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<span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_with_Moore's_law_and_computing_performance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_with_Moore's_law_and_computing_performance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relation with Moore's law and computing performance</span> </div> </a> <ul id="toc-Relation_with_Moore's_law_and_computing_performance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Breakdown_of_Dennard_scaling_around_2006" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Breakdown_of_Dennard_scaling_around_2006"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Breakdown of Dennard scaling around 2006</span> </div> </a> <ul id="toc-Breakdown_of_Dennard_scaling_around_2006-sublist" 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">As transistors become smaller, their power density remains constant</div> <p>In <a href="/wiki/Semiconductor_electronics" class="mw-redirect" title="Semiconductor electronics">semiconductor electronics</a>, <b>Dennard scaling</b>, also known as <b>MOSFET scaling</b>, is a <a href="/wiki/Scaling_law" class="mw-redirect" title="Scaling law">scaling law</a> which states roughly that, as <a href="/wiki/Transistor" title="Transistor">transistors</a> get smaller, their <a href="/wiki/Power_density" title="Power density">power density</a> stays constant, so that the power use stays in proportion with area; both <a href="/wiki/Voltage" title="Voltage">voltage</a> and <a href="/wiki/Electric_current" title="Electric current">current</a> scale (downward) with length.<sup id="cite_ref-cartesian_1-0" class="reference"><a href="#cite_note-cartesian-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The law, originally formulated for <a href="/wiki/MOSFET" title="MOSFET">MOSFETs</a>, is based on a 1974 paper co-authored by <a href="/wiki/Robert_H._Dennard" title="Robert H. Dennard">Robert H. Dennard</a>, after whom it is named.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dennard_scaling&action=edit&section=1" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For long MOS transistors (i.e. one side is significantly longer than the other two), with constant electric field inside the MOS, Dennard scaling gives<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\propto S^{-1},W\propto S^{-1},t_{ox}\propto S^{-1},V_{\mathrm {DD} }\propto S^{-1},V_{\mathrm {T} }\propto S^{-1},N_{A}\propto S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mi>W</mi> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>∝</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\propto S^{-1},W\propto S^{-1},t_{ox}\propto S^{-1},V_{\mathrm {DD} }\propto S^{-1},V_{\mathrm {T} }\propto S^{-1},N_{A}\propto S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b9e30bf026904dcfd2da304c4723148e2cac1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:61.563ex; height:3.009ex;" alt="{\displaystyle L\propto S^{-1},W\propto S^{-1},t_{ox}\propto S^{-1},V_{\mathrm {DD} }\propto S^{-1},V_{\mathrm {T} }\propto S^{-1},N_{A}\propto S}"></span> </p><p>where parameters are scaled by a factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. </p> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Property</th> <th>Symbol</th> <th>Equation</th> <th>Scaling exponent (constant field) </th> <th>Scaling exponent (fixed voltage) </th></tr> <tr> <td>Oxide Capacitance </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ox}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ox}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237dfd6b593a728563dac74e79ceeeb0dbe78457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.632ex; height:2.509ex;" alt="{\displaystyle C_{ox}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ox}/t_{ox}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ox}/t_{ox}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d68c3cbb4fa6965c4a9dd554f3d82306a0e7581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.025ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{ox}/t_{ox}}"></span> </td> <td>1 </td> <td>1 </td></tr> <tr> <td>Device Area </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\cdot L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⋅</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\cdot L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9a671c7fe1b8ca8a41b17532d5dc049b7fbbe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.697ex; height:2.176ex;" alt="{\displaystyle W\cdot L}"></span> </td> <td>-2 </td> <td>-2 </td></tr> <tr> <td>Gate Capacitance </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ceaffa7f3ca93b56b4bb68a621bf5bf573f6b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.683ex; height:2.843ex;" alt="{\displaystyle C_{g}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ox}\cdot W\cdot L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> <mo>⋅</mo> <mi>W</mi> <mo>⋅</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ox}\cdot W\cdot L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9795ea4179e7dfba78c1de5a0247578fb4eebce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.008ex; height:2.509ex;" alt="{\displaystyle C_{ox}\cdot W\cdot L}"></span> </td> <td>-1 </td> <td>-1 </td></tr> <tr> <td>Transconductance </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2b988ea630d2c5571afe47efa3d3b251708acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.192ex; height:2.509ex;" alt="{\displaystyle K_{n}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}C_{ox}W/L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}C_{ox}W/L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9ae3d953fa929054b1cc049fb38f60b5832482" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:2.843ex;" alt="{\displaystyle \mu _{n}C_{ox}W/L}"></span> </td> <td>1 </td> <td>1 </td></tr> <tr> <td>Saturation Current </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{on}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{on}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce092636f8919564149328799dce5bff9921a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.039ex; height:2.509ex;" alt="{\displaystyle I_{on}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}V_{GT}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}V_{GT}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf0102d42d68ef00985088f9f718b4a0e5b29b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.228ex; height:3.176ex;" alt="{\displaystyle K_{n}V_{GT}^{2}}"></span> </td> <td>-1 </td> <td>1 </td></tr> <tr> <td>On Resistance </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{on}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{on}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effa57a9fb439d1181f220f5c33cac022b46439b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.78ex; height:2.509ex;" alt="{\displaystyle R_{on}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{DD}/I_{on}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{DD}/I_{on}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/342acd2a21efb6f937632023145d36f1c2575823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.51ex; height:2.843ex;" alt="{\displaystyle V_{DD}/I_{on}}"></span> </td> <td>0 </td> <td>-1 </td></tr> <tr> <td>Intrinsic Delay </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{pd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{pd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d8661082e350059f4d3d47311cc7cd6a2a13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.759ex; height:2.676ex;" alt="{\displaystyle t_{pd}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{on}C_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{on}C_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db740b2de10c99e289a8574fb845d650a28932de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.463ex; height:2.843ex;" alt="{\displaystyle R_{on}C_{g}}"></span> </td> <td>-1 </td> <td>-2 </td></tr> <tr> <td>Power </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{av}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{av}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa8fe67f4c4c5ec1525a17a1c21f47f4cb3663d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.391ex; height:2.509ex;" alt="{\displaystyle P_{av}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\cdot C\cdot V_{DD}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>⋅</mo> <mi>C</mi> <mo>⋅</mo> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\cdot C\cdot V_{DD}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f7bf5910623c977e5fb0d01c2f67595b31f3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.712ex; height:3.176ex;" alt="{\displaystyle f\cdot C\cdot V_{DD}^{2}}"></span> </td> <td>-2 </td> <td>1 </td></tr> <tr> <td>Power Density </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25caf35e7f0024ee63d61e86295ecdb1f3739d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.67ex; height:2.176ex;" alt="{\displaystyle PD}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{av}/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>v</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{av}/A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcce26e049c45e166e01f58526f3c1f4c5d392c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.297ex; height:2.843ex;" alt="{\displaystyle P_{av}/A}"></span> </td> <td>0 </td> <td>3 </td></tr></tbody></table> <p>Explanation of symbols: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>: Scaling factor: a factor by which all the device dimensions and voltages are scaled down.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>: Area of the transistor.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>: Width of the transistor channel.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>: Length of the transistor channel.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ox}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{ox}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237dfd6b593a728563dac74e79ceeeb0dbe78457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.632ex; height:2.509ex;" alt="{\displaystyle C_{ox}}"></span>: Oxide capacitance: the capacitance per unit area of the gate dielectric layer (the oxide layer).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ox}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ox}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7af96f82e6bbde68546d473ce7546aac936852a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.053ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{ox}}"></span>: Permittivity of the oxide layer: a measure of how well the oxide layer can store electrical energy.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{ox}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{ox}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1775d439ed71f8b90908c64a6fd2270e5ae90e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.809ex; height:2.343ex;" alt="{\displaystyle t_{ox}}"></span>: Thickness of the oxide layer.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ceaffa7f3ca93b56b4bb68a621bf5bf573f6b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.683ex; height:2.843ex;" alt="{\displaystyle C_{g}}"></span>: Total capacitance of the gate electrode.