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Semiconductor statisticsSemiconductor: a system with electron orbitals grouped into two energy bands separated by an energy gap (band gap)Lower band is the valence band, upper band is the conduction bandIn Si: energy gap is 1.1 eV, room temperature is 25 meVIn a pure semiconductor at zero temperature, all VB orbitals are occupied, all CB orbitals are empty and semiconductor is an insulatorFinite conductivity is due to occupied orbitals in CB (conduction electrons) or unoccupied orbitals in VB (holes)Two mechanisms are possible: 1). Thermal excitation of electrons from VB to CB2). Thermal ionization of impurities that provide additional orbitalsand can donate (donors) or accept (acceptors) electronsvcgεεε−=Semiconductor statisticsIn pure (intrinsic) semiconductors, concentration of conduction electrons is equal to the concentration of holes:Electrical neutrality condition:In semiconductors with impurities (doped semiconductors), impurities may be ionized; donors become positively charged and acceptors become negatively chargedNet ionized donor (doping) concentration:Electrical neutrality condition:Electron and hole concentrations can be found from the F-D distribution functions:Chemical potential (Fermi level) is found from electrical neutrality condition:henn =−+−≡Δadnnn−+−=Δ=−adhennnnnVNnfNffVNnfNfhhVBhheheeCBeee=⇒=⇒+−=−≡=⇒=⇒+−=∑∑)(1])(exp[1)(1)()(1])(exp[1)(ετεμεεετμεεnnnheΔ=− )()(μμClassical regimeClassical regime implies low orbital occupancies:This happens when the Fermi level lies deep in the energy gap, so that:Therefore:Such a semiconductor is called nondegenerate and is described by classical distribution functions: 1)( ,1)( <<<<εεheff0)( and 0)( >−>−vcεμμε])(exp[)( ];)(exp[)(τεμετμεε−−≅−−≅heff1])(exp[1])(exp[11])(exp[1)(1])(exp[1])(exp[11])(exp[1)(<<−−⇒>>−⇒<<+−=<<−−⇒>>−⇒<<+−=τεμτεμτεμετμετμετμεεvvhcceffClassical regimeTotal number of electrons in CB:Total number of holes in VB:Orbitals close to the band edge make dominant contributions, orbitalsdeep in the bands can be ignoredNc(Nv) is a partition function for one free particle with spin ½ and the density-of-states effective mass:444344421cCBccCBeNN∑∑−−−−=−−= ])(exp[])(exp[])(exp[τεετμετμε])(exp[τμε−−≡cceNN444344421vVBvvVBhNN∑∑−−−−=−−= ])(exp[])(exp[])(exp[τεετεμτεμ])(exp[τεμvvhNN −−=VmNVmNVnZhvecQ232*232*1)2(2 ;)2(22 hhπτπτ==⇒=Law of mass actionQuantum concentration for electrons and holes (effective density of states of CB and VB) :Carrier concentrations then become:Law of mass action:In classical regime product of concentrations is independent on the Fermi level and impurity concentration:In a pure (intrinsic) semiconductor:In general: (mass action law)232*232*)2(2)2(2hhπτπτhvveccmVNnmVNn=≡=≡])(exp[])(exp[τεμτμεvvhccennnn−−=−−=]exp[])(exp[])(exp[])(exp[τετεετεμτμεgvcvcvcvvcchennnnnnnn−=−−=−−⋅−−=]2exp[)(21τεgvcinnn −=⇒==ihennn⇒≠henn2ihennn =Intrinsic Fermi levelIntrinsic