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1 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) 0 for (v GS V T )/α 0 v DS (cutoff) i D K(v GS V T ) 2 /2 for 0 (v GS V T )/α v DS (saturation) K(v GS V T αv DS /2)αv DS for 0 v DS (v GS V T )/α (linear) with K (W/αL)µ e C ox, V T = V FB 2φ p-si + [2ε Si qn A ( 2φ p-si v BS )] 1/2 /C ox and α = 1 + [(ε Si qn A /2( 2φ p-si v BS )] 1/2 /C ox (frequently α 1) The factor α: 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 modeling i D,sub-threshold (φ(0)), then i D,s-t (v GS, v DS ) [with v BS = 0] Stepping back and looking at the equations Clif Fonstad, 10/22/09 Lecture 12 - Slide 1

2 Final comments on α The Gradual Channel result ignoring α and valid for v BS " 0, and v DS # 0 is: i G (v GS,v DS,v BS ) = 0, i B (v GS,v DS,v BS ) = 0, and i D (v GS,v DS,v BS ) = K 2 v GS "V T (v BS ) [ ] < 0 <v DS 0 for v GS "V T (v BS ) [ ] 2 for 0 < [ v GS "V T (v BS )] <v DS # K v GS "V T (v BS ) " v DS $ % 2 & ' ( v DS for 0 <v DS < v GS "V T (v BS ) [ ] with K ) W L µ C * e ox and C * ox ) * ox t ox We noted last lecture that these simple expressions without α are easy to remember, and refining them to include α involves easy to remember substitutions: v DS " # v DS L " # L K "K # What we haven't done yet is to look at α itself, and ask what it means. What is it physically? 1/x DT (V BS ) G " #1+ 1 * C ox $ Si qn A [ ] = 2 2% p&si & v BS =1+ " Si x DT =1+ " Si t ox " ox t ox " ox C ox + $ Si qn A 2$ Si [ 2% p&si & v BS ] * C ox =1+ C * DT = C * DT * * ε ox ε Si B ε ox /t ox ε Si /x DT x DT C ox C Clif Fonstad, 10/22/09 GB Lecture 12 - Slide 2 Look back at Lec. 10.

3 Foil 7 from Lecture 10 MOS Capacitors: the gate charge as v GB is varied q " G = # qn % Si A 1+ 2C " 2 ox v GB $V ( ( FB ) $1 " C ' ox & # Si qn * A ) q " " G = C ox qn AP X DT V FB q G * [coul/cm 2 ] q " " G = C ox ( ) v GB #V T Inversion + qn AP X DT Layer Charge V T Depletion Region Charge v GB [V] ( ) Accumulation v GB #V FB Layer Charge The charge expressions: C " ox # $ ox, " t C ox ( v GB #V FB ) for v GB $ V ox FB. % q " G (v GB ) = Si qn A 1+ 2C ". & 2 ox ( v GB #V ) FB ) - #1 " C ( ox ' % Si qn + for V FB $ v GB $ V T. A *. " / C ox ( v GB #V T ) + qn A X DT for V T $ v GB Clif Fonstad, 10/22/09 Lecture 12 - Slide 3

4 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, J h and J e are zero in steady state so the carrier populations are in equilibrium with the potential barriers, φ(x), as they are in thermal equilibrium, and we have: n(x) = n i e q"(x ) kt #q"(x) kt and p(x) = n i e Once again this means we can find φ(x), and then n(x) and p(x), by solving Poisson's equation: d 2 "(x) dx 2 = # q [ $ n i e #q"(x )/ kt # e q"(x )/ kt ( ) + N d (x) # N a (x)] This version is only valid, however, when φ(x) -φ p. When φ(x) > -φ p we have accumulation and inversion layers, and we assume them to be infinitely thin sheets of charge, i.e. we model them as delta functions. Clif Fonstad, 10/22/09 Lecture 12 - Slide 4

5 Foil 9 from Lecture 10 Poisson's Equation calculation of gate charge Calculation compared with depletion approximation model for t ox = 3 nm and N A = cm -3 : t ox,eff 3.2 nm t ox,eff 3.3 nm We'll look in this vicinity today. We've ignored sub-threshold charge in our MOSFET i-v modelling thus far. Clif Fonstad, 10/22/09 Lecture 12 - Slide 5 Plot courtesy of Prof. Antoniadis

