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1、6.012-ElectronicDevicesandCircuitsFall2003TheGradualChannelApproximationfortheMOSFET:WearemodelingtheterminalcharacteristicsofaMOSFETandthuswantiD(vDS,vGS,vBS),iB(vDS,vGS,vBS),andiG(vDS,vGS,vBS).WerestrictourmodeltovDS≥0andvBS£0,sothediodesatthesourceanddrainarealwaysreverse
2、biased;inthiscaseiBª0.Becauseoftheinsulatingnatureoftheoxidebeneaththegate,wealsohaveiG=0,andourproblemreducestofindingiD(vDS,vGS,vBS).Themodelweuseiswhatiscalledthe"gradualchannelapproximation",anditissonamedbecauseweassumethatthevoltagesvarygraduallyalongthechannelfromthed
3、raintothesource.Atthesametime,theyvaryquicklyperpendicularlytothechannelmovingfromthegatetothebulksemiconductor.Inthemodelweassumewecanseparatetheproblemintotwopieceswhichcanbeworkedassimpleone-dimensionalproblems.Thefirstpieceisthex-directionproblemrelatingthegatevoltagetot
4、hechannelchargeandthedepletionregion;thisistheproblemwesolvedwhenwestudiedtheMOScapacitor.Thesecondpieceisthey-directionprobleminvolvingthecurrentin,andvoltagedropalong,thechannel;thisistheproblemwewillconsidernow.Tobeginweassumethatthevoltageonthegateissufficienttoinvertthe
5、channelandproceed.NoticethatiD(vDS,vGS,vBS)isthecurrentinthechannel;thisisadriftcurrent.Thereisaresistivevoltagedrop,vCS(y),alongthechannelfromvCS=vDSatthedrainendofthechannel,y=L,tovCS=0atthesourceendof1thechannel,y=0.Atanypoint,y,alongthechannelwewillhave:iD=-qN*(y)sy(y)WT
6、hecurrentisnotafunctionofy,-q*(y)isthechannelNsheetchargedensityaty,-qN*(y)=-Cox*[vGB-VT(y)]withCox*≡eo/to,andsy(y)isthenetvelocityofthechargecarriersinthey-directionaty,whichformodestelectricfieldsislinearlyproportionaltothefield:-!dvCS(y)sy(y)=-µeEy(y)=-µedyThecurrentisthe
7、ndvCS(y)iD=WµeCox*[vGB-VT(y)]dyToproceed,wemustexaminethefactor[vGB-VT(y)].WearereferencingourvoltagestothesourcesowefirstwritevGB=vGS-vBS.NextwelookatVT(y);whyisitafunctionofy?ToanswerthisquestionwemustnotethatthepictureisabitdifferentintheMOSFETthanitwasbeforewiththeisolat
8、edMOSstructurebecausenowthechannel(inversionlayer)canhaveavoltagerelativeto
