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第 29卷?第 2期2008 年 2 月半?导?体?学?报JOURNAL OF SEMICONDUCTORSVol.29?No.2Feb.,2008*This investigation and Jie Binbin have been supported by the CTSAH Associates(CTSA)founded by the late Linda Su?Nan Chang Sah.?Corresponding author.Email:bb_jie and tom_sah ?Received 10 January 2008,revised manuscript received 20 January 2008?2008 Chinese Institute of ElectronicsThe Bipolar Field?Effect Transistor:IV.Short Channel Drift?Diffusion Current Theory(Two?MOS?Gates on Pure?Base)*Jie Binbin1,?and Sah Chih?Tang1,2,3,?(1 Peking University,Beijing?100871,China)(2 University of Florida,Gainesville,Florida?32605,USA)(3 Chinese Academy of Sciences,Foreign Member,Beijing?100864,China)Abstract:This paper gives the short channel analytical theory of the bipolar field?effect transistor(BiFET)with the driftand diffusion currents separately computed in the analytical theory.As in the last?month paper which represented thedrift and diffusion current by the single electrochemical(potential?gradient)current,the two?dimensional transistor ispartitioned into two sections,the source and drain sections,each can operate as the electron or hole emitter or collectorunder specific combinations of applied terminal voltages.Analytical solution is then obtained in the source and drain sec?tions by separating the two?dimensional trap?free Shockley Equations into two one?dimensional equations parametricallycoupled via the surface?electric?potential and by using electron current continuity and hole current continuity at the bound?ary between the emitter and collector sections.Total and the drift and diffusion components of the electron?channel andhole?channel currents and output and transfer conductances,and the electrical lengths of the two sections are computedand presented in graphs as a function of the D.C.terminal voltages for the model transistor with two identical and con?nected metal?oxide?silicon?gates(MOS?gates)on a thin pure?silicon base over practical ranges of thicknesses of the siliconbase and gate oxide.Deviations of the two?section short?channel theory from the one?section long?channel theory aredescribed.Key words:bipolar field?effect transistor theory;MOS field?effect transistor;simultaneous electron and hole surface andvolume channels;surface potential;two?section short?channel theory;double?gate pure?basePACC:7340Q?EEACC:2560S;2560BCLCnumber:TN386?1?Document code:A?Article ID:0253?4177(2008)02?0193?081?IntroductionThe silicon MOS field?eff ect transistor(FET)technology is advancing into the nanometer dimen?sions in the form of double?gate and thin pure?base,fin?like structure,known as the FinFETs 1.Theirexperimental electrical current?voltage characteris?tics,recently reported by IMEC with Assignees fromfour companies 2,could not be accounted for 3bythe current MOSFET theory that was developed fromthe traditional 55?year?old unipolar field?ef fect theoryof the p/n junction?gate f ield?effect transistor(JG?FET)inventedandtheorizedbyShockleyin1952 4,5,and reduced to practice by Dacey andRoss 6.This 1952?Shockley theory was f ollowed byrecent device theorists and engineers to compact mod?el(i)the MOS?gate and Insulator?Gate FETs(MOS?FET and IGFET)with single?gate on semi?inf inite?thick?base,?bulk transistors 7,8and(ii)also theFinFETs which was extensively and carefully re?viewed by Ortiz?Conde,Garcia?Sanchez,Liou andstudents in January 2007 9.It was soon theorized anddemonstrated in March 2007 3and later presented byus 3,10 14that the observed experimental current?voltage characteristics reported by IMEC+Assign?ees 2showed distinct bipolar behavior,namely thesimultaneous presence of both electron and hole sur?f ace?inversion?channel currents,even a hint of vol?ume?channel currents,giving six channels in theirtwo?MOS?gates on thin?base FinFET structures.Thebipolar nature of the FET and our theory were namedthe 100%Bipolar Field Effect Transistor(BiFET)Theory 3,10 14,since it includes all four currents,thedrift and diffusion currents of both electrons andholes.In contrast,the now 55?year?old Shockley!s1952 Unipolar Field?Ef fect Transistor Theory(Uni?FET)is a 25%FET theory because its theory and an?alytical solution 4 6considered only one current,thedrift of one carrier species.This 25%UniFET theorywas employed by 1964?Sah 15in the constant thresh?old?voltage?based MOS transistor model with parabol?ic voltage dependence of the current,and it was used半?