6.973 Semiconductor OptoelectronicsTutorial: Statistical MechanicsRajeev J. RamElectrical EngineeringMassachusetts Institute of TechnologyOutline:•Counting States• Microscopic Definition of Temperature• Microscopic Definition of Chemical Potential• Equilibrium Between Two Systems• Quasi-EquilibriumE=0 g=1E=1 g=3E=2 g=3E=3 g=1Total Energy # of MicrostatesMicrostates and CountingMicrostates and CountingEnsemble of 3 ‘2Ensemble of 3 ‘2--level’ Systemslevel’ SystemsAs we shall see, gis related to the entropy of the system…E=0 g=1E=1 g=4E=2 g=6E=3 g=4E=4 g=1Total Energy # of MicrostatesE=2E=2Microstates and CountingMicrostates and CountingEnsemble of 4 ‘2Ensemble of 4 ‘2--level’ Systemslevel’ Systems0501001502002503000246810Number of MicrostatesTotal Energy010203040500246810Number of MicrostatesTotal EnergyMicrostates and CountingMicrostates and CountingFor most mesoscopic and macroscopic systems (N large), gis a monotonically increasing function of EThe larger the systems, the stronger the dependence on EN=3N=4N=5N=10N=10N=5System + Reservoir MicrostatesSystem + Reservoir MicrostatessystemreservoirExampleConsider a system of 3 ‘2-levels’ + a reservoir of 10 ‘2-levels’Gibb’s Postulate = all microstates are equally likelyProbability of finding:Es= 0 45/78Es= 1 30/78Es= 2 3/78Most electrons are in the ground state so reservoir entropy is maximized!System + Reservoir MicrostatesSystem + Reservoir MicrostatessystemreservoirFor sufficiently large reservoirs….…we only care about the most likely microstate for S+RNow we have a tool to look at equilibrium…System + Reservoir in EquilibriumSystem + Reservoir in EquilibriumsystemreservoirEquilibrium is when we are sitting in this max entropy (g) state…is the same for two systems in equilibriumSystem + Reservoir in EquilibriumSystem + Reservoir in EquilibriumsystemreservoirWe observe that two systems in equilibrium have the same temperature, so we hypothesize that…This microscopic definition of temperature is a central result of stat. mech.BoltzmannBoltzmannDistributionsDistributionsSis the thermodynamic entropy of a systemBoltzmann observed that…and…so he hypothesized thatBoltzmannBoltzmannDistributionsDistributionsreservoir controls system distributionSystem + Reservoir in EquilibriumSystem + Reservoir in EquilibriumsystemreservoirNow we allow system and reservoir to exchange particles as well as energy…System + Reservoir in EquilibriumSystem + Reservoir in EquilibriumsystemreservoirEntropy of reservoir can be expanded for each case…Difference in entropy of the two configurations is…..where µis the electrochemical potentialSystem + Reservoir in EquilibriumSystem + Reservoir in EquilibriumChemical potential is change in energy of system if one particle is added without changing entropySystem + Reservoir in EquilibriumSystem + Reservoir in EquilibriumExample: FermiExample: Fermi--DiracDiracStatisticsStatisticsConsider that the system is a single energy level which can either be…occupied:unoccupied:Normalized probability…Two Systems in EquilibriumTwo Systems in Equilibriumsystem 1reservoirsystem 2Particles flow from 1 to 2…Particles flow from 2 to 1…In equilibrium…E3E2E1Near Equilibrium Electron DistributionsNear Equilibrium Electron DistributionsOptical ExcitationOptical ExcitationE3E2E1Intraband scattering: electron-electronelectron-acoustic phononInterband scattering: electron-holeelectron-phonon with defectsWhat are f1, f2, & f3under illumination (non-equilibrium) ?Rate Equation FormalismRate Equation Formalismnumber of electrons = number of states x probability of occupancyassume total number of electrons in N1, N2, & N3 is contantRate Constants in EquilibriumRate Constants in EquilibriumDetailed BalanceDetailed BalanceIn equilibrium:Detailed balance:In equilibrium, each scattering process balances with its inverseRate EquationsRate EquationsAssume the rate constants don’t change out of equilibrium…SteadySteady--State SolutionsState SolutionsNonNon--equilibriumequilibriumFor example when intraband scattering is much faster than interbandscattering…SteadySteady--State SolutionsState SolutionsNonNon--equilibriumequilibriumEquilibrium Fermi-Diracdistribution:Non-equilibrium Quasi-Fermi-Diracdistribution:Intraband states have same chemical potentialin ‘equilibrium’ with each other because of fast intraband scatteringSteadySteady--State SolutionsState SolutionsNonNon--equilibriumequilibriumInterband states have different chemical potentialsunlessCounting in NonCounting in Non--equilibrium Semiconductorsequilibrium SemiconductorsEquilibrium