6 Foil 10 from Lecture 10 MOS Capacitors: Sub-threshold charge Assessing 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: q N(inversion) v GB * ( ) = "C ox ( v GB "V T ) But how good an approximation is this? To see, we calculate the electron charge below threshold (weak inversion): q N(sub"threshold ) v GB ( ) = " q n i e q#(x)/ kt dx x d 0 $ ( ) φ(x) is a non-linear function of x, making the integral difficult, v GB ( ) 2 "(x) = " p + qn A x - x d 2# Si but if we use a linear approximation for φ(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

7 Foil 11 from Lecture 10 Sub-threshold electron charge, cont. We begin by saying "(x) # "(0) + ax where a $ d"(x) where dx x= 0 = % 2qN A ["(0) % " p ] & Si With this linear approximation to φ(x) we can do the integral and find q N(sub"threshold ) ( v GB ) # q kt q n(0) a = "q kt q $ Si 2qN A [%(0) " % p ] n i eq%(0) kt To proceed it is easiest to evaluate this expression for various values of φ(0) below threshold (when its value is -φ p ), and to also find the corresponding value of v GB, from [ ] v GB "V FB = #(0) " # p + t ox 2$ Si qn A #(0) " # p $ ox This 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

8 Foil 12 from Lecture 10 Sub-threshold electron charge, cont. 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

9 MOSFETs: Conventional strong inversion operation, V GS > V T v GS > V T G S + v DS > 0 D i D n+ n+ p-si v BS + B High concentration of electrons in a strong inversion layer drifting to the drain because of field due to v DS. 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. Clif Fonstad, 10/22/09 Lecture 12 - Slide 9

10 MOSFETs: Sub-threshold operation, V GS < ~ V T v GS < V T G S ~ + v DS > 0 D i D n+ n+ p-si A small number of electrons surmount the barrier and diffuse to drain. v BS+ B The electrons diffuse and do not feel v DS until they get to the edge of the depletion region. No surface channel; diffusion flux from source to drain when v DS > 0 For any v GB > V FB some electrons in the source can surmount the barrier and diffuse to the drain. Though always small, this flux can become consequential as v GS approaches V T. Clif Fonstad, 10/22/09 Lecture 12 - Slide 10

11 MOSFETs: Sub-threshold operation, V GS < ~ V T What do we mean by "consequential"? When is this current big enough to matter? There are at least three places where it matters: 1. It can limit the gain of a MOSFET linear amplifier. In Lecture 21 we will learn that we achieve maximum gain from MOSFETs operating in strong inversion when we bias as close to threshold as possible. This current limits how close we can get. 2. It is a major source of power dissipation and heating in modern VLSI digital ICs. When you have millions of MOSFETs on an IC chip, even a little bit of current through the half that are supposed to be "off" can add up to a lot of power dissipation. We'll see this in Lecture It can be used to make very low voltage, ultra-low power integrated circuits. In Lecture 25 we'll talk about MIT/TI research on sub-threshold circuits with 0.3 V supplies and using µw's of power. Clif Fonstad, 10/22/09 Lecture 12 - Slide 11

12 Sub-threshold Operation of MOSFETs: finding i D Begin by considering the device illustrated below: S G D -t ox 0 t n+ n+ n+ p -t ox 0 x 0 B L y - Set v GS = V FB, and v DS = v BS = 0. - The potential profile vs. y, φ(y) at any x between 0 and t n+ is then: S v GS = V FB G v DS = 0 n+ n+ p t n+ vbs = 0 B x 0 L D y φ p φ(y) φ n+ 0 L y Clif Fonstad, 10/22/09 Lecture 12 - Slide 12

13 Sub-threshold Operation of MOSFETs, cont. - Now consider φ(y) when v GS = V FB, v BS = 0. and v DS > 0: -t ox 0 S v GS = V FB G v DS > 0 n+ n+ p D φ(y) φ n+ φ n+ + v DS v DS t n+ x v BS = 0 B 0 L y φ p 0 L y - So far this is standard MOSFET operating procedure. We could apply a positive voltage to the gate and when it was larger than V T we would see the normal drain current that we modeled earlier. Rather than do this, however, consider forward biasing the substrate-source diode junction, I.e, v BS > 0 Clif Fonstad, 10/22/09 Lecture 12 - Slide 13

