1Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06-Electron Energy Band- Fermi Level-Electrostatics of device charges2Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06The Simplified Electron Energy Band Diagram3Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06xElectron EnergyE-field+-ECEV21xElectron EnergyE-field+-ECEV21Electric potential φ(2) < φ(1)Electric potential φ(2) > φ(1)Energy Band Diagram with E-fieldElectron concentration n kT/)]1()2([qkT/)1(qkT/)2(qeee)1(n)2(nφ−φφφ==4Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06Probability of available states at energy E being occupied f(E) = 1/ [ 1+ exp (E- Ef) / kT] where Ef is the Fermi energy and k = Boltzmann constant=8.617 ∗ 10-5eV/KThe Fermi-Dirac Distribution (Fermi Function)T=0K0.5E -Eff(E)5Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06(2) Probability of available states at energy E NOT being occupied 1- f(E) = 1/ [ 1+ exp (Ef-E) / kT]Properties of the Fermi-Dirac Distribution(1) f(E) ≅ exp [- (E- Ef) / kT] for (E- Ef) > 3kTNote:At 300K,kT= 0.026eV•This approximation is called Boltzmann approximationProbabilityof electron state at energy Ewill be occupied6Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06EcEvEiEf(n-type)Ef(p-type)q|ΦF|Let qΦF≡ Ef-Ein = niexp [(Ef-Ei)/kT]∴n = niexp [qΦF /kT]How to find Efwhen n(or p) is known7Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06Dependence of Fermi Level with Doping ConcentrationEi≡ (EC+EV)/2 Middle of energy gapWhen Si is undoped, Ef= Ei ; also n =p = ni8Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06At thermal equilibrium ( i.e., no external perturbation),The Fermi Energy must be constant for all positionsThe Fermi Energy at thermal equilibriumMaterial AMaterial BMaterial CMaterial DEFPosition xElectron energy9Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06AftercontactformationBeforecontactformationElectron Transfer during contact formationESystem 1System 2EF1EF2eSystem 1System 2EF---+++ESystem 1System 2EF1EF2eSystem 1System 2EF---+++Net negative chargeNet positive charge10Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06Fermi level of the side which has a relatively higher electric potential will have a relatively lower electron energy ( Potential Energy = -q • electric potential.) Only difference of the E 's at both sides are important, not the absolute position of the Fermi levels.Side 2Side 1Ef1f2EVa> 0qVaVaSide 2Side 1Ef1f2EVa< 0q| |+- -+Potential difference across depletion region= Vbi-VaApplied Bias and Fermi Level11Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06PN junctionsComplete Depletion Approximation used for charges inside depletion regionr(x) ≈ ND+(x) – NA-(x)n-Sip-SiDepletion regionND+onlyNA-onlyρ(x) is +ρ(x) is -ND+ and nρ(x) is 0NA-and pρ(x) is 0E-field++++----Quasi-neutralregionQuasi-neutralregionn-Sip-SiDepletion regionND+onlyNA-onlyρ(x) is +ρ(x) is -ND+ and nρ(x) is 0NA-and pρ(x) is 0E-field++++----Quasi-neutralregionQuasi-neutralregionhttp://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.htmlThermal Equilibrium12Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S061) Summation of all charges = 0Electrostatics of Device Charges xρ1Semiconductorρ(x)ρ2xd1xd2Semiconductor--x=0p-typen-type2) E-field =0 outside depletion regionsρ2• xd2= ρ1• xd1E = 0E = 0E ≠013Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S063) Relationship between E-field and charge density ρ(x)d [ε E(x)] /dx = ρ(x) “Gauss Law”4) Relationship between E-field and potential φE(x) = - dφ(x)/dx14Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06Example Analysis : n+/ p-Si junctionx-qNaxdx=0+Q'n+ Sip-Siρ(x)E (x)xd1) ⇒ Q’ = qNaxd3) ⇒Slope = qNa/εs4) ⇒ Area under E-field curve = voltage across depletion region = qNaxd2/2εsEmax=qNaxd/εsDepletionregion2) ⇒ E = 0Depletion regionis very thin and is approximated as a thin sheet charge15Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06x-x=0-qNA+qNDρ(x)-xp+xnx=0xρ1(x)-xp-qNAQ=+qNAxpxx=0ρ2(x)+qND+xnQ=-qNAxpSuperposition PrincipleIf ρ1(x) ⇒ E1(x) and V1(x)ρ2(x) ⇒ E2(x) and V2(x) thenρ1(x) + ρ2(x) ⇒ E1(x) + E2(x) and V1(x) + V2(x)+16Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06x-x=0Slope =-qNA/εsE1(x)-xpx=0xρ1(x)-xp-qNAQ=+qNAxpxx=0ρ2(x)+qND+xnQ=-qNAxp+x-x=0+xnE2(x)Slope =+qND/εs17Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06x-x=0Slope = - qNA/εsE(x) = E1(x)+ E2(x)-xp+xnSlope = + qND/εsEmax = -qNA xp/εs= -qND xn/εsSketch of E(x)18Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06xx=0Q'Metalρ(x)Oxidex=xox−ρxdx=0Q'Metal Semiconductorρ(x)Oxidex=xox=xo+Q'-x−ρxdx=0Q'Metal Semiconductorρ(x)Oxidex=xox=xo+SemiconductorDepletion Mode :Charge and Electric Field Distributionsby Superposition Principle of Electrostaticsxxdx=0Metal SemiconductorE(x)Oxidex=xox=xo+xxdx=0Metal SemiconductorE(x)Oxidex=xox=xo+xxdx=0Metal SemiconductorE(x)Oxidex=xox=xo+=+19Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06Approximation assumesVSidoes not change muchOXOXCnV∆∝∆SiOXFBGVVVV++=Picks up all the changesin VGJustification:If surface electron densitychanges by ∆n EiEf()kTennlnδ∝∆∆∝δδbut the change of VSi changes only by kT/q [ ln (∆n)] – small!Why xdmax~ constant beyond onset of strong inversion ?Higher than VT20Professor N Cheung, U.C. BerkeleySemiconductor Tutorial 2EE143 S06VSi n-surface0 2.10E+040.1 9.84E+050.2 4.61E+070.3 2.16E+090.4 1.01E+110.5 4.73E+120.6 2.21E+140.7 1.04E+160.8 4.85E+170.9 2.27E+19n-surfacep-SiNa=1016/cm3EfEi0.35eVn-bulk = 2.1 • 104/cm3qVSiOnset of strong inversion(at VT)N (surface) = n (bulk) • exp [
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