6.973 Semiconductor OptoelectronicsLecture 3: Semiconductor StatisticsRajeev J. RamElectrical EngineeringMassachusetts Institute of TechnologyOutline:•Effective Mass Hamiltonian•Density of states•Relating N and P to EFCourse website: http://rleweb.mit.edu/sclaser/6.973.htmDielectric 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--xxAsAsBandstructureBandstructureBandstructureBandstructure• the wavefunction for an electron in an energy state is with the Bloch amplitudes being approximately the orbital wavefunctions• the eigenenergies are thebandstructureBandstructureBandstructurefor ‘Perturbed’ Crystalsfor ‘Perturbed’ CrystalsNew wavefunction is a superposition of old wavefunctions…So long as Uextvaries slowly enough not to change the ‘orbitals’ …BandstructureBandstructurefor ‘Perturbed’ Crystalsfor ‘Perturbed’ CrystalsFor slowly varying Uext(r), instead of solving the full bandstructure, we can solve the much simpler ‘Effective Mass Hamiltonian’All the complications of the crystal potential, UC(r), are incorporated within the known bandstructurehttp://www.research.ibm.com/DAMOCLES/html_files/phys.htmlDensity of States for Full BandDensity of States for Full BandFortunately, we are usually only interested in the states within a few kBT of the bandedgesClose to a bandedge,the bandstructure can be approximated as a parabola… BandstructureBandstructurefor ‘Perturbed’ Crystalsfor ‘Perturbed’ CrystalsExample: States of a Finite Size CrystalExample: States of a Finite Size CrystalApproximate boundaries as infinite potential well…Density of States in Different SolidsDensity of States in Different Solids(Scientific American)Density of States in a 3Density of States in a 3--D CrystalD CrystalShells of constant energy in k-space space. States fill 3-D space. KxKyKzEnergyDensity of States3-DWhat is the Fermi level or chemical potential ?Counting ElectronsCounting Electrons= sum of (# of states) x (probability of a state being occupied)Counting and Fermi IntegralsCounting and Fermi Integrals33--D Conduction Electron DensityD Conduction Electron Densityis the Fermi-Dirac IntegralCounting and Fermi IntegralsCounting and Fermi Integrals33--D Hole DensityD Hole DensityBoltzmannBoltzmannApproximation in EquilibriumApproximation in EquilibriumBoltzmann Approximation:Approximations for Fermi IntegralsApproximations for Fermi Integrals33--D Carrier DensitiesD Carrier DensitiesSommerfeld Approximation:Unger Approximation:whereApproximations for Fermi IntegralsApproximations for Fermi Integrals33--D Carrier DensitiesD Carrier DensitiesImage: Coldren & CorzineApproximations for Inverse Fermi IntegralsApproximations for Inverse Fermi IntegralsInverse First-order Sommerfeld Approximation:Inverse Second-order Unger Approximation:for 0.04 errorfor 0.04 errorDensity of States in a 2Density of States in a 2--D CrystalD CrystalShells of constant energy in k-space space. KxKyKzKxKyEBlack circles: Contours of constant E on surfaces of allowed states (parabaloids). Three subbands are plotted.3-D 2-DDensity of States in a Finite CrystalDensity of States in a Finite CrystalEnergyDensity of States3-D2-D3-D 2-DCounting and Fermi IntegralsCounting and Fermi Integrals22--D Conduction Electron DensityD Conduction Electron DensityExact solution !Inhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in EquilibriumConsider a solid with a spatially varying impurity concentration…In equilibrium, the carrier concentration is balanced by an internal electrostatic potential…02 10164 10166 10168 10161 10170 0.5 1 1.5 2Electron Concentration (cm-3)Microns00.010.020.030.040.050.060 0.5 1 1.5 2Electrostatic Potential (V)MicronsInhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in EquilibriumDividing solid into slices where φiis uniform…If electrostatic potential varies slowly compared to wavepacket…Inhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in EquilibriumDividing solid into slices where φiis uniform……the envelope function has solutions of the form……therefore the eigenenergies are…If electrostatic potential varies slowly compared to wavepacket…Inhomogeneous Semiconductors in EquilibriumInhomogeneous Semiconductors in EquilibriumBoltzmann approx.Given the modified energy levels, the 3-D DOS becomes….…in equilibrium the carrier concentration is…The slowly varying electrostatic potential can be incorporated inSummary of Key ConceptsSummary of Key Concepts• So long as Uextvaries slowly enough not to change the ‘orbitals’ …• Close to a bandedge,the bandstructure can be approximated as a parabola… • In 3-D, electron and hole densities are given in terms of Fermi-Dirac Integrals.• Fermi-Dirac Integrals can’t be invered, to go from N to EFneed approximations• The slowly varying electrostatic potential can be incorporated