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SSPD_Chapter 7_Part3_Electrical properties of MOS continued2
7.3.4. Theoretical Formulation of Threshold Voltage in (E)NMOS.
In SSPD_Chapter 6_Part 4_concluded_MOS and its Physics we have dealt with the theoretical formulation of the Threshold Voltage. The expression comes as follows:
7.3.18
Where The Flat Band Voltage is = V _{FB} = ϕm – ϕs = ϕm – [χ + E _{g} /(2q) + ψ _{B} ]
χ = electron affinity in semi-conductor, E _{g} = energy band-gap of Silicon and ψ _{B} = potemtial diiference between Fermi-level(Ef) and intrinsic level(Ei) in Si-Bulk.
Therefore qV _{FB} = qϕm – qϕs = qϕm – [qχ + E _{g} /(2) + qψ _{B} ];
If source to bulk voltage (V _{SB} ) is not zero then this also has to be accounted. This is called Body Effect.
Q _{SS} = Q _{it} (interface trapped charges) + Q _{ot} (oxide trapped charges) + Q _{m} (mobile charges in the oxide) + Q _{f} (fixed oxide charges).
Generally substrate and source are connected together and threshold voltage is at the minimum.
But in certain applications, substrate is kept at reverse bias with respect to source. NMOS is built in P tub and PMOS is built in N tub.
Hence in NMOS, substrate is at negative bias with respect to source and in PMOS, substrate is at positive bias with respect to source.
Substrate bias gives rise to increase in threshold voltage. Hence Body-effect is represented by the following empirical expression:
7.3.17
V _{TO} = Threshold voltage with zero substrate bias;
And γ = body effect parameter (√V);
2ψ _{B} = surface potential parameter.
Lower is the substrate doping, lower will be body effect.
The Typical values of Body effect is given in Table 7.3.1.
Table 7.3.1. Body effect in (E)NMOS and (D)NMOS
Device | Substrate biasV _{SB} | ThresholdV _{Th} | |
(E)NMOS | 0V | 0.2V _{DD} | 1V for V _{DD} = +5V |
5V | 0.3V _{DD} | 1.5V for V _{DD} = +5V | |
(D)NMOS | 0V | -0.7V _{DD} | -3.5V for V _{DD} = +5V |
5V | -0.6V _{DD} | -3V for V _{DD} = +5V |
7.3.5. Theoretical formulation of Transconductance (g m )
Transconductance is the partial derivative of output current (I _{ds} ) with respect to input voltage (V _{gs} ) with output voltage constant i.e. V _{DS} is held constant.
7.3.18
From Eq.7.3.3,
And Transit Time is given by equation 7.3.7,
Using these two equations incremental change in output current:
7.3.19
But channel charge induced depends on gate capacitance and gate voltage:
Therefore
Hence
7.3.21
In saturation from Eq.7.3.15,
.
is the actual drop across the conical channel no matter what V _{ds} is.
Hence in saturatin region:
( 7.3.22
From Eq.7.3.13 ,
Substituting this in Eq.7.3.22:
( = (
Or
Or
7.3.23Where
From Eq.7.3.23 it is evident that transconductance can be improved by reducing Channel Length.
Transconductance can be improved by increasing Channel Width. But both these methods have their drawbacks.
7.3.6. Theoretical Formulation of unity current gain band-width.
In BJT unity current gain band-width is defined as transit frequency ω _{T} . It is derived by determining the Current Gain Band Width Product.
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