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2b988ea630d2c5571afe47efa3d3b251708acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.192ex; height:2.509ex;" alt="{\displaystyle K_{n}}"></span>: Transconductance parameter: a measure of how much the drain current changes in response to a change in the gate voltage.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267d03f9351dcc8d3d3ac7cad59ea3ba4fecbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.62ex; height:2.176ex;" alt="{\displaystyle \mu _{n}}"></span>: Electron mobility in the channel: a measure of how easily electrons can move through the channel.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{on}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{on}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce092636f8919564149328799dce5bff9921a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.039ex; height:2.509ex;" alt="{\displaystyle I_{on}}"></span>: Saturation current: the maximum current that can flow through the transistor when it is turned on.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{GT}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{GT}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2811e48074853cb2144e225af04f0fdb5ff61f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.036ex; height:2.509ex;" alt="{\displaystyle V_{GT}}"></span>: Gate overdrive voltage: the difference between the gate voltage and the threshold voltage.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{on}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{on}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effa57a9fb439d1181f220f5c33cac022b46439b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.78ex; height:2.509ex;" alt="{\displaystyle R_{on}}"></span>: Resistance of the transistor when it is turned on.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{DD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{DD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58134c1a0a1c794cc6ae8d8d1866cd5138010ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.309ex; height:2.509ex;" alt="{\displaystyle V_{DD}}"></span>: Supply voltage: the voltage that is applied to the transistor.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{pd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{pd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d8661082e350059f4d3d47311cc7cd6a2a13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.759ex; height:2.676ex;" alt="{\displaystyle t_{pd}}"></span>: Intrinsic delay: the time it takes for the transistor to switch from on to off or vice versa.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{av}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{av}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa8fe67f4c4c5ec1525a17a1c21f47f4cb3663d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.391ex; height:2.509ex;" alt="{\displaystyle P_{av}}"></span>: Average power consumption: the average amount of power that the transistor consumes.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: Operating frequency: the frequency at which the transistor is switching.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25caf35e7f0024ee63d61e86295ecdb1f3739d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.67ex; height:2.176ex;" alt="{\displaystyle PD}"></span>: Power density: power consumption per unit area.</li></ul> <p>In fixed voltage scaling, the supply voltage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{DD}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>D</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{DD}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58134c1a0a1c794cc6ae8d8d1866cd5138010ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.309ex; height:2.509ex;" alt="{\displaystyle V_{DD}}"></span> is held constant (at ~5V) instead of scaling like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {DD} }\propto S^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {DD} }\propto S^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812f0ed143304944556c19166aa9004db1a74468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.051ex; height:3.009ex;" alt="{\displaystyle V_{\mathrm {DD} }\propto S^{-1}}"></span>. This results in different scaling exponents. The clock frequency grows faster at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}"></span> instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"></span>, but at the price of rapidly increasing power density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PD\propto S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>D</mi> <mo>∝</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PD\propto S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd77304dc0aa4c51fc135522fd19aa0dc4dd250c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.344ex; height:2.676ex;" alt="{\displaystyle PD\propto S^{3}}"></span>. </p><p>Fixed voltage scaling was the common scaling regime which ended around 2005 at the "power wall", when it was too difficult to keep the chip cool. Furthermore, at constant supply voltage, the field grows like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"></span>, and the off-current grows <i>exponentially</i> with the field, resulting in high static power consumption since the 90 nm node. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dennard_scaling&action=edit&section=2" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dennard's model of MOSFET scaling implies that, with every technology generation: </p> <ol><li>Transistor dimensions could be scaled by −30% (0.7×). This has the following effects simultaneously: <ul><li>The area of an individual device reduces by 51%, because <a href="/wiki/Area#Rectangles" title="Area">area is length times width</a>.