Solution-Modern-Semiconductor-Devices-for-Integrat

Chapter1VisualizationoftheSiliconCrystal1.1(a)PleaserefertoFigure1-2.The8corneratomsaresharedby8unitcellsandthereforecontribute1atom.Similarly,the6faceatomsareeachsharedby2unitcellsandcontribute3atoms.And,4atomsarelocatedinsidetheunitcell.Hence,therearetotal8siliconatomsineachunitcell.(b)Thevolumeoftheunitcellis3223831060110435435cm.cm.A.Vcellunit,andoneunitcellcontains8siliconatoms.Theatomicdensityofsiliconis322100058cmatoms)(silicon.VatomssiliconNcellunitSi.Hence,thereare5.001022siliconatomsinonecubiccentimeter.(c)Inordertofindthedensityofsilicon,weneedtocalculatehowheavyanindividualsiliconatomisg/atom.atoms/mole.g/mole.MassatomSi232311067410026128.Therefore,thedensityofsilicon(Si)ing/cm3is3atomSi1SiSicm/g2.33MassNρ.FermiFunction1.2(a)AssumeE=EfinEquation(1.7.1),f(E)becomes½.Hence,theprobabilityis½.(b)SetE=Ec+kTandEf=EcinEquation(1.7.1):27011111.eef(E)/kTEkTEcc.TheprobabilityoffindingelectronsinstatesatEc+kTis0.27.©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.*ForProblem1.2Part(b),wecannotuseapproximationssuchasEquations(1.7.2)or(1.7.3)sinceE-EfisneithermuchlargerthankTnormuchsmallerthan-kT.(c)f(E)istheprobabilityofastatebeingfilledatE,and1-f(E)istheprobabilityofastatebeingemptyatE.UsingEquation(1.7.1),wecanrewritetheproblemaskT)Ef(EkT)Ef(Ecc31/kTEkTE/kTEkTEfcfcee311111where/kTEkTE/kTEkTE/kTEkTE/kTEkTE/kTEkTEfcfcfcfcfceeeee333331111111/kTEkTEfce311.Now,theequationbecomes/kTEkTE/kTEkTEfcfcee31111.ThisistrueifandonlyiffcfcEkTEEkTE3.Solvingtheequationabove,wefindkTEEcf2.1.3(a)AssumeE=EfandT0KinEquation(1.7.1).f(E)becomes½.Hence,theprobabilityis½.(b)f(E)istheprobabilityofastatebeingfilledatE,and1-f(E)istheprobabilityofastatebeingemptyatE.UsingEquation(1.7.1),wecanrewritetheproblemas)Ef(E)Ef(Evc1/kTEE/kTEEfvfcee11111where©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458./kTEE/kTEE/kTEE/kTEE/kTEE/kTEEfvfvfvfvfvfceeeeee111111111.Now,theequationbecomes/kTEE/kTEEfvfcee1111.ThisistrueifandonlyiffvfcEEEE.Solvingtheequationabove,wefind2vcfEEE.(c)TheplotoftheFermi-DiracdistributionandtheMaxwell-Boltzmanndistributionisshownbelow.00.511.522.533.544.5500.10.20.30.40.50.60.70.80.91TheBoltzmanndistributionconsiderablyoverestimatestheFermidistributionforsmall(E-Ef)/kT.Ifweset(E-Ef)/kT=AinEquations(1.7.1)and(1.7.2),wehaveAAe.e11101.SolvingforA,wefind3121110ln11101011..A.e.eeAAA.Therefore,theBoltzmannapproximationisaccuratetowithin10%for(E-Ef)/kT2.31.ProbabilityFermi-DiracDistributionMaxwell-BoltzmannDistribution(E-Ef)/kT©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic,mechanical,photocopying,recording,orlikewise.Forinformationregardingpermission(s),writeto:RightsandPermissionsDepartment,PearsonEducation,Inc.,UpperSaddleRiver,NJ07458.1.4(a)PleaserefertotheexampleinSec.1.7.2.Theratioofthenitrogenconcentrationat10kmabovesealeveltothenitrogenconcentrationatsealevelisgivenby)/kTE(E/kTE/kTELevelSeakmLevelSeakmLevelSeakmeeeNNNN1010)()(2102wheregravityofonacceleratimoleculeNofmassaltitudeEELevelSeakm210.10564980106612810142246erg.scmg.cmTheratiois300)()(2112731038110564210211614.eeNNNN.K)Kerg.erg)/(.(LevelSeakm.Sincenitrogenislighterthanoxygen,thepotentialenergydifferencefornitrogenissmaller,andconsequentlytheexponentialtermfornitrogenislargerthan0.25foroxygen.Therefore,thenitrogenconcentrationat10kmismorethan25%ofthesealevelN2concentration.(b)Weknowthat25.0)()(2102LevelSeakmONON,30.0)()(2102LevelSeakmNNNN,and4)()(22LevelSeaLevelSeaONNN.Then,kmLevelSeaLevelSeaLevelSeaLevelSeakmkmkmONONONNNNNNNONNN1022222102102102)()()()()()()()(8.425.01430.0.ItismoreN2-richthanatsealevel.1.5/kTEEEfffeEEf1111/kTEEE/kTEEEffffee1/kTEEEffe11©2010PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.ThispublicationisprotectedbyCopyrightandwrittenpermissionshouldbeobtainedfromthepublisherpriortoanyprohibitedreproduction,storagei