (along kz) light mass (along kz)BandstructureBandstructureof of GaAsGaAsSpin-orbit splittingSummary of Key Summary of Key GaAsGaAsBandstructureBandstructureConceptsConcepts• k is the ‘crystal momentum’ & labels the energy levels within a band• Near a minima the energy is parabolic in k: where m* is the effective mass• the wavefunction for an electron in an energy state is with the Bloch amplitudes being approximately the orbital wavefunctions• Electronic band width, orbital overlap, and orbital orientation are all related • The conduction band is ‘s-like’ with a minima at k=0• The valence band is a ‘p-like’ with the heavy-hole and light-hole band maxima at k=0Semiconductor AlloysSemiconductor AlloysLinear interpolation (Vegard’s Law) for lattice constant nearly always holds…For a perfectly disordered alloy……virtual crystal approximation…Bandgap is……in general, there will be some ordering which introduces ‘bowing’…Dielectric constant (static)12.90-2.84xDielectric constant (high frequency)10.89-2.73xEffective electron mass me0.0637+0.083x mo(x<0.45)Effective hole masses mh0.50+0.29x moEffective hole masses mlp0.087+0.063x moLattice constant 5.6533+0.0067x AngstromsBandgap (direct) 1.424+1.247xEg(eV)GaAs AlAs1.4242.293.13ΓΧ~0.45Material Parameters for AlMaterial Parameters for AlxxGaGa11--xxAsAshttp://www.ioffe.rssi.ru/SVA/NSM/Electronic Archive of Material ParametersElectronic Archive of Material ParametersshutterssourcessubstrateGrowth of Growth of HeterostructureHeterostructureSolidsSolidsMBE of MBE of GaAs/AlGaAsGaAs/AlGaAsSpecies of Species of HeterojunctionsHeterojunctionshttp://www.utdallas.edu/~frensley/technical/hetphysType IType IType IIType IIIType IIIEcGaAsAlGaAsEfEvEc∆Ev∆AlGaAsE1E2Type IAlxGa1-xAs HeterojunctionsHeterojunctionsInconsistent values in