Summary: SSPD_Chapter 7_Part 3 is the continuation of the Electrical Properties of MOS. In this module theoretical formulation of threshold voltage, transconductance, unity gain BW, output incremental conductance and figure of merit of NMOS is derived.
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:
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:
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.
From Eq.7.3.3,
And Transit Time is given by equation 7.3.7,
Using these two equations incremental change in output current:
But channel charge induced depends on gate capacitance and gate voltage:
Therefore
Hence
In saturation from Eq.7.3.15,
Hence in saturatin region:
From Eq.7.3.13 ,
Substituting this in Eq.7.3.22:
Or
Or
Where
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.
Short Circuit Current Gain is derived as:
Where
Therefore
In exactly the same manner short circuit current gain of NMOS is derived and set to unity.
Where
In Eq.7.3.25, ω_{0} = 5 ω_{u} therefore, in working range of frequencies, Equation 7.3.25 can be simplified to:
If Equation 7.3.26 is equated to Unity then its corresponding Unity Gain Frequency is:
From Eq.7.3.27 it is evident that increase in transconductance gives a higher frequency range of opeartaion.
7.3.7. Theoretical Formulation of Output Conductance of (E)NMOS.
Theoretical expression of drain current in saturation region is given by Equation 7.3.16:
As is evident from the above Equation, I_{ds} has no dependence on V_{ds} and I_{ds}-V_{ds} family of curves are perfectly horizontal and parallel to one another. Horizontal I-V curve implies infinite output impedance of the active device. But in practice it is not so. Real MOS devices have slopes in their family of I-V curves and this slope becomes pronounced as we scale the devices for the different generation of Technology. This is known as Channel Length Modulation effect.
In BJT we have Base Width Modulation also known as Early Effect. This Early Effect is the cause of the slope in output family of curves of CB BJT and CE BJT. Due to these slopes we have h_{ob} and h_{oe} parameters in the two circuit configurations. Since Early Effect is more pronounced in CE BJT hence h_{oe} = 1/40kΩ is greater than h_{ob }= 1/2MΩ. Channel Length Modulation is analogous to Early Effect and its degradation of family of output curves of NMOS is clealrly brought out in Figure 7.3.2. The physics of this degradation is that we have assumed that after saturation, I_{ds}(sat) becomes constant at:
In Equation 7.3.2 it is assumed that with increase in V_{DS}, the voltage drop across the conical channel is constant at V_{DS}^{*} and excess voltage drops across the pinched off region. The second assumption is that the resistance of the conical section is constant at :
R_{0}/3 where R_{0} is the resistance of the parallelopied channel.
This assumption does not remain valid with the scaling of devices. As the the device is scaled, the variation in V_{ds} leads to significant change in the resistance of the conical channel because as pinched off region increases the axial length of the conical channel reduces and hence resistance offered decreases and I_{DS(sat)} increases with the increase in V_{DS}.
Channel Length Modulation is included in the saturated drain current in the following manner:
Where λ = channel length modulation parameter which is dependent on channel length L and its typical values are:
Therefore the partial derivative of I_{ds} with respect to V_{ds} with gate voltage constant gives the the reciprocal of the incremental output resistance of (E)NMOS:
The overall dependence of 1/r_{ds} is :
The incremental model of MOSFET incorporating transconductance and channel length modulation is given in Figure 7.3.1.
7.3.8. Figure of Merit of MOSFET.
The Unity Gain Bandwidth of MOSFET defines the Figure of Merit of MOSFET. Using Equation 7.3.27:
Transconductance is given as:
Or
Where
Gate Capacitance is given as:
Substituting Eq.7.3.23 and 7.3.13 in Eq.7.3.31 we get:
From Equation 7.3.7
Therefore Figure of Merit is the reciprocal of the transit time across the channel.
Larger is the electron mobility, better will be the Figure of Merit. Hence <100> orientation Si Substrate is chosen for fabrication of CMOS circuis. The mobility of electron and hole is always much larger in <100> orientation substrate than that in <111> orientation substrate. In Table 7.3.2 a comparative study of the electron mobility in <111> and <100> substrate is given.
Table 7.3.2. Comparative study of mobilities in <100> and <111>orientation substrate in thin channel and in bulk.
Bulk mobility<100> | 2D channel mobility<100> | 2D Channel mobility<111> | |
µ_{n} | 1250cm^{2}/(V-sec) | 650 cm^{2}/(V-sec) | 500 cm^{2}/(V-sec) |
µ_{p} | 480 cm^{2}/(V-sec) | 240 cm^{2}/(V-sec) | 216 cm^{2}/(V-sec) |
For fast CMOS circuits fabrications, <100> Si Substrate is the choice of material in Industries.