14 Sub-threshold Operation of MOSFETs, cont. - Apply v BS > 0, keep the same v DS > 0, and adjust v GS such that the potential at the oxide-si interface, φ(0,y), equals φ p + v BS. - Now consider φ(x,y): v GS s.t. φ(0,y) = φ p + v BS G v DS > 0 S -t ox D φ(y) φ n+ + v DS 0 φ n+ Electron Injection n+ p n+ and t n+ x Diffusion v BS > 0 B 0 L - With this biasing the structure is being operated as a lateral BJT! The drain/collector current is: i D /C " W t n + qn i 2 y 0 L φ p + v BS φ p p',n' v BS n'(0 + ) = n po (e qvbs/kt -1) D e e qv BS / ( kt #1) N Ap L eff 0 L y y Clif Fonstad, 10/22/09 Lecture 12 - Slide 14 - This is not sub-threshold operation yet.

15 Sub-threshold Operation of MOSFETs, cont. - Now again make v BS = 0, but keep the same v DS and v GS so that the potential at the oxide-si interface, φ(0,y), is still > φ p. - Now φ(x,y) is different for 0 < x < x D, φ(0,y) and x D < x < t n+ : -t ox 0 x D v GS s.t. φ(0,y) > φ p G v DS > 0 S Injection n+ n+ p D -φ p φ(x) φ n+ φ n+ + v DS 0 φ(x)-φ p L y t n+ vbs = 0 B φ p x 0 L - Now there is lateral BJT action only along the interface. - The drain current that flows in this case is the sub-threshold drain current. y φ(y) φ n+ + v DS φ p φ n+ 0 L y Clif Fonstad, 10/22/09 Lecture 12 - Slide 15 - This is sub-threshold operation!

16 Sub-threshold Operation of MOSFETs, cont. -t ox 0 x D t n+ - The barrier at the n + -p junction is lowered near the oxide-si interface for any v GS > V FB. - The barrier is lowered by φ(x) - φ p for 0 < x < x D. (This is the effective v BE on the lateral BJT between x and x + dx.) Plot vs y at fixed x, V FB < v GS < V φ(0,y) T 0 < x < x G v S DS > 0 D. D φ n+ + v DS x - Injection n+ n+ p v BS = 0 B 0 L The barrier lowering (effective forward bias) (1) is controlled by v GS, and (2) decreases quickly with x. vbe,eff (x) y v GS = [φ(x) φ p ] - φ p φ(0) Clif Fonstad, 10/22/09 Injection occurs over this range. Lecture 12 - Slide 16 -t ox φ φ(x) p -φ p φ(x) 0 φ p φ n+ x d φ(x)-φ p L Plot vs x at fixed y, 0 < y < L. y x

17 Sub-threshold Operation of MOSFETs, cont. - To calculate i D, we first find the current in each dx thick slab: S V FB < v GS < V T G v DS > 0 D -t ox 0 x x+dx n+ n+ p x D v BS = 0 v DS > 0 x n' x,0 0 L ( ) $ n i e q"(x,v GS )/ kt ( ) = n i e q"(x,v GS )/ kt #1 n'(x,0) " n'(x,l) di D (x) = qd e L Clif Fonstad, 10/22/09 y n' ( x,l) " n i e q#(x,v GD )/ kt W dx # W L D qn e i eq$(x,v GS )/ kt 1" e "qv DS )/ kt ( )dx Lecture 12 - Slide 17

18 Sub-threshold Operation of MOSFETs, cont. - Then we add up all the contributions to get i D : i D = W $ L D e& %& 0 # x d ' q n i e q"(x,v GS )/ kt dx) () 1* e*qv DS / kt ( ) - This is what we called q N(sub-threshold) in Lecture 9 and today on Foil 7. Substituting the expression we found for this (see Foil 7), we have: i D(sub"threshold ) = W L D e % ' q kt &' q ( # Si 2qN A [ $(0,v GS ) " $ p ] n ie q $(0,vGS ) kt * )* 1" e"qv DS / kt ( ) - Using the Einstein relation and replacing n i with N A e qφp/kt, we obtain: i D(sub"th ) = W L µ # * kt & e C ox % ( $ q ' 2 1 * 2C ox 2q) Si N A *(0,vGS )" "* [*(0,v GS ) " * p ] eq p { [ ]} kt 1" e "qv DS / kt ( ) - To finish (we are almost done) we need to replace φ(0,v GS ) with v GS since we want the drain current's dependence on the terminal voltage. Clif Fonstad, 10/22/09 Lecture 12 - Slide 18

