VDS - nanoHUB.org

Virtual Source Model:
part 2
Professor Mark Lundstrom
Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA
[email protected]
11/14/14
the MIT VS Model
 32 nm technology 
2
Lundstrom 11.13.14
MOSFET IV: low VDS
I D = W Cox ! sat (VGS " VT )
VGS
gate-voltage
controlled resistor
ID =
W
µ C (V ! VT )VDS
L n ox GS
Lundstrom 11.13.14
level 0’ model
1) I D W = !Qn (VG ) " (VDS )
2)
VGS ! VT : Qn (VGS ) = 0
VGS > VT : Qn (VGS ) = "Cox (VGS " VT )
VT = VT 0 ! " VDS
3)
! (VDS ) = FSAT (VDS )! sat
4) FSAT (VDS ) =
5) VDSAT =
VDS VDSAT
There are only 6 device-specific
parameters in this model:
Cox ,VT ,! ," sat , µn , L
1/ !
"1+ (VDS VDSAT )! $
#
%
! sat L
µn
+ !
4
Lundstrom 11.13.14
below threshold
Qn (VGS )
VGS > VT :
Need to treat subthreshold region.
VGS ! VT : Qn (VGS ) = 0
Qn (VGS ) = !Cox (VGS ! VT )
VGS
VT
5
Lundstrom 11.13.14
subthreshold characteristics
log10 {Qn (VGS )}
VGS > VT :
Qn (VGS ) = !Cox (VGS ! VT )
VGS < VT :
Qn (VGS ) ! "eq(VGS "VT ) mkBT
VT
Lundstrom 11.13.14
VG
6
charge vs. gate voltage
Qn (VGS )
Qn = ! Cox (VGS ! VT )
VGS
VT
"k T%
Qn (VGS ) = ! ( m ! 1) Cox $ B ' eq(VGS !VT ) mkBT
# q &
m = 1+ C D Cox ! 1
7
empirical treatment
(
Qn (VGS ) = !Cox m ( kBT q ) ln 1+ eq(VGS !VT ) mkBT
)
VGS << VT :
ln (1 + x ) ! x
Qn (VGS ) ! "Cox m ( k BT q ) eq(VGS "VT ) mkBT
Qn (VGS ) = ! ( m ! 1) Cox ( k BT q ) eq(VGS !VT ) mkBT
correct
G. T. Wright, Threshold modelling of MOSFETs for CAD of CMOS VLSI,
Electron Lett., 21, pp. 223–224, Mar. 1985.
8
empirical treatment
(
Qn (VGS ) = ! Cox m ( kBT q ) ln 1+ eq(VGS !VT ) mkBT
)
VGS > VT :
ln (1 + x ) ! ln ( x )
Qn (VGS ) ! "Cox (VGS " VT )
Qn (VGS ) = ! Cox (VGS ! VT )
correct
G. T. Wright, Threshold modelling of MOSFETs for CAD of CMOS VLSI,
Electron Lett., 21, pp. 223–224, Mar. 1985.
9
Level 1 model
1) I D W = !Qn (VGS ) " (VDS )
(
q(V
2) Qn (VGS ) = ! Cinv m ( kBT q ) ln 1+ e
GS !VT
VT = VT 0 ! " VDS
3) ! (VDS ) = FSAT (VDS )! sat
4) FSAT (VDS ) =
5) VDSAT =
VDS VDSAT
mkBT
)
There are only 7 devicespecific parameters in
this model:
Cox ,VT ,! , m," sat , µn , L
1/ !
"1+ (VDS VDSAT )! $
#
%
! sat L
µn
)
+ !,"
10
intrinsic vs. extrinsic voltages
VD
RD
VG! = VG
D
VG = VG!
VDS
!
G
VGS
!
