1. Long Channel MOSFETs

2. Long Channel MOSFETs : DRAIN CURRENT MODEL
General drain current model
Simplified!!!
Charge Sheet approximation: analytical expression for S/D current
Conventionally
V source: Grounded
Vds, Vg : > 0V
Vbs: < 0 (initially assumed ‘0’
P-type: Na
f(x, y): band bending, or intrinsic potential w.r.t. bulk intrinsic potential
V(y): electron quasi-Fermi potential at y w.r.t. Fermi potential of the n+ source

3. Long Channel MOSFETs : DRAIN CURRENT MODEL
No bias between Source and Substrate  ff(source) = ff (bulk)
Channel to substrate diode (n+/p-)
Quasi Fermi potential stays essentially constant across the depletion region
V(y) does not change with x  no current flow:  V(y=L) =Vds
Inversion charge density as a function of Quasi Fermi Potential, V(y)
and 
Under surface inversion 

4. Long Channel MOSFETs : DRAIN CURRENT MODEL
Gradual-Channel Approximation
1-D MOSFET Model: Gradual channel approximation (GCA)
Assuming
dE/dy << dE/dx, valid for channel region except beyong the pinch-off
Hole current, generation & recombination current are negligible
Current continuity equation is to electron current in the y-direction
Ids is constant along y
V(y) ~ quasi Fermi potential, playing the role of fn
Ids(y) at the position, y
Qi(y) at the position, (y, z)

5. Long Channel MOSFETs : DRAIN CURRENT MODEL
Gradual-Channel Approximation
PAO & SAH’s Double Integral
PAO & SAH’s Double Integral

6. Long Channel MOSFETs : MOSFET I-V
PAO & SAH’s Double Integral
Boundary Conditions 1. 2.
Charge Sheet Approximation : for analytical solution
For an analytical solution
Inversion layer thickness is zero (like a sheet of charge)
No potential drop or band bending across the inversion layer
Depletion approximation is applied to the bulk depletion region

7. Long Channel MOSFETs : MOSFET I-V
Since
I-V using charge sheet model
For a given Vg, Ids first increases linearly with the Vds  linear (tiode) region
Then, gradually levels off to a saturated value (saturation region)

8. Long Channel MOSFETs : MOSFET I-V
Linear (Triode) Region
If Vds is small, from power of Vds, keep only 1st order of Vds
Vt is the gate voltage when the surface potential or band bending reaches 2yB and the Si charge is the bulk depletion charge for that potential.
Typically 0.6 ~ 0.9V
If Vg < Vt : little current flow  MOSFET subthreshold region

9. Long Channel MOSFETs : MOSFET I-V
Linear region  MOSFETs act like a R:
 Modulated by Vg
Vt can be determined from Ids-Vg plot at low Vds
Extrapolated intercept of the linear portion of the Ids(Vg ) plot with the Vg axis at low Vds
This value is slightly higher than “2yB” Vt due to the inversion layer capacitance and other effects
Ids(Vg ) is not linear near Vt  CSA is not valid in this regime

10. Long Channel MOSFETs : MOSFET I-V
Characteristics in the Saturation Region
For a large Vds , 2nd term in power series of Vds,
m: body effect coefficient, typically 1.1<m<1.4
As Vds increase, Ids followed a parabolic curve until a maximum or saturation value is reached when Vds=Vdsat=Vt-Vg
Valid only Vds<Vdsat
Beyond Vds=Vdsat(Vds>Vdsat)

11. Long Channel MOSFETs : MOSFET I-V
Characteristics in the Saturation Region
Valid only Vds<Vdsat
Beyond Vds=Vdsat(Vds>Vdsat)

12. Long Channel MOSFETs : MOSFET I-V
Onset of pinch-off and current saturation
Drain current saturation can be understood from the inversion charge density.
When V<2yB
Shaded area
If Vds is small (linear region)
Qi(drain end) ~ Qi (source end) (slightly lower)
As Vds↑, Ids↑ and Qi(drain end)↓ to zero when Vds = Vdsat = (Vg-Vt)/m
Ids is at its maximum
Surface c hannel vanishes at the drain end
Pinch-off

13. Long Channel MOSFETs : MOSFET I-V
Onset of pinch-off and current saturation
As Vds↑(over Vdsat), pinch-off point moves toward source
 Ids remains essentially the same because for Vds>Vdsat, V at pinch-off point remains at Vdsat.  L changes to L’ (L’<L)  channel length modulation
Drain current:
Integrating from 0 to y after multiplying by dy
Substituting Ids from

14. Long Channel MOSFETs : MOSFET I-V
Onset of pinch-off and current saturation
V(y) and –Qi/mCox=(Vg-Vt)/m-V(y) for several Vds
At low Vds  V(y) ↑ LINEARLY
As Vds ↑  Qi(L) ↓ due to electron quasi-EF ↓
 dV/dy ↑ to maintain current continuity
Vds =Vdsat=(Vg-Vt)/m  Qi(L) =0 V(y)/dy=∞
The Ey changes more rapidly than Ex,
GCA breaks down
Beyond Pinch-Off (Vds>2yB)
Carriers not confined to the surface channel
2-D Poisson;s equation for carrier injection from
pinch-off into the drain depletion region
Not power series in Vds
General form from Qi=0 and V=Vdsat(dIds/dVds=0) from
Vds (several X >2yB), slight change ≈5% in Idsat compared to
CSA 

15. Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS
On a linear scale Id approach ‘0’ immediately below Vt.
On a log scale, Id remains at nonnegligible levels for several tenths of a volt below Vg. (yB<yS<2yB)
The inversion charge density does not drop to zero abruptly
Exponential dependence on ys or Vg.
Important in low-voltage, low power application

16. Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS
Drift and Diffusion Components of Drain Current
Strong inversion region , 2yB<yS  Drift current dominates
Subthreshold conduction , yB<yS<2yB  Diffusion current dominates
Both Diffusion and Drift included
Ratio varies along the channel
At low Vds, diffusion and drift components CAN BE separated
If qV.kT<<1: only the 1st order terms of V kept Qi(V) replaced by its 0th order value Qi(V=0)
Current continuity  V must vary linearly from S to D
Total current dV/dy  the drift current E-field or dyS/dy (or dyS/dV(y) since constant dV/dy)
 The drift fraction is given by the change of yS(band bending) w.r.t. quasi-Fermi potential, dyS/dV
If V 0

17. Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS
Drift and Diffusion Components of Drain Current
Voltage drop across the oxide given by the last term of
Weak inversion, yB<yS<2yB
Numerator << 1
DIFFUSION component dominates
Strong inversion, yS>2yB
dyS/dV≈1
DRIFT current dominattes

18. Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS
Subthreshold Current Expression
By Gauss’s Law
In weak inversion, <<
Expanding into a power series: 0th order=-Qd, 1st order term = -Qi
Surface potential yS is related to the Vg through
Since the inversion charge density is small, yS can be considered as a function of Vg only, independent of V.
The electric field along the channel direction is small. The drift current is negligible.
Substituting Qi into

19. Long Channel MOSFETs : SUBTRESHOLD CHARACTERISTICS
Subthreshold Current Expression
yS can be expressed in terms of Vg
yS only slightly deviates from the threshold value, 2yB.
 | yS - 2yB | << 2yB and expand the square-root term around yS = 2yB :