19 Sub-threshold Operation of MOSFETs, cont. - The relationship relating φ(0,v GS ) and v GS is: [ ] + 1 v GS = V FB + "(0) # " p C ox [ ] * 2$ Si qn A "(0) # " p - From this we can relate a change in v GS to a change in φ(0), which is what we really need. To first order the two are linearly related: "v GS # dv ' ) + GS 1 2% "$(0) = 1+ Si qn A ) (, "$(0). n "$(0) * d$(0) *) 2C ox [ $(0) & $ p ] -) " n - In the current equation we have the quantity {φ(0,v GS ) - [-φ p ]}. - φ p is simply φ(0,v T ), the potential at x = 0 when the gate voltage is V T, so {"(0,v GS ) # [#" p ]} = "(0,v GS ) # "(0,V T ) i D(sub"threshold ) # W L µ C $ * kt ' e ox & ) % q ( { } = { v GS #V T } n - Using this and the definition for n, we arrive at: 2 ( ) ( n "1) e q { v GS "V T } n kt 1" e "qv DS / kt Clif Fonstad, 10/22/09 Lecture 12 - Slide 19

20 Sub-threshold Operation of MOSFETs, cont. - To fully complete our modeling, we must add two more points: 1. The dependences on v BS and v DS : v BS : The threshold voltage depends on v BS. φ(0,v T ) does also, i.e. φ(0,v T ) = - φ p -v BS, and so do the junction barriers. Taking this all into account we find that the only change we need to make is to acknowledge that n and V T both depend on v BS. v DS : The drain to source voltage introduced a factor (1 - e -qvds/kt ) 1. This is discussed in the handout posted on Stellar. The complete expression for i D is: i D,s"t (v GS,v DS,v BS ) # W L µ $ * kt ' e C ox & ) % q ( 2 ( ) [ n(v BS ) "1] e q { v GS "V T (v BS )} n kt 1" e "qv DS / kt 2. The factor n: The value of n depends on φ(0,v GS ). Notice, however, that the subthreshold current is largest as φ(0,v GS ) approaches -φ p -v BS, so it makes sense to evaluate it there and take that as its value for all v GS : & ( n " ' 1+ 1 )( * 2C ox * 2# Si qn A ( & + [ $(0) % $ p ],( - ( ' 1+ 1 * )( C ox # Si qn A [ ] 2 %2$ p % v BS Clif Fonstad, 10/22/09 Lecture 12 - Slide 20 ** Notice that this is exactly the same expression as that for α! * ( +,(

21 Sub-threshold Operation of MOSFETs, cont. - Comparing current levels above and below threshold: The ranges of the two models do not overlap, but is it still interesting to compare the largest possible value of the subthreshold drain current model (v GS - V T = 0 V),* with the strong inversion model at v GS - V T = 0.06 V, 0.1 V, and 0.2 V: i D(sub"threshold ) K i D(strong inversion ) K # $ & % kt q ' ) ( 2 ( n "1) e q v GS "V T { } n kt v BS = 0 (0.025) = 1.56 x 10-4 V 2 ( ) 2 " 1 2# v $V GS T (0.06) 2 = 1.5 x 10-3 V (0.1) 2 = 4 x 10-3 V 2 (0.2) 2 = 1.6 x 10-2 V 2 We see that the current in strong inversion drift current quickly becomes much larger, although only grows quadratically. * This is pushing the model, particularly with regard to the Clif Fonstad, 10/22/09 diffusion current model, beyond it's range of strict validity, and is probably somewhat of an over-estimate. Lecture 12 - Slide 21

22 Sub-threshold Operation of MOSFETs, cont. - Plotting our models for the earlier device: N A = cm -3, t ox = 3 nm: v BS = 0 Clif Fonstad, 10/22/09 Lecture 12 - Slide 22

