Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral

<table>The first derivative <svg height="14.7125" id="M432" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 80.875 14.7125" width="80.875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan>
<tspan style="font-size: 12.50px; " x="6.6515961" y="0">𝑢</tspan>
</text>
<text transform="matrix(1,0,0,-1,-59.37,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑆</tspan>
</text>
<text transform="matrix(1,0,0,-1,-52.57,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑦</tspan>
<tspan style="font-size: 12.50px; " x="10.364987" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="14.528486" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="18.00432" y="0">𝑑</tspan>
<tspan style="font-size: 12.50px; " x="24.655916" y="0">𝜉</tspan>
<tspan style="font-size: 12.50px; " x="30.682362" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="34.84586" y="0">𝑦</tspan>
<tspan style="font-size: 12.50px; " x="41.047348" y="0">)</tspan>
</text>
</g>
</g>
</svg>  versus drain voltage <svg height="14.2375" id="M433" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 18.5375 14.2375" width="18.5375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.2375)">
<g transform="translate(72,-60.61)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑉</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.76,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝐷</tspan>
</text>
</g>
</g>
</svg> for different <svg height="13.25" id="M434" style="vertical-align:-2.34499pt" version="1.1" viewbox="0 0 17.424999 13.25" width="17.424999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.25)">
<g transform="translate(72,-61.4)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑉</tspan>
<tspan style="font-size: 12.50px; " x="7.1892252" y="0">𝑔</tspan>
</text>
</g>
</g>
</svg> bias. <svg height="17.887501" id="M435" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 103.7625 17.887501" width="103.7625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,17.8875)">
<g transform="translate(72,-57.69)">
<text transform="matrix(1,0,0,-1,-71.95,60.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
<text transform="matrix(1,0,0,-1,-61.31,57.74)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝐴</tspan>
</text>
<text transform="matrix(1,0,0,-1,-51.06,60.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.279387" y="0">0</tspan>
</text>
<text transform="matrix(1,0,0,-1,-26.52,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
<tspan style="font-size: 8.75px; " x="4.3759999" y="0">7</tspan>
</text>
<text transform="matrix(1,0,0,-1,-15.19,60.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">c</tspan>
<tspan style="font-size: 12.50px; " x="5.551332" y="0">m</tspan>
</text>
<text transform="matrix(1,0,0,-1,0.09,65.87)">
<tspan style="font-size: 8.75px; " x="0" y="0">−</tspan>
<tspan style="font-size: 8.75px; " x="5.99512" y="0">3</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.7125" id="M436" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 51.349998 14.7125" width="51.349998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-68.04,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">o</tspan>
<tspan style="font-size: 8.75px; " x="4.1922078" y="0">x</tspan>
</text>
<text transform="matrix(1,0,0,-1,-55.51,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.291889" y="0">0</tspan>
</text>
</g>
</g>
</svg> nm.</table>

Active and Passive Electronic Components

fig16

Figure 16

Figure 16: Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral 