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Unformatted text preview: 6.012  Microelectronic Devices and Circuits Lecture 11  MOSFETs II; Large Signal Models  Outline • Announcements
On Stellar  2 writeups on MOSFET models • The Gradual Channel Approximation (review and more)
MOSFET model: gradual channel approximation
iD ≈ 0
K
K (Example: nMOS)
for (vGS – VT)/α ≤ 0 ≤ vDS (cutoff)
2 /2α
(vGS – VT)
for 0 ≤ (vGS – VT)/α ≤ vDS (saturation)
(vGS – VT – αvDS/2)vDS for 0 ≤ vDS ≤ (vGS – VT)/α (linear) with K ≡ (W/L)µeCox*, VT = VFB – 2φpSi + [2εSi qNA(2φpSi – vBS)]1/2/Cox*
and α = 1 + [(εSi qNA/2(2φpSi – vBS)]1/2 /Cox (frequently α ≈ 1) • Refined device models for transistors (MOS and BJT)
Other flavors of MOSFETS: pchannel, depletion mode The Early Effect: 1. Basewidth modulation in BJTs: wB(vCE)
2. Channellength modulation in MOSFETs: L(vDS)
Charge stores:
1. Junction diodes
2. BJTs
3. MOSFETs Extrinsic parasitics: Lead resistances, capacitances, and inductances
Clif Fonstad, 3/18/08 Lecture 11  Slide 1 An nchannel MOSFET showing gradual channel axes S G
+ vGS –
n+ 0
0 iG D
iD
n+ x pSi
vBS vDS +
B iB L y Extent into plane = W Gradual Channel Approximation:
 A onedimensional electrostatics problem in the x direction is solved to find
the channel charge, qN*(y); this charge depends on vGS, vCS(y) and vBS.
 A onedimensional drift problem in the y direction then gives the channel
current, iD, as a function of vGS, vDS, and vBS.
Clif Fonstad, 3/18/08 Lecture 11  Slide 2 Gradual Channel Approximation iv Modeling
(nchannel MOS used as the example) The Gradual Channel Approximation is the approach typically
used to model the drain current in field effect transistors.*
It assumes that the drain current, iD, consists entirely of carriers
flowing in the channel of the device, and is thus proportional
to the sheet density of carriers at any point and their net
average velocity. It is not a function of y, but its components
in general are:
iD = " W # "q # n * ( y ) # sey ( y )
ch
In this expression, W is the width of the device, q is the
charge on each electron, n*ch(y) is sheet electron concentration in the channel (i.e. electrons/cm2) at y, and sey(y) is the net
!
electron velocity in the ydirection.
If the electric field is not too large, sey(y) = µeEy(y), and iD = " W # q # n * ( y ) # µe E y ( y ) = W # q # n * ( y ) # µe
ch
ch dvCS ( y )
dy Cont. * Junction FETs (JFETs), MEtal Semiconductor FETs (MESFETs1), and Heterojunction FETs 2
Clif Fonstad, 3/18/08 (HJFETs ), as well as Metal Oxide Semiconductor FETs (MOSFETs). Lecture 11  Slide 3 1. Also called Shottky Barrrier FETs (SBFETs). 2. Includes HEMTs, TEGFETs, MODFETs, SDFETs, HFETs, PHEMTs, MHEMTs, etc. ! GCA iv Modeling, cont. S –
n+ We have: G
+ vGS
0
0 dvCS ( y )
iD = W " q " n ( y ) " µe
dy vDS iG vBS iD
n+ x
L pSi *
ch D +
B y iB To eliminate the derivative from this equation we integrate both
sides with respect to y from the source (y = 0) to the drain (y
= L). This corresponds to integrating the right hand side with
respect to vCS from 0 to vDS, because vCS(0) = 0 to vCS(L) = vDS:
L L dvCS ( y )
*
" iD dy = W # µe # q # " nch ( y ) dy dy = W # µe # q #
0
0 v DS n * (vCS ) dvCS
" ch
0 The left hand integral is easy to evaluate; it is simply iDL. Thus
we have:
L W
" iD dy = iD L # iD = L $ µe $ q $
0
Clif Fonstad, 3/18/08 v DS n * (vCS ) dvCS
" ch
0
Cont. Lecture 11  Slide 4 GCA iv Modeling, cont. The various FETs differ primarily in the nature of their channels
and thereby, the expressions for n*ch(y).
For a MOSFET we speak in terms of the inversion layer charge,
qn*(y), which is equivalent to  q·n*ch(y). Thus we have:
v DS
W
iD = " µe # q* (vGS , vCS , v BS ) dvCS
n
L
0
We derived qn* earlier by solving the vertical electrostatics
problem, and found:
*
q* (vGS , vCS , v BS ) = " Cox [vGS " vCS " VT (vCS , v BS )]
n
! { [ with VT (vCS , v BS ) = VFB " 2# p " Si + 2$SiqN A 2# p " Si " v BS + vCS
} 1/ 2 *
Cox Using this in the equation for iD, we obtain: !
W
iD (vGS , v DS , v BS ) = µe
L v DS # {C [v
*
ox GS " vCS " VT (vCS , v BS )]} dvCS 0 At this point we can do the integral, but it is common to simplify
the expression of VT(vCS,vBS) first, to get a more useful result. ! Clif Fonstad, 3/18/08 Cont. Lecture 11  Slide 5 GCA  dealing with the nonlinear dependence of VT on vCS Approach #1  Live with it
Even though VT(vCS,vBS) is a nonlinear function of vCS, we can
still put it in this last equation for iD:
W
iD = µe
L + *%
t
,Cox 'vGS " vCS " VFB + 2# p " Si " ox 2$SiqN A 2# p " Si " v BS + vCS
1&
$ox
0 v DS [ (.
*/ dvCS
)0 and do the integral, obtaining: *
W
v'
*$
iD (v DS , vGS , v BS ) =
µe Cox +& vGS " 2# p " VFB " DS )v DS
L
2(
,% ( 3
.
+
2SiqN A 0 2# p + v DS " v BS
/
2 ! ) 3/2 ( " 2# p " v BS ) 3/2 14
35
26 The problem is that this result is very unwieldy, and difficult to
work with. More to the point, we don't have to live with it
because it is easy to get very good, approximate solutions
that are much simpler to work with.
Clif Fonstad, 3/18/08 Cont. Lecture 11  Slide 6 GCA  dealing with the nonlinear dependence of VT on vCS Approach #2  Ignore it
Early on researchers noticed that the difference between VT at
0 and at y, i.e. VT(0,vBS) and VT(vDS,vBS), is small, and that using
VT(0,vBS) alone gives a result that is still quite accurate and is
very easy to use:
v DS
W
*
iD (vGS , v DS , v BS ) = µe # {Cox [vGS " vCS " VT (0, v BS )]} dvCS
L
0
2
$
W
v DS '
*
= µe Cox %[vGS " VT (0, v BS )] v DS "
(
L
2)
& The variable, vCS,
is set to 0 in VT. This result looks much simpler than the result of Approach #1,
and it is much easier to use in hand calculations. It is, in
! fact, the one most commonly used by the vast majority of
engineers. At the same time, the fact that it was obtained by
ignoring the dependence of VT on vCS is cause for concern,
unless we have a way to judge the validity of our approximation. We can get the necessary metric through Approach #3.
Clif Fonstad, 3/18/08 Cont. Lecture 11  Slide 7 GCA  dealing with the nonlinear dependence of VT on vCS Approach #3  Linearize it (i.e. expand it, keep first order term)
In this approach we leave the variation of VT with vCS in, but
linearize it by doing a Taylor's series expansion about vCS = 0:
#VT
VT [vCS , v BS ] " VT (0, v BS ) +
$ vCS
#vCS v = 0
CS Taking the derivative and evaluating it at vCS = 0 yields: t ox
VT [vCS , v BS ] " VT (0, v BS ) +
!