导?体?学?报第 29 卷by the initial circuit simulator SPICE 7.It was soonextended to account for the non?parabolic voltage de?pendence of the current by including the voltage?dependence of the threshold voltage from immobileimpurity ions in the surface space?charge layer withvoltage?dependentthicknessby1965?Sah?Pao 16,known as?bulk?charge.This was immediately usedby the second generation SPICE 7.These were fur?ther extended in the 1966 Sah?Pao theory 16,17f or thebulk!or semi?infinite?thick MOSFET to include boththe drift and the diffusion currents via the electro?chemical potential gradient as the ef fective or totalfield that drives the current,but again for only onecarrier species,the electron or the hole,but notboth,thus,it is the 50%BiFET or the 100%UniFETtheory.This was again extended in 1996?Sah and2005?Jie?Sah 18 20to provide rigorous 2?D analyticalseparation of the drift current due to electric f ield andthe diffusion current due to charge?carriers!concen?tration gradient,in order to give the underlying phys?ics of the multiple drift current terms,and to meetthe preference of the more popular current transporttheory via drift and diffusion currents,rather thanelectrochemical current.This paper completes thepreceding three analytical expositions of the BiFETtheory 12 14by providing the analytical solutions andnumerical results for the more popular analytically?separated drift and diffusion current long?channeltheory 18 20,now for electrically short channels,soas to extend the results we just reported f or the elec?trochemical current in electrically short channels 14.Following the presentation f ormat given for the ana?lytically?separateddrif t?dif fusionlongchannel 13,families of graphs are presented for the total and thedrift?diffusion components of the drain current andalso the output and transfer conductances,as well asthe voltage dependences of the section length,y0?VGSand y0?VDS.Fractional or percent deviations of thelong channel theory from the electrically short chan?nel theoryare alsocomputedandpresentedingraphs.2?Short Channel Theory of the Two?Gateon Pure?Base Transistors?The voltage and current equations of the UniFETtheory with single?gate on thick or semi?infinite im?pure?base from the Electrochemical Potential Theorygivenin1966?Sah?Pao 16,17andfromtheDrif t?DiffusionTheorygivenin1996?2005?Sah?Jie 18 20were modified f or the BiFET with two identical gateson a thin pure?base 3,10 14.They are immediately ap?plicable to each of the two sections separated by theboundary y=y0(x).The complete analytical solutionalso requires the additional equations from matchingthe electric?potential and electric?current?density so?lutions in the two sections at their boundary.From both the integration form and dif ferentialform ofthe electrochemical current theory 14,amatch of the electron and hole currents at the bound?ary between the two sections of the BiFET gives theunipolar initial guess solution of the flatband liney0 14.From the drift and diffusion current theory,the electron and hole currents read:IDN=-(W/y0)?q?nNEY+qDn(N/y)x y?(x=0 to xB;y=0 to y0)(1)IDP=-W/(L-y0)?q?pPEY+qDp(P/y)x y(x=0 to xB;y=y0to L)(2)ID=IDN+IDP=-IS?(3)whereournotationandcoordinatesysteminRef.14 are used.In the unipolar initial guess forthe bipolar gate?voltage?equations,Equations(11),(13)and(14)of Ref.14 are also used here.It isvery important to note that when the flatband line islocated inside the base region(0 y0 L)away fromits two physical boundaries y=0 and y=L,the cor?rect quasi?Fermi potential boundary values are UNS=USBand UPD=UDBwhen USB USBor UDS 0 which is the case of initial e?lectron surface inversion channel and later hole sur?f ace inversion channel when the electron current issaturated after UDS UGSor UDG 0 or UGD 0 to in?duce the surface?inversion hole channel in pure base,or surface accumulation hole channel in p?type base):When UGB#USB,y0=0IDN=0(4)IDP=?p(W/L)CO(kT/q)2 UGB-US(L)2-UGB-US(0)2Parabolic drift P1?+2 US(L)-US(0)+(XB/2)(CO/CD)(UGB-US(L)2-(UGB-US(0)2+(XB/2)(CD/CO)exp(USB-US(0)-exp(UDB-US(L)(/y)(EX)2or drift E2X+Dp(W/L)2CO(kT/q)US(L)-US(0)?Linear diffusion D1?(5)194第 2 期Jie and Sah:?The Bipolar Field?Effect Transistor:IV.Short Channel Drift?Diffusion Current%When USB UGB UDB,0 y0 LIDN=?n(W/y0)CO(kT/q)2 UGB-US(0)2Parabolic drift P1?+2 UGB-US(0)+(XB/2)(CO/CD)UGB-US(0)2+(XB/2)(CD/CO)1-exp(US(0)-USB)(/y)(EX)2or drift E2X?+Dn(W/y0)2CO(kT/q)UGB-US(0)Linear diffusion D1?(6)IDP=?p W/(L-y0)CO(kT/q)2 UGB-US(L)2Parabolic drift P1?+2 US(L)-UGB+(XB/2)(CO/CD)UGB-US(L)2+(XB/2)(CD/CO)1-exp(UDB-US(L)(/y)(EX)2or drift E2X?+Dp W/(L-y0)2CO(kT/q)US(L)-UGBLinear diffusion D1?(7)When UGB&UDB,y0=LIDN=?n(W/L)CO(kT/q)2 UGB-US(0)2-UGB-US(L)2Parabolic drift P1?+2 US(L)-US(0)+(XB/2)(CO/CD)(UGB-US(0)2-(UGB-US(L)2+(XB/2)(CD/CO)exp(US(L)-UDB)-exp(US(0)-USB)(/y)(EX)2or drift E2X?+Dn(W/L)2CO(kT/q)US(L)-US(0)Linear diffusion D1?(8)IDP=0(9)3?Computed Variations of the ElectricalLengths with Electrical Potential?The numerical solutions of the analytical formulasgiven in the previoussection are given in sixteen figures inthis section.They are from the initial solution for the Bi?FET theory which is dominated by the majority carrier ineach of the two sections,so the minority carriersare neg?lected in the initial solution.The higher order solutionsincluding both carrier species and their approach to thebipolar solution is reported in a future paper to assess thedeviations of the initial solutions and convergence speedsfor optimization of deviation and computation time.Toillustrate the voltage dependences of electrical length ofthe two sections,the normalized electrical lengths,y0/Land 1-(y0/L),and their voltage derivativesare graphedrespectively in Figs.1 and 2 as a function of the fractionalcurrent?saturation voltages,VGS/VDSfrom 0 to 1 and1-(VGS/VDS)from 0 to 1.These two figuresshow theFig?1?Source section electrical length y0/L iscontinuousat VGS=0.Its derivative with respect to the drain voltage(/UDS)(y0/L)is continuous at VGS=0.Its derivative with respect to the gate volt?age(/UGS)(y0/L)diverges at VGS=0 as shown in this log?logplot.continuous electrical lengths through the current satura?tion point,and the discontinuous voltage derivatives.These are indicative of the match of the analytical solu?tions from the two sections at the electrical boundary ofthe two sections,which was defined 14as the flatbandpoint(or line or plane),y0.4?Computed Current?Voltage Characteris?tics and Channel Section Lengths?Selected results from computed terminal current?volt?age and conductance?voltage characteristics and theirvoltage and oxide?base thicknesses dependences are illus?trated in twelve figures from Figs.3 to 14,in four groupsof three figures each,versus VGSor VDS.These are simi?lar to those in our second BiFET theory report 13whichwere obtained,without taking into account of the electri?cally shortened channels,by letting y0=L for the sourceelectron emitter and L-y0=L for the drain hole emit?ter.These 12 figures show increases of currents and con?ductances by the shortened channel,but not the generaldependences on the terminal voltage,VGSand VDS.TheFig?2?Drain section electrical length(L-y0)/L is continuous atVGS=VDS.Its derivatives with respect to the gate voltage and drainvoltage,(/UGS)(y0/L)and(/UDS)(y0/L)both diverge atVGS=VDS.195半?导?体?学?报第 29 卷Fig?3?The DC transfer characteristics at two Drain?Source volta?ges,VDS=0?50V and 1?0V.(a)The total drain current IDand e?lectron and hole drift and diffusion channel current components;(b)The fractional electron or hole parabolic drift current P1,lineardiffusion current D1,and(/y)(E2X)current normalized to thetotal electron or hole channel currents,IDNor IDP.(c)The nor?malized electron emitter electrical length y0/L and hole emitterelectrical length(L-y0)/L.quantitative details in percentage deviations are reportedin Figs.15 and 16.4.1?Current?Voltage and Conductance?Voltage Characteris?tics?The VGSand VDSvariationsof the total drain currentandthedriftanddiffusioncomponentsofthedraincurrent,theirtransconductancesandoutputFig?4?The D.C.transconductance?gate?voltage characteristics attwo VDS=0?5V and 1?0V.(a)The total transconductance gmandelectron and hole drift and diffusion components;(b)The fractionalelectron or hole transconductance from the parabolic drift currentP1,linear diffusion current D1,and(/y)(E2X)current normal?ized to the total electron or hole transconductances gmNor gmP.(c)The derivative of emitter length(/UGS)(y0/L).The discontinu?ity at VGS=0 and VDSis not significant.Fig?5?The D.C.output?conductance?gate?voltage characteristics attwo VDS=0?5V and 1?0V.(a)The total output conductance gdandelectron and hole drift and diffusion components;(b)The fractionaloutput conductance from the parabolic drift current P1,linear dif?fusion current D1,and(/y)(E2X)current normalized to the totaloutput conductances gdNor gdP.(c)The derivative of emitter sec?tion electrical length(/UDS)(y0/L).The discontinuity at VGS=VDSis apparent.conductances,and the electron and hole electrical chan?nel lengths,are given in six figures,from Figs?3 to 8.Figures 3,4,and 5 are the currents and twoconductances andtheirdriftanddiffusioncompo?nents,as well as the length of the two electrical sectionsversus VGS.Figures 6,7,and 8 are the currents and two con?ductances and their drift and diffusion components,aswell as the length of the two electrical sections versusVDS.Fig?6?The DC output current?voltage characteristics at two gate?source voltages VGS=1?0V and 1?5V.(a)The total drain currentIDand the electron and hole drift and diffusion channel currentcomponents;(b)The fractional electron or hole parabolic drift cur?rent P1,linear diffusion current D1,and the(/y)(E2X)driftcurrent normalized to the total electron or hole current,IDNor IDP.(c)The electron emitter length y0and the hole emitter length(L-y0).196第 2 期Jie and Sah:?The Bipolar Field?Effect Transistor:IV.Short Channel Drift?Diffusion Current%Fig?7?The transconductance?drain?voltage characteristics at twoVGS=1?0V,1?5V.(a)The total transconductance gmandelectronand hole drift and diffusion components;(b)The fractionalelectronor hole transconductance from the parabolic drift current P1,lineardiffusion current D1,and(/y)(E2X)current normalized to thetotal electron or hole transconductances gmNor gmP.(c)The deriv?ative of emitter length(/UGS)(y0/L).The discontinuity at VGS=VDSis not significant.4.2?Gate?Oxide and Base Thickness DependencesThe VGSand VDSvariations of the variables justshown in the six figures,Figs.3 8,at x0=1?5nm andxB=30nm,are given in six more figures in Figs.9 14,for two base thicknesses,xB=3nm and 300nm and twooxide thickness,xO=2nm and 1nm.Fig?8?The output?conductance?drain?voltage characteristics at twoVGS=1?0V and 1?5V.(a)The total output conductance gdand theelectron or hole drift and diffusion components;(b)The fractionalelectron or hole output conductance from the parabolic drift currentP1,linear diffusion current D1,and the(/y)(E2X)current nor?malized to the total electron or hole output conductance gdNor gdP.(c)The derivative of emitter electrical length(/UDS)(y0/L).The discontinuity at VGS=VDSis apparent.Fig?9?The DC output current?voltage characteristics at two basethicknesses,xB=300nm and 3nm.(a)The total drain current IDand electron or hole drift and diffusion channel current compo?nents;(b)The fractional parabolic electron or hole parabolic driftcurrent P1,linear diffusion current D1,and the(/y)(E2X)cur?rent,normalized to the total electron or hole channel currents,IDNor IDP.