6.012 Microelectronic Devices and Circuits Formula Sheet for Exam One, Fall 2009 Parameter Values: Periodic Table: ! q = 1.6x10"19Coul#o= 8.854 x10"14 F/cm#r,Si= 11.7, #Si$ 10"12 F/[email protected][ ]$ 1010cm"3kT /q $ 0.025 V; kT /q( )ln10 $ 0.06 V1µm = 1x10"4cm! III IV V B C N Al Si P Ga Ge As In Sn SbDrift/Diffusion: Electrostatics: ! Drift velocity : s x= ±µmEx Conductivity : "= q µen + µhp( ) Diffusion flux : Fm= #Dm$Cm$xEinstein relation : Dmµm=kTq! "dE (x)dx=#(x) E (x) =1"#(x)dx$%d&(x)dx= E(x)&(x) = % E (x)dx$%"d2&(x)dx2=#(x)&(x) = %1"#(x)dxdx$$The Five Basic Equations: ! Electron continuity : "n(x,t)"t#1q"Je(x,t)"x= gL(x,t) # n(x,t) $ p(x,t) # ni2[ ]r(T) Hole continuity : "p(x,t)"t+1q"Jh(x,t)"x= gL(x,t) # n(x,t) $ p(x,t) # ni2[ ]r(T)Electron current density : Je(x,t) = qµen(x,t)E (x,t) + qDe"n(x,t)"x Hole current density : Jh(x,t) = qµhp(x,t)E(x,t) # qDh"p(x,t)"x Poisson's equation : "E(x,t)"x=q%p(x,t) # n(x,t) + Nd+(x) # Na#(x)[ ]Uniform doping, full ionization, TE ! n - type, Nd>> Na no" Nd# Na$ ND, po= ni2no, %n=kTqlnNDnip - type, Na>> Nd po" Na# Nd$ NA, no= ni2po, %p= #kTqlnNAniUniform optical excitation, uniform doping ! n = no+ n' p = po+ p' n' = p'dn'dt= gl(t) " po+ no+ n'( )n' rLow level injection, n',p' << po+no: dn'dt+n'#min= gl(t) with #min$ por( )"1Flow problems (uniformly doped quasineutral regions with quasi-static excitation and low level injection; p-type example): Short base, infinite lifetime limit: ! Minority carrier excess : d2n'(x)dx2"n'(x)Le2= "1DegL(x) Le# De$eMinority carrier current density : Je(x) % qDedn'(t)dxMajority carrier current density : Jh(x) = JTot" Je(x) Electric field : Ex(x) %1qµhpoJh(x) +DhDeJe(x)& ' ( ) * + Majority carrier excess : p'(x) % n'(x) +,qdEx(x)dx! Minority carrier excess : d2n'(x)dx2" #1DegL(x) $ n'(x) " #1DegL(x)dxdx%%Non-uniformly doped semiconductor sample in thermal equilibrium ! d2"(x)dx2=q#nieq"(x ) kT$ e$q"(x ) kT[ ]$ Nd(x) $ Na(x)[ ]{ }no(x) = nieq"(x ) kT, po(x) = nie$q"(x ) kT, po(x)no(x) = ni2Depletion approximation for abrupt p-n junction: Ideal p-n junction diode i-v relation: ! "(x) =0#qNApqNDn0 for for for forx < #xp#xp< x < 00 < x < xnxn< x$ % & & ' & & NApxp= NDnxn(b)(n#(p=kTqlnNDnNApni2w(vAB) =2*Si(b# vAB( )qNAp+ NDn( )NApNDn Epk=2q(b# vAB( )*SiNApNDnNAp+ NDn( ) qDP(vAB) = #AqNApxpvAB( )= #A 2q*Si(b# vAB( )NApNDnNAp+ NDn( )! n(-xp) =ni2NApeqvAB/ kT, n'(-xp) =ni2NApeqvAB/ kT"1( ); p(xn) =ni2NDneqvAB/ kT, p'(xn) =ni2NDneqvAB/ kT"1( ) iD= Aq ni2DhNDnwn,eff+DeNApwp,eff# $ % & ' ( eqvAB/ kT-1[ ] wm,eff= wm" xm if Lm>> wmLmtanh wm" xm( )Lm[ ] if Lm~ wm Lm if Lm<< wm) * + , + qQNR,p -side= Aq n'(x)dx, -wp-xp-qQNR,n -side= Aq p'(x)dx, Note : p'(x) . n'(x) in QNRs xnwn-Small Signal Linear Equivalent Circuit for a p-n Diode (n+-p doping assumed for Cd) ! gd"#iD#vABQ=qkTISeqVAB/ kT$q IDkT, Cd= Cdp+ Cdf, where Cdp(VAB) = Aq%SiNAp2&b' VAB( ), and Cdf(VAB) =q IDkTwp' xp[ ]22De= gd(d with (d"wp' xp[ ]22DeLarge signal BJT Model in Forward Active Region (FAR): (npn with base width modulation) ! iBvBE,vCE( )= IBSeqvBE/ kT"1( ) iCvBE,vBC( )=#FiBvBE,vCE( )1+$vCE[ ]=#FIBSeqvBE/ kT"1( )1+$vCE[ ]with : IBS%IES#F+ 1( )=Aqni2#F+ 1( )DhNDEwE,eff+DeNABwB,eff& ' ( ( ) * + + , #F%,F1",F( ), and $%1VAAlso, ,F=1"-B( )1+-E( ) and #F.1"-B( )-E+-B( ) with -E=DhDe/NABNDE/wB,effwE,eff and -B=wB,eff22 LeB2When -B. 0 then ,F.11+-E( ) and #F.1-EMIT OpenCourseWarehttp://ocw.mit.edu 6.012 Microelectronic Devices and Circuits Fall 2009 For information about citing these materials or our Terms of Use, visit:
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