Sub-threshold MOSFET Operation - MIT OpenCourseWare

1 0 iD K(vGS VT with K (W/ L) - Microelectronic Devices and Circuits Lecture 12 - Sub-threshold MOSFET Operation - Outline Announcement Hour exam two: in 2 weeks, Thursday, Nov. 5, 7:30-9:30 pm ALSO: sign up for an iLab account!! Review MOSFET model: gradual channel approximation (Example: n-MOS) for (vGS VT)/ 0 vDS (cutoff) K(vGS VT)2 /2 for 0 (vGS VT)/ vDS (saturation) vDS/2) vDS for 0 vDS (vGS VT)/ (linear) **Cox , VT= VFB 2 p-Si + [2 Si qNA(|2 p-Si| vBS)]1/2/Cox and = 1 + [( Si qNA/2(|2 p-Si| vBS)]1/2 /Cox (frequently 1) The factor.)

2 What it means physically Sub-threshold Operation - qualitative explanation Looking back at Lecture 10 ( Sub-threshold electron charge) Operating an n-channel MOSFET as a lateral npn BJT The Sub-threshold MOSFET gate-controlled lateral BJT Why we care and need to quantify these observations Quantitative Sub-threshold modelingiD, Sub-threshold ( (0)), then iD,s-t(vGS, vDS) [with vBS = 0] Stepping back and looking at the equations Clif Fonstad, 10/22/09 Lecture 12 - Slide 1 Final comments on The Gradual Channel result ignoring and valid for is: !

3 IG(vGS,vDS,vBS)=0, iB(vGS,vDS,vBS)=0, andiD(vGS,vDS,vBS)=0forvGS"VT(vBS)[]<0<v DSK2vGS"VT(vBS)[]2for 0<vGS"VT(vBS)[]<vDSKvGS"VT(vBS)"vDS2# $ % & ' ( vDSfor 0<vDS<vGS"VT(vBS)[] with K)WL eCox* and Cox*)*oxtox! vBS"0, and vDS#0We noted last lecture that these simple expressions without are easy to remember, and refining them to include involves easy to remember substitutions: ! vDS"#vDSL"#LK"K#What we haven't done yet is to look at itself, and ask what it means. What is it physically? ! "#1+1 Cox*$SiqNA22%p&Si&vBS[]=Cox*+$SiqNA2$Si2 %p&Si&vBS[]Cox*1/xDT(VBS) G ox ox/tox Si !

4 =1+"SixDT"oxtox=1+"Si"oxtoxxDT=1+CDT*Cox *=CDT*CGB* Si/xDT B Clif Fonstad, 10/22/09 Lecture 12 - Slide 2 Look back at Lec. 10. Foil 7 from Lecture 10 MOS Capacitors: the gate charge as vGB is varied Clif Fonstad, 10/22/09 Lecture 12 - Slide 3 vGB [V] VT VFB qG* [coul/cm2] qNAPXDT ! qG"=Cox"vGB#VT() +qNAPXDTI nversion LayerCharge ! qG"(vGB)=Cox"vGB#VFB() for vGB$VFB%SiqNACox"1+2 Cox"2vGB#VFB()%SiqNA#1& ' ( ( ) * + + for VFB$vGB$VTCox"vGB#VT()+qNAXDT for VT$vGB, - .. / .. The charge expressions: ! qG"=#SiqNACox"1+2 Cox"2vGB$VFB()#SiqNA$1% & ' ' ( ) * * DepletionRegionCharge !)

5 QG"=Cox"vGB#VFB()Accumulation Layer Charge ! Cox"#$oxtox Foil 8 from Lecture 10 MOS Capacitors: How good is all this modeling? How can we know? Poisson's Equation in MOS As we argued when starting, Jh and Je are zero in steady state so the carrier populations are in equilibrium with the potential barriers, (x), as they are in thermal equilibrium, and we have: ! n(x)=nieq"(x)kTandp(x)=nie#q"(x)kTOnce again this means we can find (x), and then n(x) and p(x), by solving Poisson's equation: !

6 D2"(x)dx2=#q$nie#q"(x)/kT#eq"(x)/kT()+Nd (x)#Na(x)[]This version is only valid, however, when | (x)| - p. When | (x)| > - p we have accumulation and inversion layers, and we assume them to be infinitely thin sheets of charge, we model them as delta functions. Clif Fonstad, 10/22/09 Lecture 12 - Slide 4 Foil 9 from Lecture 10 Poisson's Equation calculation of gate charge Calculation compared with depletion approximation model for tox = 3 nm and NA = 1018 cm-3: Clif Fonstad, 10/22/09 Lecture 12 - Slide 5 tox,eff nm tox,eff nm We'll look in this vicinity today.

7 We've ignored Sub-threshold charge in our MOSFET i-v modelling thus far. Plot courtesy of Prof. Antoniadis Foil 10 from Lecture 10 MOS Capacitors: Sub-threshold chargeAssessing how much we are neglecting Sheet density of electrons below threshold in weak inversion In the depletion approximation for the MOS we say that the charge due to the electrons is negligible before we reach threshold and the strong inversion layer builds up: !

8 QN(inversion)vGB()="Cox*vGB"VT()But how good an approximation is this? To see, we calculate the electron charge below threshold (weak inversion): ! qN(sub"threshold)vGB()="qnieq#(x)/kTdxxd vGB()0$ (x) is a non-linear function of x, making the integral difficult, ! "(x)="p+qNA2#Six-xd()2but if we use a linear approximation for (x) near x = 0, where the term in the integral is largest, we can get a very good approximate analytical expression for the integral. Clif Fonstad, 10/22/09 Lecture 12 - Slide 6 Foil 11 from Lecture 10 Sub-threshold electron charge, cont.

9 We begin by saying ! "(x)#"(0)+ax where a$d"(x)dxx=0=%2qNA"(0)%"p[]&Siwhere With this linear approximation to (x) we can do the integral and find ! qN(sub"threshold)vGB()#qkTqn(0)a="qkTq$S i2qNA%(0)"%p[]nieq%(0)kTTo proceed it is easiest to evaluate this expression for various values of (0) below threshold (when its value is - p), and to also find the corresponding value of vGB, from ! vGB"VFB=#(0)"#p+tox$ox2$SiqNA#(0)"#p[]Th is has been done and is plotted along with the strong inversion layer charge above threshold on the following foil. Clif Fonstad, 10/22/09 Lecture 12 - Slide 7 Foil 12 from Lecture 10 Sub-threshold electron charge, cont.

10 6 mV Neglecting this charge in the electrostatics calculation resulted in only a 6 mV error in our estimate of the threshold voltage value. Today we will look at its impact on the Sub-threshold drain current. Clif Fonstad, 10/22/09 Lecture 12 - Slide 8 MOSFETs: Conventional strong inversion Operation ,VGS > VT p-SiBG+vGS > VTn+Dn+S vDS > 0vBS+iDHigh concentration of electrons in a strong inversion layer drifting to the drain because of field due to vDS. n-type surface channel; drift flux from source to drain In our gradual channel approximation modeling we have assume a high conductivity n-type channel has been induced under the gate.