23 Sub-threshold Operation of MOSFETs, cont. - Zooming into a lower current scale: N A = cm -3, t ox = 3 nm: v BS = 0 Clif Fonstad, 10/22/09 Lecture 12 - Slide 23

24 Sub-threshold Operation of MOSFETs, cont. - Repeating the plot with a log current scale: N A = cm -3, t ox = 3 nm: Slope = 60 x n mv/decade* v BS = 0 Clif Fonstad, 10/22/09 Lecture 12 - Slide 24 * n = 1.25 here so 75 mv/decade

25 Sub-threshold Output Characteristic - We plot a family of i D vs v DS curves with (v GS - V T ) as the family variable, after first defining the sub-threshold diode saturation I S,s"t # W 2 current, I S,s-t : L µ $ * kt ' e C ox & ) [ n "1] = K o V 2 t [ n "1] % q ( 10-1 I S,s-t log i D,s-t Note : V t " kt q, K o " W L µ * e C ox (v -V GS T ) = -0.06xn Volts 10-2 I S,s-t 10-3 I S,s-t (v GS -V T ) = -0.12xn Volts (v GS -V T ) = -0.18xn Volts (v GS -V T ) = -0.24xn Volts 10-4 I S,s-t v DS i D,s"t (v GS,v DS ) # I S,s"t e q { v GS "V T } n kt 1" e "qv DS ( / kt ) v BS = 0 Note: The device we modeled had n = 1.25, so it Clif Fonstad, 10/22/09 follows a "75 mv rule" [i.e. 60 x n = 75]. Lecture 12 - Slide 25

26 Sub-threshold Output Characteristic, cont. - To compare this with something we've already seen, consider the BJT and plot a family of i C vs v CE curves with v BE as the family variable log i C α F I ES v BE = 0.66 Volts α F I ES α F I ES α F I ES v BE = 0.60 Volts v BE = 0.54 Volts v BE = 0.48 Volts i C (v BE,v CE ) " # F I ES e qv BE kt 1$ e $qv CE / kt ( ) - The two biggest differences are (1) the magnitudes of the I S 's, and (2) the factor of "n" in the MOSFET case. The totality of v BE reduces the barrier, whereas only a fraction 1/n of v GS does. - A third difference is that a BJT has a base current.* Clif Fonstad, 10/22/09 Lecture 12 - Slide 26 * This is the price paid for having n = 1 in a BJT. v CE

27 Large Signal Model for MOSFET Operating Sub-threshold - The large signal model for a MOSFET operating in the weak inversion or sub-threshold region looks the same model as that for a device operating in strong inversion (v GS > V T ) EXCEPT there is a different equation relating i D to v GS, v DS, and v BS : We will limit our model to v GS " V T, v DS > 3kT /q and v BS = 0. D i D i G,s"t (v GS,v DS,v BS ) = 0 G i G (= 0) S,B ( ) i D,s"t (v GS,v DS,0) # I S,s"t ( 1" $v DS )e q { v GS "V To } n kt 1" e "qv DS / kt Early effect i D (v GS, v DS ) 1 for v DS > 3 kt/q Clif Fonstad, 10/22/09 Lecture 12 - Slide 27

28 Microelectronic Devices and Circuits Lecture 12 - Sub-threshold MOSFET Operation - Summary Sub-threshold operation - qualitative explanation Look back at Lecture 10 (Sub-threshold electron charge) BJT action in depletion/weak inversion layer along oxide the interface MOSFET gate-controlled lateral BJT Important in/for 1. power dissipation in normally-off logic gates 2. limiting the gain of strong inversion linear amplifiers 3. realizing ultra-low power, very low voltage electronics Quantitative sub-threshold modeling This gives us a precise description of the voltage dependence It also gives us the information on I S,s-t and n we need for device design i D,s"t (v GS,v DS,v BS ) # I S,s"t e q { v GS "V T (v BS )} n kt 1" e "qv DS / kt with: I S,s"t # W L µ e C ox $ * kt ' & ) 2 [ n "1] and & ( n " ' 1+ 1 ( ) # Si qn A [ ] * % q ( )( C ox 2 $2% p $ v BS,( Clif Fonstad, 10/22/09 Lecture 12 - Slide 28 * ( + = -

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