Lecture #4Important ConstantsDopant Ionization (Band Model)Carrier Concentration vs. TemperatureElectrons and Holes (Band Model)Thermal EquilibriumAnalogy for Thermal EquilibriumFermi FunctionEffect of Temperature on f(E)Boltzmann ApproximationEquilibrium Distribution of CarriersSlide 12Equilibrium Carrier ConcentrationsSlide 14Intrinsic Carrier ConcentrationN-type MaterialP-type MaterialDependence of EF on TemperatureSummarySlide 20Lecture #4OUTLINE• Energy band model (revisited)• Thermal equilibrium• Fermi-Dirac distribution– Boltzmann approximation• Relationship between EF and n, pRead: Chapter 2 (Section 2.4)EE130 Lecture 4, Slide 2Spring 2007Important Constants•Electronic charge, q = 1.610-19 C•Permittivity of free space, o = 8.85410-14 F/cm•Boltzmann constant, k = 8.6210-5 eV/K•Planck constant, h = 4.1410-15 eV-s•Free electron mass, mo = 9.110-31 kg•Thermal voltage kT/q = 26 mVEE130 Lecture 4, Slide 3Spring 2007Dopant Ionization (Band Model)EE130 Lecture 4, Slide 4Spring 2007Carrier Concentration vs. TemperatureEE130 Lecture 4, Slide 5Spring 2007Electrons and Holes (Band Model)•Electrons and holes tend to seek lowest-energy positions–Electrons tend to fall–Holes tend to float up (like bubbles in water)electron kinetic energyhole kinetic energyIncreasing electron energyIncreasing hole energyEcEvEE130 Lecture 4, Slide 6Spring 2007Thermal Equilibrium•No external forces are applied:–electric field = 0, magnetic field = 0–mechanical stress = 0–no light•Dynamic situation in which every process is balanced by its inverse process–Electron-hole pair (EHP) generation rate = EHP recombination rate•Thermal agitation electrons and holes exchange energy with the crystal lattice and each other Every energy state in the conduction band and valence band has a certain probability of being occupied by an electronEE130 Lecture 4, Slide 7Spring 2007Analogy for Thermal Equilibrium•There is a certain probability for the electrons in the conduction band to occupy high-energy states under the agitation of thermal energy (vibrating atoms)DishVibrating TableSand particlesEE130 Lecture 4, Slide 8Spring 2007Fermi Function•Probability that an available state at energy E is occupied:•EF is called the Fermi energy or the Fermi levelThere is only one Fermi level in a system at equilibrium.If E >> EF : If E << EF : If E = EF :kTEEFeEf/)(11)(EE130 Lecture 4, Slide 9Spring 2007Effect of Temperature on f(E)EE130 Lecture 4, Slide 10Spring 2007Boltzmann ApproximationProbability that a state is empty (occupied by a hole):kTEEFFeEfkTEE/)()( ,3 IfkTEEFFeEfkTEE/)(1)( ,3 IfkTEEkTEEFFeeEf/)(/)()(1EE130 Lecture 4, Slide 11Spring 2007Equilibrium Distribution of Carriers•Obtain n(E) by multiplying gc(E) and f(E)Energy banddiagramDensity ofStatesProbabilityof occupancyCarrier distributionEE130 Lecture 4, Slide 12Spring 2007•Obtain p(E) by multiplying gv(E) and 1-f(E)Energy banddiagramDensity ofStatesProbabilityof occupancyCarrier distributionEE130 Lecture 4, Slide 13Spring 2007Equilibrium Carrier Concentrations•Integrate n(E) over all the energies in the conduction band to obtain n:•By using the Boltzmann approximation, and extending the integration limit to , we obtain band conduction of topcEc(E)f(E)dEgn2/32* /)(22 wherehkTmNeNnnckTEEcFcEE130 Lecture 4, Slide 14Spring 2007•Integrate p(E) over all the energies in the valence band to obtain p:•By using the Boltzmann approximation, and extending the integration limit to -, we obtain 1band valenceof bottomvEvdEf(E)(E)gp2/32* /)(22 wherehkTmNeNppvkTEEvvFEE130 Lecture 4, Slide 15Spring 2007Intrinsic Carrier Concentration 2 / /)( /)( /)( ikTEvckTEEvckTEEvkTEEcneNNeNNeNeNnpGvcvFFc 2/ kTEvciGeNNnEE130 Lecture 4, Slide 16Spring 2007N-type MaterialEnergy banddiagramDensity ofStatesProbabilityof occupancyCarrier distributionEE130 Lecture 4, Slide 17Spring 2007P-type MaterialEnergy banddiagramDensity ofStatesProbabilityof occupancyCarrier distributionEE130 Lecture 4, Slide 18Spring 2007Dependence of EF on Temperature10131014101510161017101810191020 300K400K400K300K kTEEcFceNn/)( nNkTEEccFlnNet Dopant Concentration (cm-3)EcEvEF for donor-dopedEF for acceptor-dopedEE130 Lecture 4, Slide 19Spring 2007Summary•Thermal equilibrium:–Balance between internal processes with no external stimulus (no electric field, no light, etc.)–Fermi function •Probability that a state at energy E is filled with an electron, under equilibrium conditions.•Boltzmann approximation:For high E, i.e. E – EF > 3kT: For low E, i.e. EF – E > 3kT: kTEEFeEf/)(11)(kTEEFeEf/)()(kTEEFeEf/)()(1EE130 Lecture 4, Slide 20Spring 2007•Relationship between EF and n, p :•Intrinsic carrier concentration : /)( kTEEvvFeNp /)( kTEEcFceNn 2/
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