1. EE 466: VLSI Design
Lecture 03

2. Channel Length Modulation
The inversion layer charge QI represents the total mobile electron charge on the surface and its expression at the source end of the channel is:
QI at x=0=-Cox(VGS-VT0) and the inversion layer charge at the drain end of the channel is expressed as: QI at x=L=-Cox(VGS-VT0-VDS)
At the edge of saturation when the drain-to-source voltage reaches saturation (VDS=VDSAT=VGS-VT0) the inversion layer charge at the drain end becomes zero.
We can approximate the inversion layer charge at the drain end by QI at x=L=0 (even though this is not quite true)
When VDS=VDSAT, the channel is pinched off at the drain end.
Further increase of the drain-to-source voltage (VDS>VDSAT) results in even a larger pinched-off portion of the channel.
The effective channel length is reduced to: L’=L-DL where DL is the length of the channel where QI=0.
The pinch-off point moves from the drain end of the channel towards the source end with increasing VDS.

3. Channel Length Modulation
For L’<x<L the channel voltage is Vc at x=L’=VDSAT.
The electrons traveling from source to drain traverse the inverted channel segment of length L’ and then they are injected into the depletion region of length DL that separates the pinch-off point from the drain edge.
The gradual channel approximation is valid in this region and is given by: IDSAT=(mnCox)/2(W/L’)[VGS-VT0]2
The effective channel length for the MOSFET operating in saturation is now L’ and the above equation accounts for the actual shortening of the channel.
Shortening of the channel is also known as channel length modulation.
If L’ is replaced by L in the equation we can show that the computed saturation current using L’ is greater than the new IDSAT computed using L.

4. Channel Length Modulation
We must modify the equation for saturation current so that it reflects the dependency on VDS. Note that the saturation current will increase with increasing VDS since L’ decreases with increasing VDS.
The first term of the equation accounts for the channel length modulation effect.
Let 1-DL1-lVDS, with l being an empirical model parameter called the channel length modulation coefficient.
Assume that lVDS<<1 then the saturation current becomes:
The above equation can be used with sufficient confidence for most first order hand calculations

5. The Threshold Voltage
Any gate-to-source voltage less than VT0 is not sufficient to establish an inversion layer.
The MOSFET conducts no current between its source and drain terminals unless VGS is greater than VT0.
Increasing the gate-to-source voltage above and beyond VT0 will not affect the surface potential and the depletion region depth.
There are 4 physical properties that affect the threshold voltage namely (i) the work function difference between the gate and the channel, (ii) the gate voltage component to change the surface potential, (iii) the gate voltage component to offset the depletion region charge and (iv) the voltage component to offset the fixed charges in the gate oxide and in the silicon oxide interface.
Oxide (SiO2)
P-type Semiconductor (Si)
Metal (Al) Ec Ei FF
2FF
EFp
-FF Ev
qVT0

6. The Threshold Voltage
The work function difference (Fgate-to-channel) reflects the built in potential of the MOS structure which consists of the p-type substrate, the thin silicon dioxide layer and the gate electrode.
FGC = FF(substrate) –FM if the gate material is metal (Aluminum)
If polysilicon is the gate material: FGC = FF(substrate) – FF (gate)
An externally applied voltage must be changed to achieve surface inversion i.e. to change the surface potential by -2FF.
The depletion region charge density at surface inversion (FS = -FF) is QB0=-(2qNASi|-2FF|)-1/2
Assuming that the substrate is biased at a different voltage level than the source (at ground) then the depletion region charge density can be expressed as a function of the source-to-body voltage VSB: QB=-(2qNASi|-2FF|+ VSB)-1/2
The third component offsets the depletion region charge and is equal to –QB/Cox (Cox=ox/tox).

7. The Threshold Voltage
The gate voltage that is necessary to offset to the fixed positive charge density Qox (at the interface between the gate oxide and the silicon substrate) is –Qox/Cox.
Combining all the voltage components we can determine the threshold voltage. For zero substrate bias VT0 is given by: VT0=FGC-2FF-QB0/Cox-Qox/Cox
Thus determine the expression for nonzero substrate bias.

