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

<table>The drift <svg height="14.375" id="M341" style="vertical-align:-3.24037pt" version="1.1" viewbox="0 0 14.725 14.375" width="14.725" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.375)">
<g transform="translate(72,-60.5)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑑</tspan>
</text>
</g>
</g>
</svg> (–-) and diffusion <svg height="14.35" id="M342" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 23.375 14.35" width="23.375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.35)">
<g transform="translate(72,-60.52)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">d</tspan>
<tspan style="font-size: 8.75px; " x="4.3759999" y="0">i</tspan>
<tspan style="font-size: 8.75px; " x="6.8090558" y="0">ﬀ</tspan>
</text>
</g>
</g>
</svg> (....) currents  versus <svg height="14.2375" id="M343" 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> in log scale. <svg height="14.375" id="M344" style="vertical-align:-3.24037pt" version="1.1" viewbox="0 0 19.5375 14.375" width="19.5375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.375)">
<g transform="translate(72,-60.5)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑑</tspan>
<tspan style="font-size: 8.75px; " x="4.656064" y="0">1</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.35" id="M345" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 28.85 14.35" width="28.85" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.35)">
<g transform="translate(72,-60.52)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">d</tspan>
<tspan style="font-size: 8.75px; " x="4.3759999" y="0">i</tspan>
<tspan style="font-size: 8.75px; " x="6.8090558" y="0">ﬀ</tspan>
</text>
<text transform="matrix(1,0,0,-1,-53.84,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
</g>
</g>
</svg> strong inversion: <svg height="14.375" id="M346" style="vertical-align:-3.24037pt" version="1.1" viewbox="0 0 19.5375 14.375" width="19.5375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.375)">
<g transform="translate(72,-60.5)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑑</tspan>
<tspan style="font-size: 8.75px; " x="4.656064" y="0">2</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.35" id="M347" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 28.85 14.35" width="28.85" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.35)">
<g transform="translate(72,-60.52)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐼</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.95,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">d</tspan>
<tspan style="font-size: 8.75px; " x="4.3759999" y="0">i</tspan>
<tspan style="font-size: 8.75px; " x="6.8090558" y="0">ﬀ</tspan>
</text>
<text transform="matrix(1,0,0,-1,-53.84,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> weak inversion. <svg height="18" id="M348" style="vertical-align:-3.22281pt" version="1.1" viewbox="0 0 161.5 18" width="161.5" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,18)">
<g transform="translate(72,-57.6)">
<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>
<text transform="matrix(1,0,0,-1,10.96,60.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,20.08,57.74)">
<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,-1,32.61,60.86)">
<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<svg height="3.925" id="M349" style="vertical-align:-1.76814pt" version="1.1" viewbox="0 0 4.0374999 3.925" width="4.0374999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,3.925)">
<g transform="translate(72,-68.86)">
<text transform="matrix(1,0,0,-1,-71.95,70.67)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
</text>
</g>
</g>
</svg> and <svg height="10.9375" id="M350" style="vertical-align:-0.20064pt" version="1.1" viewbox="0 0 68.775002 10.9375" width="68.775002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9375)">
<g transform="translate(72,-63.25)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑊</tspan>
<tspan style="font-size: 12.50px; " x="11.452748" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="14.928582" y="0">𝐿</tspan>
<tspan style="font-size: 12.50px; " x="27.244038" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="39.284428" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="45.535927" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="51.787426" y="0">.</tspan>
</text>
</g>
</g>
</svg></table>

Active and Passive Electronic Components

fig12

Figure 12

Figure 12: Taylor Expansion of Surface Potential in MOSFET: Application to Pao-Sah Integral 