#ox #SiqN A
& vCS
2 2$ p % v BS ( ) With this qn* is ! '
t
*
q* (vGS , vCS , v BS ) " # Cox )vGS # vCS + VT (0, v BS ) # ox
n
)
$ox
( $SiqN A
2 2% p # v BS ( ) *
& vCS ,
,
+ *
= # Cox [vGS #  vCS + VT (v BS )] where ! Clif Fonstad, 3/18/08 ' t ox
*
" # )1 +
$SiqN A 2 2% p & v BS , and VT (v BS ) # VT (0, v BS )
( $ox
+
Lecture 11  Slide 8 ( ) Cont. GCA  dealing with the nonlinear dependence of VT on vCS Using this result in the integral in the expression for iD gives:
v DS
W
*
iD (vGS , v DS , v BS ) = µe $ {Cox [vGS " # vCS " VT (0, v BS )]} dvCS
L
0
2
%
W
v DS (
*
= µe Cox &[vGS " VT (v BS )] v DS " #
)
L
2*
' Except for α this is the
Approach 2 result. Plotting this equation for increasing values of vGS we see that it
traces inverted parabolas as shown below. ! iD inc.
vGS
Note: iD saturates
after its peak value
(solid lines), rather
than decreasing
(dashed lines). vDS
Clif Fonstad, 3/18/08 Cont. Lecture 11  Slide 9 Gradual Channel Approximation, cont.
The drain current expression, cont:
The point at which iD reaches its peak value and saturates is
easily found. Taking the derivative and setting it equal to
1
zero we find: " iD
=0
when
v DS =
[vGS $ VT (v BS )]
" v DS
#
What happens physically at this voltage is that the channel (inversion) at the drain end of the channel disappears: ! *
q* ( L) " # Cox {vGS # VT (v BS ) # $ v DS }
n
1
= 0 when v DS = [vGS # VT (v BS )]
$ For vDS > [vGSVT(vBS)]/α, all the additional draintosource voltage
appears across the high resistance region at the drain end of
!
the channel where the mobile charge density is very small,
and iD remains constant independent of vDS:
iD (vGS , v DS , v BS ) =
Clif Fonstad, 3/18/08 ! 1W
2
*
µe Cox [vGS # VT (v BS )] for
2" L v DS > 1
[vGS # VT (v BS )]
"
Lecture 11  Slide 10 Gradual Channel Approximation, cont. The full model:
With this drain current expression, we now have the complete
set of Gradual Channel Model expressions for the MOSFET
terminal characteristics in the three regions of operation:
Valid for v BS " 0, and v DS # 0 :
iG (vGS , v DS , v BS ) = 0 and
iB (vGS , v DS , v BS ) =
&
,
0
,
,
1W
2
*
iD (vGS , v DS , v BS ) = '
µe Cox [vGS $ VT (v BS )]
2%L
,
v)
,W
*&
µe Cox 'vGS $ VT (v BS ) $ % DS * % v DS
, %L
(
2+
(
with VT (v BS )  VFB Clif Fonstad, 3/18/08 { [ 0 for [vGS $ VT (v BS )] < 0 < % v DS
0 < [vGS $ VT (v BS )] < % v DS for 0 < % v DS < [vGS $ VT (v BS )] for 1
$ 2. p $ Si + * 2/SiqN A 2. p $ Si $ v BS
Cox &
1,
/SiqN A
%  1+ * '
Cox , 2 2. p $ Si $ v BS
( [ )1 / 2
,
*
,
+
} C* ox 1/ 2 /ox
t ox
Lecture 11  Slide 11 iD Gradual Channel Approximation, cont. ! $
*
0
*
*
K
2
iD (vGS , v DS , v BS ) =!
%
[vGS " VT (v BS )]
2
*
* K $v " V (v ) " # v DS ' # v
( DS
T
BS
* % GS
&
2)
& Linear or
Triode
Region Clif Fonstad, 3/18/08 iD iB G+ iB (vGS , v DS , v BS ) = 0 iG (vGS , v DS , v BS ) = 0 + vDS iG The full model, cont: D + B vBS vGS
– for S
[vGS " VT (v BS )] < 0 < # v DS for 0 < [vGS " VT (v BS )] < # v DS Saturation for 0 < # v DS < [vGS " VT (v BS )] Linear or
Triode increasing with vGS K+ Cutoff W
*
µe Cox
#L Saturation or
Forward Active
Region Cutoff Region vDS Lecture 11  Slide 12 The operating regions of MOSFETs and BJTs: Comparing an nchannel MOSFET and an npn BJT iD MOSFET D
+ Linear
iD or
Triode iG Saturation (FAR) vDS G+ iD ! K [vGS  V T(vBS)]2/2! vGS
– iC BJT C – S + Cutoff vDS iB
vCE B+ Saturation
iC i vBE
– B – E FAR iB ! IBSe qV BE /kT
vCE > 0.2 V Cutoff 0.6 V Input curve
Clif Fonstad, 3/18/08 vBE Forward Active Region
iC ! !F iB 0.2 V Cutoff vCE Output family
Lecture 11  Slide 13 pchannel MOSFET's: The other "flavor" of MOSFET
pchannel S – vGS G
+ iG vDS p+ D
iD
p+ Structure:
nSi
vBS The voltage progression:
Inversion +
B Depletion VT
Gradual channel model*: iB Accumulation v GS 0 V FB For enhancement mode pchannel:
VT ( i.e. v GS at threshold) < 0, VFB > 0.. Valid for v SB " 0, and v SD # 0 : iG (v SG , v SD , v SB ) = 0 and
iB (v SG , v SD , v SB ) = 0
&
,
0
for [v SG $ VT (v SB ) ] < 0 < % v SD
,
,
2
1W
*
$ iD (v SG , v SD , v SB ) = '
µe Cox [v SG $ VT (v SB ) ]
for 0 < [v SG $ VT (v SB ) ] < % v SD
2%L
,
v)
,W
*&
µe Cox 'v SG $ VT (v SB ) $ % SD * % v SD for 0 < % v SD < [v SG $ VT (v SB ) ]
, %L
(
2+
(
1
1/ 2
VT (v SB ) = VFB " 2# n " Si " $ [2# n " Si " v SB ]
with $ % * [2&SiqN D ] 1 / 2
Cox
Clif Fonstad, 3/18/08
Lecture 11  Slide 14
! * Enhancement mode only, VT ( i.e. v GS at threshold) < 0. pchannel MOSFET's: cont. S G
+ vGS – vDS iG p+ pchannel Structure:
nSi + Symbol: vDS iG iB G+ + vBS B FAR model:
vGS < VT
iG
vBS > 0
vDS < 0 G +
– – + Symbol: vSG vSB
– – B
iB iG vSD
iD – D FAR model:
vSG > VT
vSB < 0
vSD > 0 + iB vDS i D(vGS ,vDS,vBS )
B
iB +
vBS vGS S
Oriented as found in circuits:
S
+ +
B D iD vBS vGS Clif Fonstad, 3/18/08 iD
p+ Symbol and FAR model:
Oriented with source down like nchannel:
iD D G D – S
S
+ + vSB
iB vSG
– – B
i D(vSG ,vSD,vSB )
vSD G
iG iD – D Lecture 11  Slide 15 Depletion mode MOSFET's: The very last MOSFET variant
It is possible to have nchannel MOSFETs with VT < 0.
In this situation the channel exists with vGS = 0, and a
negative bias must be applied to turn it off.
This type of device is called a "depletion mode" MOSFET.
Devices with VT > 0 are "enhancement mode."
iD increasing vGS
vGS = 0 vGS ! V T vDS For a pchannel depletion mode MOSFET, VT > 0.
The expressions for iD(vGS, vDS, vBS) are exactly the same
for enhancement mode and depletion mode MOSFETs.
Clif Fonstad, 3/18/08 Lecture 11  Slide 16 BJT Characteristics (npn) Saturation iC iB FAR Forward Active Region iB ! IBS e qV BE /kT
vCE > 0.2 V iC ! !F iB Cutoff vBE 0.6 V 0.2 V Input curve
BJT MODELS
BJT Models Output family
–– BC
vBC IIES
ES vBE vBE iFF
i
"!iRiR
RR
– Clif Fonstad, 3/18/08 CC
+ ICS
ICS
iRiR + "FiiF
!F F B+
B+ vCE Cutoff – E – E Forward active region:
vBE > 0.6 V
vCE > 0.2 V CC
!F!bFiB
i (i.e. vBC < 0.4V) iR is negligible vCE
Other regions
Cutoff:
vBE < 0.6 V
Saturation:
vCE < 0.2 V i
iB B
B+
B + IBS
IBS vvBE
BE
– – EE
Lecture 11  Slide 17 MOSFET Characteristics Linear
iD or
Triode Saturation (FAR) (nchannel) iD ! K [vGS  V T(vBS )]2/2! Also:
iG ≈ 0
iB ≈ 0
K = (W/L)µeCox* Cutoff vDS Output family
Output family Model valid for vBS ≤ 0 and vDS ≥ 0, insuring
iG(vGS, vDS,vBS) ≈ 0, iB(vGS, vDS,vBS) ≈ 0
0 MOSFET
D
circuit
model iD +
vDS iG
G + vGS i D(vGS ,vDS,vBS ) ≈
iB + B
vBS
–– S
Clif Fonstad, 3/18/08 for (vGS – VT) ≤ 0 ≤ αvDS (cutoff) (W/2αL)µeCox*(vGS – VT)2
for 0 ≤ (vGS – VT) ≤ αvDS (saturation) (W/αL)µeCox*(vGS – VT – αvDS/2)αvDS
for 0 ≤ αvDS ≤ (vGS – VT) (linear) with VT = VFB – 2φpSi + [2εSi qNA(2φpSi – vBS)]1/2/Cox*
α = 1 + [(εSi qNA/2(2φpSi – vBS)]1/2 /Cox
(frequently α ≈ 1) Lecture 11  Slide 18 The Early Effect:
(exaggerated for clarity*) BJT: iC iC npn 0.2 V 1/! = V A MOSFET: iD vvCE
CE iD nchannel 1/! = V A
Clif Fonstad, 3/18/08 * Typically the Early effect is far more important
in smallsignal applications than large signal. vDS
v DS Lecture 11  Slide 19 Active Length Modulation  the Early Effect: MOSFET
"Channel length modulation" MOSFET:
We begin by recognizing that the channel length decreases
with increasing vDS and writing this dependence to first order
in vDS:
1 [1 + $(v DS # VDSat )]
L " Lo [1 # $(v DS # VDSat )] and
"
L
Lo
W
*
K=
µe Cox
%L
Inserting the channel length variation with vDS into K we have: ! K " K o [1 + #(v DS $ VDSat )] where Ko % W
*
µe Cox
& Lo Thus, in saturation:
iD " Ko
2
(vGS # VT ) [1 + $(v DS # VDSat )]
2 !
Note: λ is the inverse of the Early Voltage, VA (i.e., λ = 1/VA). Clif Fonstad, 3/18/08 Lecture 11  Slide 20 ! Active Length Modulation  the Early Effect: BJT
"Base width modulation" BJT:
We begin by recognizing that the base width decreases
with increasing vCE and writing this dependence to first order
in vCE:
1
1
w * " w * (1 # $vCE ) and
" * (1 + $vCE )
B
Bo
w * w Bo
B
Then we recall that in a modern BJT the base defect, δB, is
negligible and βF depends primarily on the emitter defect, δE,
and can be written:
!
(1 + #B ) $ 1 = De N DE w *
E
"F =
(#E + #B ) #E Dh N AB w *
B
Inserting the base width variation with vCE into βF we have: *
De N DE wE
"F ! "Fo(1 + $vCE ) where "Fo %
#
*
Dh N AB wBo
Thus, in the F.A.R.:
iC " # Fo (1 + $vCE ) iB
Note: λ is the inverse of the Early Voltage, VA (i.e., λ = 1/VA).
!
Clif Fonstad, 3/18/08
Lecture 11
!  Slide 21 Large signal models*:
i BJT:
npn Saturation iC FAR
Forward Active Region
iC ! !F iB iB ! IBSe qV BE /kT
vCE > 0.2 V Cutoff iC ≈ βF(1 + λvCE)iB 0.6 V MOSFET:
nchannel vBE Linear
iD or
Triode iD ≈ K[vGS  VT(vBS)  vDS/2]vDS 0.2 V Saturation (FAR)
! v [vGS  V T(vBS 2 2/2!
iD ≈iDK[K GSVT(vBS)])][1+λ(vDSVDSat]/2 Cutoff
Clif Fonstad, 3/18/08 vCE Cutoff vDS * The Early effect is included, but barely visible. Lecture 11  Slide 22 Large signal models: when will we use them?
