Quantitative MOSFET. Step 1. Connect the MOS capacitor results for the electron charge in the inversion layer Q N to the drain current.


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1 Quantitative MOSFET Step 1. Connect the MOS capacitor results for the electron charge in the inversion layer Q N to the drain current. V DS _ n source polysilicon gate y = y * 0 x metal interconnect to gate n polysilicon gate y ptype metal interconnect to bulk (a) W y* t ox E y (y * ) n drain gate oxide x channel electrons at position y = y * drifting with velocity v y (y * ) from source to drain x (b)
2 Drift Current Equation Drift current for electrons in the channel: J y ( x y) = Ðqnx ( y)v y () y The drain current at position y is the integral of the drift current density across the cross section. Since the conventional direction of is opposite to the direction of the y axis we insert a minus sign: x = ÐW J y ( x y)dx = 0 x Wv y () y qn( x y)dx 0 The integral is the negative of the electron charge in the channel per unit area at point y. The symbol for this quantity is  Q N (y): = ÐWv y ()Q y N () y Note that isnõt a function of the position in the channel
3 MOSFET DC Model: a First Pass metal interconnect to gate V GS _ n polysilicon gate Start simple  small V DS makes the channel uniform V DS (< 0.1 V) n source 0 x y Q N ptype metal interconnect to bulk y = L n drain Channel charge: MOS capacitor in inversion with V GB =. Q N = ÐC ox ( V GB Ð V Tn ) = ÐC ox ( Ð V Tn ) Drift velocity: electric Þeld is just E y =  V DS / L so v y =  µ n (V DS / L) Drain current equation for V DS ÒsmallÓ... say less than 0.1 V. W = µ n C ox  ( VGS Ð V L Tn )V DS Note that is proportional to V DS with channel resistance under gate control. This voltage controlled resistor region is sometimes useful.
4 Triode Region metal interconnect to gate n polysilicon gate Increase V DS  channel charge becomes a function of position y. V DS _ n source 0 y x Q N (y) ptype metal interconnect to bulk n drain y = L First pass: approximate the drain current equation by taking averages of the channel charge and the drift velocity (Second pass: Section 4.4 (not assigned)) ÐWQ N v y Average drift velocity: still use µ n (V DS / L)  which is a very rough approximation.
5 Triode Region (Cont.) Next approximate the average channel charge by averaging Q N (y=0) at the source end and Q N (y=l) at the drain end of the channel: Q N ( y=0) = ÐC ox ( Ð V Tn ) At the drain end the positive drain voltage reduces the magnitude of the channel charge... why? The effect can be approximated by using V GD (the drop from drain to channel at y = L)  Q N ( y=l) = ÐC ox ( V GD Ð V Tn ) = ÐC ox ( Ð V DS Ð V Tn ) Note that V GD =  V DS > V Tn in order for there to be a channel left at the drain end. Substituting we derive the equation for the triode region which is defined by  V DS > V Tn and > V Tn. = µ n C W ox  ( VGS Ð V L Tn Ð V DS 2)V DS
6 Drain Characteristics Example: µ n C ox (W/L) = 50 µa/v 2 V Tn = 1 V and (W/L) = 4. (µa) = 4 V SAT = 3 V 200 = 2 V < V Tn V DS (V) What happens when V DS >  V Tn = V DS(sat)? Q N (y = L) = 0! Initial thought is that the lack of a channel at the drain end means that must drop to zero... WRONG! Drain terminal loses control over channel > drain current ÒsaturatesÓ and remains constant (to Þrst approximation) at the value given by V DS = V DS(sat).
7 Saturation Region When > V Tn and V DS > V DS(sat) =  V T the drain current is: W = ( sat ) = µ n C ox ( VGS Ð V 2L Tn ) 2 V DSSAT metal interconnect to gate I V DSAT GS _ n polysilicon gate n source 0 y n drain x Q N (y = L) = 0 ptype metal interconnect to bulk Full model: (µa) triode region Eq. (4.17) SAT = 4 V vs. V DSSAT constant current (saturation) region Eq. (4.21) = 3 V 200 = 2 V V DS (V) < 1 V
8 MOSFET Circuit Models nchannel MOSFET drain current in cutoff triode and saturation: = 0 A ( V Tn ) = µ n C ox ( W L) [ Ð V Tn Ð ( V DS 2) ]( 1 λ n V DS )V DS ( VGS V Tn V DS Ð V Tn ) = µ n C ox ( W ( 2L) )( Ð V Tn ) 2 ( 1 λ n V DS ) ( VGS V Tn V DS Ð V Tn ) Numerical values: µ n is a function of along the channel and is much less than the mobility in the bulk (typical value 215 cm 2 /(Vs) )  therefore we consider that µ n C ox is a measured parameter. Typical value: µ n C ox = 50 µav 2 λ n sometimes called the channel length modulation parameter increases as the channel length L is reduced: 0.1µmV Ð1 λ n L The triode region equation has (1 λ n V DS ) added in order to avoid a jump at the boundary with the saturation region. For hand calculation of DC voltages and currents this term is usually omitted from. V Tn = threshold voltage = V typically for an nchannel MOSFET.
9 Backgate Effect The threshold voltage is a function of the bulktosource voltage V BS through the backgate effect. V Tn = V TOn γ n ( ÐV BS Ð2φ p Ð Ð2φ p ) where V TO is the threshold voltage with V BS = 0 and γ is the backgate effect parameter γ n = ( 2qε s N a ) C ox Physical origin: V BS (a negative voltage to avoid forward biasing the bulktosource pn junction) increases the depletion width which increases the bulk charge and thus the threshold voltage. = ( V DS V BS ) since V Tn = V Tn (V BS ) Common situation is that V BS = 0 by electrically shorting the source to the bulk (either the substrate or a deep diffused region called a well) source and bulk terminals are shorted together > no backgate effect p n source p well n substrate n drain For this case V Tn = V TOn.
10 pchannel MOSFETs Structure is complementary to the nchannel MOSFET In a CMOS technology one or the other type of MOSFET is built into a well  a deep diffused region  so that there are electrically isolated ÒbulkÓ regions in the same substrate nchannel pchannel MOSFET MOSFET (a) A A common bulk contact for all nchannel MOSFETs (to ground or to the supply) isolated bulk contact with pchannel MOSFET shorted to source (b) p n source n drain p drain p source n ptype substrate n well
11 pchannel MOSFET Models DC drain current in the three operating regions:  > 0 ÐI D = 0 A ( V SG ÐV T ) ÐI D = µ p C ox ( W L) [ V SG V Tp Ð ( V SD 2) ]( 1 λ p V SD )V SD ( VSG ÐV Tp V SD V SG V Tp ) ÐI D = µ p C ox ( W ( 2L) )( V SG V Tp ) 2 ( 1 λ p V SD ) ( VSG ÐV Tp V SD V SG V Tp ) The threshold voltage with backgate effect is given by: Numerical values: V Tp = V TOp Ðγ p (( ÐV SB 2φ n ) Ð 2φ n ) µ p C ox is a measured parameter. Typical value: µ p C ox = 25 µav µmV Ð1 λ p L V Tp = 0.7 to 1.0 V which should be approximately V Tn for a wellcontrolled CMOS process
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