DERIVATION OF MOSFET IDS VS. VDS + VGS
Derive the current expressions in the MOSFET:
[(VGS − VT H )VDS −
Linear Region: ID = µCox W
L
2
VDS
]
2
W
Saturation Region: ID = µCox 2L
(VGS − VT H )2
1. Linear Region
Figure 1. Concentration Contours in Linear Region. A uniform narrow channel exists.
KVL:
VG − VS = VG − VC + VC − VS
VG − VS = VGS
VG − VC = VGC
VC − VS = V (x)
VGS = VGC + V (x) or VGS − V (x) = VGC
1
2
DERIVATION OF MOSFET IDS VS. VDS + VGS
Total charge density at x on capacitor(COX ) is QT (x):
QT (x) = VGC COX = (VGS − V (x))COX
QT (x) = Q(x)mobile + Q(x)depletion
Q(x)mobile =mobile electron charge in channel at x
Q(x)mobile = [VGS − V (x) − VT H ]COX
Use mobile charge to get current:
Jn = qµnE + qDn dn
= qµnE (no diffusion current in the channel)
dx
qn(x) = Q(x)mobile = Qm (x)
Jn = Qm (x)µE , but E = − dV
dx
, substitute for Qm (x)
Jn = −Qm (x)µ dV
dx
Jn = µCox (VGS −V (x)−VT H ) dV
, separate variables and neglect (-) sign. Consider
dx
only the magnitude.
Jn dx = µCox [(VGS − VT H ) − V (x)]dV
Due to continuity, Jn = constant (no hole current or no generation, recombination). Integrating from source to drain or from x=0 to x=L, where L=gate length:
Jn
RL
0
dx = µCox
R V (L)
V (0)
[(VGS − VT H ) − V (x)]dV
V(L) = VDS , V(0)=0
Jn
RL
0
dx = µCox
R VDS
0
[(VGS − VT H ) − V (x)]dV
Jn L = µCox [(VGS − VT H )V −
Jn =
µCox
[(VGS
L
− VT H )VDS −
V 2 VDS
]
2 0
2
VDS
]
2
DERIVATION OF MOSFET IDS VS. VDS + VGS
3
ID = Jn W (W=Device Width)
Jn for channel is Amp/cm since Qm = Charge/cm2
[(VGS − VT H )VDS −
ID for Linear Region: ID = µCox W
L
2
VDS
]
2
2. Saturation Region
When VDS ≥ (VGS − VT H ) channel pinches off. This means that the channel
current near the drain spreads out and the channel near drain can be approximated
as the depletion region. After this occurs, at VDS = (VGS − VT H ), if you make VDS
larger, the current ID does not change (to zero approximation). This is because
any additional VDS you add will get dropped across the depletion region and won’t
change the current ID .
So for VDS ≥ (VGS − VT H ) we find ID by setting VDS = (VGS − VT H ) substituting
into the linear equation.
ID = µCox W
[(VGS − VT H )(VGS − VT H ) −
L
ID for Saturation Region:
W
ID = µCox 2L
(VGS − VT H )2
(VGS −VT H )2
]
2
4
DERIVATION OF MOSFET IDS VS. VDS + VGS
source
drain
n=10^17
n=10^15
Figure 2. Concentration Contours in Linear Region. A uniform narrow channel exists.
source
drain
n=10^17
n=10^15
Figure 3. Concentration Contours in Saturation Region. Channel
narrow near source and spreads out and widens near drain, said to be
“pinched off”.