VGS
VD! = VD " I D (VG! , VS!, VD! ) RD
VDS
VS! = VS + I D (VG! , VS!, VD! ) RS
silicon
S
RS
VS
11
Lundstrom 11.13.14
Level 1’ model
1) I D W = !Qn (VGS" ) # (VDS" )
(
q(V ! "V ) mk T
2) Qn (VGS! ) = " Cinv m ( kBT q ) ln 1+ e
GS
VT = VT 0 ! " VDS
#
3) ! (VDS" ) = FSAT (VDS" )! sat
4) FSAT (VDS! ) =
5) VDSAT =
VDS
! VDSAT
B
)
There are only 8 devicespecific parameters in
this model:
Cox ,VT ,! , m," sat , µn , L,
RSD = RS + RD
1/ "
"
#1+ (VDS
! VDSAT ) %&
$
! sat L
µn
T
+ !,"
12
outline
1) 
2) 
3) 
4) 
5) 
Traditional MOSFET theory
VS model: above threshold
VS model: subthreshold
VS model: quasi-ballistic and ballistic
Summary
Lundstrom 11.13.14
13
MOSFETs
I D W = !Qn (VGS ) " (VDS )
electrostatics
transport
! y = " µnE y
! y = ! sat
Lundstrom 11.13.14
14
mobility
ideal contacts
cross-sectional
area, A
L
n-type semiconductor
I
! d = " µ nE y
!V +
Mobility is a concept that describes long channels
(many MFP’s long). Lundstrom 11.13.14
15
mobility
Dn =
!T "
cm 2 s
2
Dn k BT
=
µn
q
µn =
L >> !
!T "
cm 2 V-s
2 ( k BT q )
Lundstrom 11.13.14
16
velocity cm/s --->
velocity saturation in bulk semiconductors
107
! = ! sat
! = µ nE
105
104
electric field V/cm --->
17
Lundstrom 11.13.14
Velocity (cm/s) 
velocity overshoot
EC
( µm )
! SAT
( µm )
D. Frank, S. Laux, and M. Fischetti, Int. Electron Dev. Mtg., Dec., 1992.
18
Lundstrom 11.13.14
The MVS model
1
1
!
µn
µapp
“apparent mobility”
! sat " !inj
“injection velocity”
19
Lundstrom 11.13.14
apparent mobility
1
1 1
= +
! app ! L
L
0
VGS > VT
VDS
n-Si
n-Si
!T "
cm 2 V-s
2 ( k BT q )
1
1
1
=
+
µ app µ n µ B
p-Si
y=0
µn =
y
µB =
!T L
cm 2 V-s
2 ( kBT q )
The MFP cannot be
“ballistic mobility”
longer than the channel
length.
Lundstrom 11.13.14
20
ballistic limit (low VDS)
ID =
L << ! , µ B << µ n , µ app " µ B
W
µ C (V ! VT )VDS
L app ox GS
µB =
1
1
1
=
+
µ app µ n µ B
!T L
cm 2 V-s
2 ( k BT q )
I D = WCox
!T
(V " VT )VDS
2 ( k BT q ) GS
ballistic MOSFET
21
Lundstrom 11.13.14
injection velocity
1 1 1 1
1 1
= + + " +
! !1 ! 2 ! 3 !1 ! 2
E
!1
!2
!3
Fn
1
1
1
=
+
!inj !T Dn !
EC ( y )
Fn
y=0
y
Qn ! "Cox (VGS " VT )
C cm 2
Lundstrom 11.13.14
22
ballistic limit (high VDS)
I D = WCox! inj (VGS " VT )
! << ! , " inj # "T
1
1
1
=
+
!inj !T Dn !
I D = WCox!T (VGS " VT )
ballistic MOSFET
23
Lundstrom 11.13.14
The VS nanotransistor model
1) I D W = !Qn (VGS" ) # (VDS" )
(
q(V ! "V ) mk T
2) Qn (VGS! ) = " Cinv m ( kBT q ) ln 1+ e
GS
VT = VT 0 ! " VDS
#
3) ! (VDS" ) = FSAT (VDS" )! sat
4) FSAT (VDS! ) =
5) VDSAT =
VDS
! VDSAT
B
)
There are only 8 devicespecific parameters in
this model:
Cox ,VT ,! , m,"inj , µapp , L,
RSD = RS + RD
1/ "
"
#1+ (VDS
! VDSAT ) %&
$
!inj L
µapp
T
+ !,"
24
The MVS model
1
1
1
1
!
=
+
µn
µapp µn µ B
#1
1
! sat " !inj = % +
$ !T Dn
&
! ('
)1
Lundstrom 11.13.14
25
conclusions
1) Traditional textbook MOSFET models correctly
describe the shape of the IV characteristics of
nanoscale MOSFETs – because they are still
“barrier controlled devices.”
2)  To get the magnitude of the current right, the
mobility and saturation velocity need to be
relplaced by the apparent mobility and the injection
velocity.
3) These two parameters are not “fudge factors” – they
have clear physical meaning.
Lundstrom Fall 2012
26