Calculate the body-effect coefficient and the change in the threshold voltage due to an applied source-to-body voltage.

Consider an n-channel silicon MOSFET at $T=300 \mathrm{~K}$ . Assume the substrate is doped to $N_{a}=3 \times 10^{16} \mathrm{~cm}^{-3}$ and assume the oxide is silicon dioxide with a thickness of $t_{o x}=20 \mathrm{~nm}=200 \mathring{A}$ . Let $V_{S B}=1 \mathrm{~V}$ .

Question

Calculate the body-effect coefficient and the change in the threshold voltage due to an applied source-to-body voltage.

Consider an n-channel silicon MOSFET at [latex]T=300 \mathrm{~K}[/latex]. Assume the substrate is doped to [latex]N_{a}=3 imes 10^{16} \mathrm{~cm}^{-3}[/latex] and assume the oxide is silicon dioxide with a thickness of [latex]t_{o x}=20 \mathrm{~nm}=200 \mathring{A} [/latex]. Let [latex]V_{S B}=1 \mathrm{~V}[/latex].

Accepted Answer

We can calculate that

[latex]\phi_{f p}=V_{t} \ln \left(\frac{N_{a}}{n_{i}} ight)=(0.0259) \ln \left(\frac{3 imes 10^{16}}{1.5 imes 10^{10}} ight)=0.3758 \mathrm{~V}[/latex]

and

[latex]C_{o x}=\frac{\boldsymbol{\epsilon}_{o x}}{t_{o x}}=\frac{(3.9)\left(8.85 imes 10^{-14} ight)}{200 imes 10^{-8}}=1.726 imes 10^{-7} \mathrm{~F} / \mathrm{cm}^{2}[/latex]

From Equation (10.82), we find the body-effect coefficient to be

[latex]\begin{aligned}\gamma=& \frac{\sqrt{2 e \epsilon_{s} N_{a}}}{C_{o x}} \ \end{aligned}[/latex] (10.82)

[latex]\begin{aligned}\gamma &=\frac{\sqrt{2 e \epsilon_{s} N_{a}}}{C_{o x}}=\frac{\left[2\left(1.6 imes 10^{-19} ight)(11.7)\left(8.85 imes 10^{-14} ight)\left(3 imes 10^{16} ight) ight]^{1 / 2}}{1.726 imes 10^{-7}}\end{aligned}[/latex]

or

[latex]\gamma=0.5776 \mathrm{~V}^{1 / 2}[/latex]

The change in threshold voltage for [latex]V_{S B}=1 \mathrm{~V}[/latex] is found to be

[latex]\begin{aligned}\Delta V_{T} &=\gamma\left[\sqrt{2 \phi_{f p}+V_{S B}}-\sqrt{2 \phi_{f p}} ight] \&=(0.5776)[\sqrt{2(0.3758)+1}-\sqrt{2(0.3758)}] \&=(0.5776)[1.3235-0.8669]=0.264 \mathrm{~V}\end{aligned}[/latex]

Comment

Figure [latex]10.51[/latex] shows plots of [latex]\sqrt{I_{D}( ext { sat })}[/latex] versus [latex]V_{G S}[/latex] for various applied values of [latex]V_{S B}[/latex]. The original threshold voltage is assumed to be [latex]V_{T O}=0.64 \mathrm{~V}[/latex].

Question 10.9: Calculate the body-effect coefficient and the change in the ...

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