1Carrier Concentration in Equilibrium¾ Since current (electron and hole flow) is dependent on the concentration of electrons and holes in the material, we need to develop equations that describe these concentrations.¾ Furthermore, we will find it useful to relate the these concentrations to the average energy(Fermi energy) in the material.Carrier Concentrations in Equilibrium Developing the Mathematical model for Electrons and HolesG Motivation F The Density of Electrons is:Probability the state is filledNumber of states per cm-3in energy range dEProbability the state is emptyNumber of states per cm-3 in energy range dEunits of n and p are [ #/cm3] The Density of Hole is:Developing the Mathematical Model for Electrons and HolesThis is known as the Fermi-Dirac integralof order 1/2 or, F1/2(ηc)Developing the Mathematical Model for Electrons and HolesEffective density of states in CB2The effective density of states in Conduction Bandat 300 K We can further define:The effective density of statesin Valance Band This is a general relationship holding for all materials and results in:Developing the Mathematical Model for Electrons and Holes()CCFNnηπ212=()VVFNpηπ212=()kTEEwherefVV−=ηFermi-Dirac integrals can be numerically determined or read from tables or...Developing the Mathematical Model for Electrons and Holes Useful approximations to the Fermi-Dirac integral:()CCeeηηηη−−−≅+11()()kTEECCfeF−=221πη()()kTEEVfVeF−=221πηkTEEifCf3−〈kTEEwhenSimilaryVf3+〈Developing the Mathematical Model for Electrons and HolesNondegenerate Case Useful approximations to the Fermi-Dirac integral:()kTEECCfeNn−=()kTEEVfVeNp−=Developing the Mathematical Model for Electrons and Holes3 Useful approximations to the Fermi-Dirac integral:()kTEEiifenn−=()kTEEifienp−=()kTEECiCieNn−=When n = ni, Ef = Ei(the intrinsic energy), thenorand()kTEEViiVeNn−=()kTEEiVVienN−=or()kTEEiCiCenN−=Developing the Mathematical Model for Electrons and Holes Other useful relationships: n⋅p product:2innp =known as the Law of mass Action()kTEECiCieNn−=and()kTEEViiVeNn−=()kTEVCkTEEVCigVCeNNeNNn−−−==2kTEVCigeNNn2−=()kTEEiifenn−=()kTEEifiepp−=andSinceDeveloping the Mathematical Model for Electrons and Holes¾ If excess charge existed within the semiconductor, random motion of charge would imply net (AC) current flow.¨ Not possible!¾ Thus, all charges within the semiconductor must cancel. Charge Neutrality:()()[]0=−+−⋅+−nNNpqdAMobile + charge ¼Immobile - charge ¼Immobile + charge ¼Mobile - charge ¼Developing the Mathematical Model for Electrons and Holes¾ NA¯= Concentration of “ionized” acceptors = ~ NA¾ ND+= Concentration of “ionized” Donors = ~ ND Charge Neutrality: Total Ionization case()()0=−+−+−nNNpdADeveloping the Mathematical Model for Electrons and Holes4DNn ≅DiNnp2≅andADNN 〉〉iDnN 〉〉andifANp ≅AiNnn2≅andDANN 〉〉iAnN 〉〉andifor2innp =2222iADADnNNNNn +−+−=2222iDADAnNNNNp +−+−=()()0=−+− nNNpDA()()02=−+− nNNnnDAi()022=−−−iADnnNNn Charge Neutrality: Total Ionization caseandDeveloping the Mathematical Model for Electrons and Holes ExampleDNn≅DiNnp2≅andADNN 〉〉iDnN 〉〉andifDeveloping the Mathematical Model for Electrons and HolesAn intrinsic Silicon wafer has 1x1010cm-3holes. When 1x1018cm-3donors are added, what is the new hole concentration? Example2222iDADAnNNNNp +−+−=Developing the Mathematical Model for Electrons and HolesAn intrinsic Silicon wafer has 1x1010cm-3holes. When 1x1018cm-3acceptors and 8x1017cm-3donors are added, what is the new hole concentration? Example2innp =Developing the Mathematical Model for Electrons and HolesAn intrinsic Silicon wafer at 470K has 1x1014cm-3holes. When 1x1014cm-3acceptors are added, what is the new electron and hole concentrations?2222iDADAnNNNNp +−+−=5 Example¨ Intrinsic material at High TemperatureDeveloping the Mathematical Model for Electrons and HolesAn intrinsic Silicon wafer at 600K has 4x1015cm-3 holes. When 1x1014cm-3acceptors are added, what is the new electron and hole concentrations?X2222iDADAnNNNNp +−+−= Temperature behavior of Doped Materialsp-type vs Temp.movn-type vs Temp.movDeveloping the Mathematical Model for Electrons and