Fermi level:In intrinsic semiconductor Fermi level lies near the middle of the energy gap:{}ccvcvccciennnnnn)exp( ]2)(exp[)(])(exp[21τετεετμε×−−=−−⇒=⇒⎟⎟⎠⎞⎜⎜⎝⎛++=⇒++⎟⎟⎠⎞⎜⎜⎝⎛=⇒+=⇒+=⇒23**2121log2122log21} ]2)(exp[)log{()](explog[ ]2)(exp[)()(expehvcvccvvccvvccvmmnnnnnnτεεμτεετμτεετμτεετμ⎟⎟⎠⎞⎜⎜⎝⎛++=**log432ehvcmmτεεμ2vcεεμ+≅Doped semiconductorsSC with more (fewer) electrons than holes is called n (p)-typeImpurities may affect either band:Donors contribute electrons to the CB or fill holes in the VBAcceptors remove electrons from the CB or create holes in the VBWe will assume that each impurity atom contributes (removes) exactly one electron (approximation of fully ionized impurities)To find concentrations we combine:Electrical neutrality condition and law of mass action:Electron concentration:−−−+→++→AADD00eennnnnnnnehadheΔ−=⇒−=−=Δ−+24)(0)(222222ieieeieeihennnnnnnnnnnnnnn+Δ+Δ=⇒=−Δ−⇒=Δ−⇒=Doped semiconductorsHole concentration:If doping concentration is much higher than intrinsic concentration, the semiconductor is called extrinsic:For extrinsic semiconductors:Majority carrier concentration:n-type:p-type:24)(22iehnnnnnn+Δ+Δ−=Δ−=inn >>Δ||||2||)2(1||4)(2222nnnnnnnniiiΔ+Δ≅Δ+Δ=+Δiiihiiiennnnnnnnnnnnnnnnnnn>>Δ≅Δ+Δ=Δ+Δ+Δ=<<Δ=Δ+Δ+Δ−=⇒<Δ||||||2||2|||| ;||2||2||||02222;22 2202222iiihiiiennnnnnnnnnnnnnnnnnn<<Δ=Δ+Δ+Δ−≅>>Δ≅Δ+Δ=Δ+Δ+Δ≅⇒>ΔExtrinsic Fermi levelExtrinsic Fermi level:n-type:p-type:In extrinsic semiconductors at low temperatures, Fermi level approaches the band edge and classical approximation breaks downquantum concentration decreases with temperature and becomes comparable to carrier concentration (degenerate semiconductors))log()()log(])(exp[)log()()log(])(exp[hvvvhvvvhecccecccennnnnnnnnnnnτεμτεμτεμτεμτμετμε+=⇒−=⇒−−=−=⇒−=⇒−−=|)|log(||)log(nnnnnnnnvvhcceΔ+≅⇒Δ≅Δ−≅⇒Δ≅τεμτεμDegenerate semiconductorsFermi gas approach:Density of states for conduction electrons is non-zero above conduction band:Concentration:Change variables:∫=εεεdfDNe)()(cemVDεεπε−⎟⎟⎠⎞⎜⎜⎝⎛=232*222)(h∫∞+−−⎟⎟⎠⎞⎜⎜⎝⎛==cdmVNnceeeετμεεεεπ1])(exp[221232*2h)(1)exp(21)exp(221 ; ; ;021021232*2ηηπητπητεμτετεετεεInxdxxnxdxxmn xxτμεdxdxxcceeccc=+−=+−⎟⎟⎠⎞⎜⎜⎝⎛=−≡−−=−==−⇒−≡∫∫∞∞hceecnmmnπτππτ222122232*2232*=⎟⎟⎠⎞⎜⎜⎝⎛⇒⎟⎟⎠⎞⎜⎜⎝⎛=hhDegenerate semiconductorsFermi-Dirac integral:A). Classical case (the Fermi level is deep in the energy gap):cf., ideal gasrnnxdxxIce≡=+−=∫∞0211)exp(2)(ηπη])(exp[2)23(2)(1)exp(1)(021τεμππηηητμεηccxccecndxxeenInnx−==Γ≅=⇒>>−⇒>>−≡−∫∞−43421rη log≅Degenerate semiconductorsB). Degenerate case (Joyce-Dixon approximation):Approximate expression:Law of mass action for degenerate semiconductors:(if electrons are degenerate, holes remain classical)iiicerArrrrrrreInnr∑==+⎟⎟⎠⎞⎜⎜⎝⎛−−≅−=⇒+=⇒≅⇒≅=≡12...9316381log)()(loglog)(ηδδηηηη8 logrrη +≅.1043.4 ;1048.1 ;1095.4 ;1053.364433211−−−−⋅−≅⋅≅⋅−≅⋅≅


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U of M PHYS 4201 - Semiconductor statistics

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