(c)The electron emitter electrical length y0and the hole e?mitter electrical length(L-y0).4.3?Deviationsof the LongChannel Solution fromtheShortChannel Solution?The deviations of the three components of drain cur?rent,transconductance and output conductance of theone?section long?channel BiFET theory 13from the two?section short?channel BiFET theory are given in Figs.15and 16.In the one?section long?channel BiFET theory,Fig?10?Base thickness xBdependence of the transconductance?drain?voltage characteristics.(a)The total transconductance gmandelectron and hole drift and diffusion components;(b)The fractionaltransconductance from the parabolic drift current P1,linear diffu?sion current D1,and the(/y)(E2X)current,normalized to thetotal electron or hole transconductances gmNor gmP.(c)The deriv?ative of emitter electrical length(/UGS)(y0/L).The discontinu?ity at VGS=VDSis not significant.197半?导?体?学?报第 29 卷Fig?11?Base thickness xBdependence of the output?conductance?drain?voltage characteristics.(a)The total output conductance gdand electron and hole drift and diffusion components;(b)The frac?tional output conductance from the electron or hole parabolic driftcurrent P1,linear diffusion current D1,and the(/y)(E2X)cur?rent,normalized to the total electron or hole output conductancesgdNor gdP.(c)The derivative of emitter length(/UDS)(y0/L).The discontinuity at VGS=VDSis apparent.the channel section lengths are taken as the physical baselength L,not taking account of the shorten electricallengths of both electron and hole emitters.5?SummaryThe electrically short channel is a universal intrinsicproperty of all field?effect transistors.Ourprevious?month paper 14reported its effects using the electrochem?ical potential representation in which the drift andFig?12?Oxide thickness xOdependence of the DC output current?voltage characteristics.(a)The total drain current IDand the elec?tron or hole drift and diffusion channel current components;(b)The fractional electron or hole parabolic drift current P1,lineardiffusion current D1,and the(/y)(E2X)current,normalized tothe total electron or hole current,IDNor IDP.(c)The electronemitter length y0and the hole emitter length(L-y0).Fig?13?Oxide thickness xOdependence of the transconductance?drain?voltage characteristics.(a)The total transconductance gmandelectron and hole drift and diffusion components;(b)The fractionalelectron and hole transconductance from the parabolic drift currentP1,linear diffusion current D1,and the(/y)(E2X)current nor?malized to the electron or hole total transconductances gmNor gmP.(c)The derivative of emitter length(/UGS)(y0/L).The discon?tinuity at VGS=VDSis not significant.diffusion currents are combined into oneterm which isproportional to the electrochemical potential gradient.This paper reportsthe electrically short channel effectsu?sing the more popular and traditional representation inwhich the drift and the currents are each represented byits own term,the drift isproportional to the electric fieldand the diffusion is proportional to the concentration gra?dient.The electrically short channel increases the currentsFig?14?Oxide thickness xOdependence of the output?conductance?drain?voltage characteristics.(a)The total output conductance gdand electron or hole drift and diffusion components;(b)The frac?tional electron or hole output conductance from the parabolic driftcurrent P1,linear diffusion current D1,and the(/y)(E2X)cur?rent,normalized to the total electron or hole output conductancesgdNor gdP.(c)The derivative of emitter length(/UDS)(y0/L).The discontinuity at VGS=VDSis apparent.198第 2 期Jie and Sah:?The Bipolar Field?Effect Transistor:IV.Short Channel Drift?Diffusion Current%Fig?15?Deviations of the BiFET long channel solution(Xlong)from the BiFET short channel solution(Xshort)versusthe gate volt?age.(a)The three components(parabolic drift current P1,lineardiffusion current D1,and the drift(/y)E2Xcurrent)of the elec?tron channel current IDNand the hole channel current IDP;(b)Thethree components of t
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