Calculate Cox, C min′ , and C FB′ for a MOS capacitor. Consider a p-type silicon substrate at T = 300 K doped to Na = 10^16 cm^-3. The oxide is silicon dioxide with a thickness of tox = 18 nm = 180 Å, and the gate is aluminum.

Question

Calculate [latex]C_{o x}, C_{\min }^{\prime}[/latex], and [latex]C_{F B}^{\prime}[/latex] for a MOS capacitor.

Consider a p-type silicon substrate at [latex]T=300 \mathrm{~K}[/latex] doped to [latex]N_{a}=10^{16} \mathrm{~cm}^{-3}[/latex].

The oxide is silicon dioxide with a thickness of [latex]t_{o x}=18 \mathrm{~nm}=180 \mathring{A} [/latex], and the gate is aluminum.

Accepted Answer

The oxide capacitance is

[latex]C_{o x}=\frac{\epsilon_{o x}}{t_{o x}}=\frac{(3.9)\left(8.85 imes 10^{-14} ight)}{180 imes 10^{-8}}=1.9175 imes 10^{-7} \mathrm{~F} / \mathrm{cm}^{2}[/latex]

To find the minimum capacitance, we need to calculate

[latex]\phi_{f p}=V_{t} \ln \left(\frac{N_{a}}{n_{i}} ight)=(0.0259) \ln \left(\frac{10^{16}}{1.5 imes 10^{10}} ight)=0.3473 \mathrm{~V}[/latex]

and

[latex]\begin{aligned}x_{d T} &=\left\{\frac{4 \epsilon_{s} \phi_{f f}}{e N_{a}} ight\}^{1 / 2}=\left\{\frac{4(11.7)\left(8.85 imes 10^{-14} ight)(0.3473)}{\left(1.6 imes 10^{-19} ight)\left(10^{16} ight)} ight\}^{1 / 2} \ & \cong 0.30 imes 10^{-4} \mathrm{~cm}\end{aligned}[/latex]

Then

[latex]\begin{aligned} C_{ ext {min }}^{\prime} &=\frac{\epsilon_{o x}}{t_{o x}+\left(\frac{\epsilon_{o x}}{\epsilon_{s}} ight) x_{d T}}=\frac{(3.9)\left(8.85 imes 10^{-14} ight)}{180 imes 10^{-8}+\left(\frac{3.9}{11.7} ight)\left(0.30 imes 10^{-4} ight)} \ &=2.925 imes 10^{-8} \mathrm{~F} / \mathrm{cm}^{2} \end{aligned} [/latex]

We may note that

[latex]\frac{C_{\min }^{\prime}}{C_{o x}}=\frac{2.925 imes 10^{-8}}{1.9175 imes 10^{-7}}=0.1525[/latex]

The flat-band capacitance is

[latex]\begin{aligned}C_{F B}^{\prime} &=\frac{\epsilon_{o x}}{t_{o x}+\left(\frac{\epsilon_{o x}}{\epsilon_{s}} ight) \sqrt{\frac{V_{t} \epsilon_{s}}{e N_{a}}}} \ &=\frac{(3.9)\left(8.85 imes 10^{-14} ight)}{180 imes 10^{-8}+\left(\frac{3.9}{11.7} ight) \sqrt{\frac{(0.0259)(11.7)\left(8.85 imes 10^{-14} ight)}{\left(1.6 imes 10^{-19} ight)\left(10^{16} ight)}}} \ &=1.091 imes 10^{-7} \mathrm{~F} / \mathrm{cm}^{2} \end{aligned} [/latex]

We also note that

[latex] \frac{C_{F B}^{\prime}}{C_{o x}}=\frac{1.091 imes 10^{-7}}{1.9175 imes 10^{-7}}=0.569 [/latex]

Comment

The ratios of [latex]C_{\min }^{\prime} / C_{o x}[/latex] and [latex]C_{F B}^{\prime} / C_{o x}[/latex] are typical values obtained in [latex]C-V[/latex] plots.

Question 10.6: Calculate Cox, C min′ , and C FB′ for a MOS capacitor. Consi...

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