Calculate the flat-band voltage for a MOS capacitor with a p-type semiconductor substrate.

Consider a MOS capacitor with a p-type silicon substrate doped to $N_{a}=10^{16} \mathrm{~cm}^{-3}$ , a silicon dioxide insulator with a thickness of $t_{o x}=20 \mathrm{~nm}=200 \mathring{A}$ , and an $\mathrm{n}^{+}$ polysilicon gate. Assume that $Q_{s s}^{\prime}=5 \times 10^{10}$ electronic charges per $\mathrm{cm}^{2}$ .

Question

Calculate the flat-band voltage for a MOS capacitor with a p-type semiconductor substrate.

Consider a MOS capacitor with a p-type silicon substrate doped to [latex]N_{a}=10^{16} \mathrm{~cm}^{-3}[/latex], a silicon dioxide insulator with a thickness of [latex]t_{o x}=20 \mathrm{~nm}=200 \mathring{A} [/latex], and an [latex]\mathrm{n}^{+}[/latex]polysilicon gate. Assume that [latex]Q_{s s}^{\prime}=5 imes 10^{10}[/latex] electronic charges per [latex]\mathrm{cm}^{2}[/latex].

Accepted Answer

The work function difference, from Figure [latex]10.16[/latex], is [latex]\phi_{m s} \cong-1.1 \mathrm{~V}[/latex]. The oxide capacitance is found to be

[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]

The equivalent oxide charge density is

[latex]Q_{s s}^{\prime}=\left(5 imes 10^{10} ight)\left(1.6 imes 10^{-19} ight)=8 imes 10^{-9} \mathrm{C} / \mathrm{cm}^{2}[/latex]

The flat-band voltage is then determined to be

[latex]V_{F B}=\phi_{m s}-\frac{Q_{s s}^{\prime}}{C_{o x}}=-1.1 \frac{-8 imes 10^{-9}}{1.726 imes 10^{-7}}=-1.15 \mathrm{~V}[/latex]

[latex]{ }^{1}[/latex] Although we will, in general, use the primed notation for capacitance per unit area or charge per unit area, we will omit, for convenience, the prime on the oxide capacitance per unit area parameter.

Comment

The applied gate voltage required to achieve the flat-band condition for this p-type substrate is negative. If the amount of fixed oxide charge increases, the flat-band voltage becomes even more negative.

Question 10.3: Calculate the flat-band voltage for a MOS capacitor with a p...

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