Digital circuit analysis/design:
This requires use of the entire circuit, and will be the topic of
Lectures 14, 15, and 16.
Bias point analysis/design:
This uses the FAR models (below and Lec. 17ff).
C
C
iIC = β IB
C
BJT
!FiB
IB
iB
iB
B+
B+
IBS
0.6 vBE
V
vBE –
–
E
E
D MOSFET D
i D = (K/2)[vGS  V T] G G
VGS = (2IDS/K)1/2 + vVS
T vGS
–
Clif Fonstad, 3/18/08 2 G+ + S iID
D + B vBS
– S
Lecture 11  Slide 23 Charge stores in devices: we must add them to our device models Parallel plate
capacitor qA "
qA ,PP = A v AB
d ρ(x)
d/2 $qA ,PP
C pp (VAB ) #
$v AB x d/2
qB( = qA) Depletion region charge store
ρ(x) qNDn v AB = VAB qA ,DP (v AB ) = " A 2q#Si [$ b " v AB ] ! qB xp A"
d = N Ap N Dn [N Ap + N Dn ] ( = Q A ) x xn qA Cdp (VAB ) = A qNAp N Ap N Dn
q#Si
2[$ b " VAB ] [ N Ap + N Dn ] QNR region diffusion charge store
p’(x), n’(x) qAB ,DF (v AB ) " Aqn i2 p’(xn) ! qA, qB (=qA) n’(xp ) wp xp xn x Cdf (VAB ) "
Clif Fonstad, 3/18/08 A #Si
w (VAB ) Dh
[e qVAB / kT # 1]
N Dn w n ,eff Note: Approximate because we are
only accounting for the charge
store on the lightly doped side. wn = 2
w n ,eff q ID (VAB )
2 Dh
kT = 2
w n ,eff 2 Dh iD (v AB ) Lecture 11  Slide 24 Adding charge stores to the large signal models: A pn diode: qAB IBS qAB: Excess carriers on pside plus
excess carriers on nside plus
junction depletion charge. B qBC C BJT: npn
(in F.A.R.) iB’
B MOSFET:
nchannel qBE: Excess carriers in base plus EB
junction depletion charge
qBC: CB junction depletion charge IBS
E qBE
D
iD S qG: Gate charge; a function of vGS, vDS, qDB qG G Clif Fonstad, 3/18/08 !FiB’ B and vBS. qDB: DB junction depletion charge
qSB: SB junction depletion charge qSB
Lecture 11  Slide 25 6.012  Microelectronic Devices and Circuits Lecture 11  MOSFETS II; LargeSignal Models  Summary • Gradual channel approximation for FETs
General approach MOSFETS in strong inversion 1. Ignore ariation of VT along channel
v
2. Linearize variation of VT along channel: introduces α factor • Additional device model issues
The Early Effect:
1. Basewidth modulation in BJTs: wB(vCE) In the F.A.R.: iC ≈ βFo(1 + lvCE)iB 2. Channellength modulation in MOSFETs: L(vDS)
In saturation: iD ≈ Ko (vGS – VT)2 [1 + λ(vDSVDSat)]/2α Charge stores:
1. Junction diodes  depletion and diffusion charge
2. BJTs  at EB junction: depletion and diffusion charge at CB junction: depletion charge (focus on FAR) 3. MOSFETs  between B and S, D: depletion charge of n+p junctions
between G and S, D, B: gate charge (the dominant store)
in cutoff: Cgs ≈ Cgd ≈ 0; all is Cgb
linear region: Cgs = Cgd = W L Cox*/2
in saturation region: Cgs = (2/3) W L Cox*
Cgd = 0 (only parasitic overlap) Clif Fonstad, 3/18/08 Lecture 11  Slide 26 MIT OpenCourseWare
http://ocw.mit.edu 6.012 Microelectronic Devices and Circuits
Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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This note was uploaded on 11/07/2011 for the course COMPUTERSC 6.012 taught by Professor Charlesg.sodini during the Fall '09 term at MIT.
 Fall '09
 CharlesG.Sodini

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