8. Body effect
The most general form of the threshold voltage is: VT=FGC-2FF-Qox/Cox-QB/Cox
VT=VT0-(QB-QB0)/Cox
(QB-QB0)/Cox=((2qNASi)-1/2) /Cox*((|-2FF+VSB|)-1/2-(|2FF|)-1/2)
This becomes the most general expression of the threshold voltage with the parameter gamma being:

9. The Threshold Voltage
Gamma is called the substrate-bias or the body effect coefficient.
The general expression for the threshold voltage can be used for both the n-channel and p-channel devices.
The differences are as follows:
Substrate Fermi potential FF is negative in nMOS but positive for pMOS.
The depletion region charge densities QB0 and QB are negative for nMOS but positive for pMOS
Further differences:
The substrate bias coefficient g is positive for nMOS and negative for pMOS.
The substrate bias voltage VSB is positive in nMOS but negative for pMOS.
Typically the threshold voltage of an enhancement mode n-type MOSFET is a positive quantity while that of a p-type MOSFET is negative.

10. Sub-Threshold Conduction
Ideally at VGS < VT, ID = 0.
The MOS device is partially conducting for gate voltages below the threshold voltage.
This is termed sub-threshold or weak inversion conduction.
In most digital applications the presence of sub-threshold current is undesirable. Why?
….most digital applications …. Does this mean some digital applications can tolerate sub-threshold currents?
A Sub-threshold digital circuit manages to satisfy the ultra-low power requirement. How?
What type of digital applications can benefit from this ultra low power design approach?

11. Subthreshold Conduction
Below cut off current does not abruptly become zero
Falls off exponentially
Useful in low power CMOS VLSI design

12. Junction Leakage Conduction even when transistor is in cut-off
Substrate to diffusion junctions are reverse biased
However reverse biased diodes do conduct leakage current

13. Leakge Current
When the junction bias voltage is significantly more than the thermal voltage temperature) the leakage current is –Is
Junction leakage limits storage time in on-chip memory elements
Requires refreshing dynamic nodes

14. Tunneling effects Ideal MOS model Quantum mechanical effect
High input impedence
No static current flow through the gate terminal
Quantum mechanical effect
Carriers “tunnel” through insulating barriers with finite probability
Insulating barrier has to be very thin for appreciable current
Current gate oxide thickness ~10-15Å
Single atomic layer of silicon ~3Å

15. Tunneling current Current technology nodes
Tunneling current as significant as junction leakage and sub-threshold conduction
Technique to reduce tunneling current
Use high-K materials in the gate oxide layer
High dielectric constant makes high gate capacitance
Reduces the need to reduce the oxide thickness
Silicon Nitride is a good candidate for such materials

16. Temperature Effects Effect on Mobility
Carrier mobility decreases with temperature
kµ is a parameter usually in the range

17. Temperature effects Threshold voltage
Vt decreases linearly with increase in temperature
Junction leakage also increases with increase in temperature
All combined results in decrease of On current and increase of Off current

18. Temperature Issues
Circuit performance is therefore generally worse at high temperatures
Conversely cooling can enable better performance
Cooling techniques
Convection
Natural
Fans
Heat sinks
Active cooling
Water cooling
Liquid nitrogen
Cost of methods have to be justified

19. Non-Ideal I-V Effects (Summary)
Miniaturization has led to modern devices having nonideal characteristics
The saturation current increases less than quadratically with increasing VGS.
Velocity saturation and mobility degradation are two of the effects that cause the non quadratic current increase with VGS.
When carrier velocity ceases to increase linearly with field strength we have velocity saturation.
The current IDS is lower than expected at high VDS.
There are several sources of leakage that result in current flow when the transistor is expected to be OFF.
The source and drain diffusion regions are form reverse biased diodes which experience junction leakage into the substrate or well.
The current into the gate IG is ideal zero, however as gate oxide thickness is reduced electrons tunnel through the gate, causing some current.