</li> <li>The capacitance associated with the device, C, is reduced by 30% (0.7×), because <a href="/wiki/Capacitance#Capacitors" title="Capacitance">capacitance varies with area over distance</a>.</li> <li>To keep the electric field unchanged, the voltage, V, is reduced by 30% (0.7×), because <a href="/wiki/Electric_field#Uniform_fields" title="Electric field">voltage is field times length</a>.</li> <li>Characteristics such as current and transition time are likewise scaled down by 30%, due to their <a href="/wiki/Capacitance#Mutual_capacitance" title="Capacitance">relationship with capacitance and voltage</a>.</li> <li>Overall circuit delay is assumed to be dominated by transition time, so it too is reduced by 30%.</li> <li>Frequency f can increase by about 40% (1.4×), because <a href="/wiki/Frequency#Definitions_and_units" title="Frequency">frequency varies with one over delay</a>.</li></ul></li> <li>Power consumption of an individual transistor decreases by 51%, because <a href="/wiki/Active_power" class="mw-redirect" title="Active power">active power</a> is CV<sup>2</sup>f.<sup id="cite_ref-Borkar_5-0" class="reference"><a href="#cite_note-Borkar-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>As a result, power consumption per unit area remains the same for every technology generation. Alternatively, with every generation the number of transistors in a chip can be doubled with no change in power consumption.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Relation_with_Moore's_law_and_computing_performance"><span id="Relation_with_Moore.27s_law_and_computing_performance"></span>Relation with Moore's law and computing performance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dennard_scaling&action=edit&section=3" title="Edit section: Relation with Moore's law and computing performance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Moore%27s_law" title="Moore's law">Moore's law</a> says that the number of transistors on a microchip doubles approximately every two years. Combined with Dennard scaling, this means that <a href="/w/index.php?title=Performance_per_joule&action=edit&redlink=1" class="new" title="Performance per joule (page does not exist)">performance per joule</a> grows even faster, doubling about every 18 months (1.5 years). This trend is sometimes referred to as <a href="/wiki/Koomey%27s_law" title="Koomey's law">Koomey's law</a>. The rate of doubling was originally suggested by Koomey to be 1.57 years,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> but more recent estimates suggest this is slowing.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Breakdown_of_Dennard_scaling_around_2006">Breakdown of Dennard scaling around 2006</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dennard_scaling&action=edit&section=4" title="Edit section: Breakdown of Dennard scaling around 2006"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/MOSFET#Scaling" title="MOSFET">MOSFET § Scaling</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CPU_clock_speed_and_Core_count_Graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/CPU_clock_speed_and_Core_count_Graph.png/220px-CPU_clock_speed_and_Core_count_Graph.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/CPU_clock_speed_and_Core_count_Graph.png/330px-CPU_clock_speed_and_Core_count_Graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/CPU_clock_speed_and_Core_count_Graph.png/440px-CPU_clock_speed_and_Core_count_Graph.png 2x" data-file-width="3400" data-file-height="1900" /></a><figcaption>Microprocessor clock speed measures the number of pulses per second generated by an oscillator that sets the tempo for the processor. It is measured in hertz (pulses per second). The power wall is visible.</figcaption></figure> <p>The dynamic (switching) power consumption of CMOS circuits is proportional to frequency.<sup id="cite_ref-cmospower_8-0" class="reference"><a href="#cite_note-cmospower-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Historically, the transistor power reduction afforded by Dennard scaling allowed manufacturers to drastically raise clock frequencies from one generation to the next without significantly increasing overall circuit power consumption. </p><p>Specifically, leakage current and threshold voltage do not scale with size, and so the power density increases with scaling. This eventually led to a power density that is too high. This is the "power wall", which caused Intel to cancel <a href="/wiki/Tejas_and_Jayhawk" title="Tejas and Jayhawk">Tejas and Jayhawk</a> in 2004.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Since around 2005–2007 Dennard scaling appears to have broken down. As of 2016, transistor counts in integrated circuits are still growing, but the resulting improvements in performance are more gradual than the speed-ups resulting from significant frequency increases.<sup id="cite_ref-cartesian_1-1" class="reference"><a href="#cite_note-cartesian-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-retrospective_10-0" class="reference"><a href="#cite_note-retrospective-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The primary reason cited for the breakdown is that at small sizes, current leakage poses greater challenges and also causes the chip to heat up, which creates a threat of <a href="/wiki/Thermal_runaway" title="Thermal runaway">thermal runaway</a> and therefore further increases energy costs.<sup id="cite_ref-cartesian_1-2" class="reference"><a href="#cite_note-cartesian-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-retrospective_10-1" class="reference"><a href="#cite_note-retrospective-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Since 2005, the clock frequency has stagnated at 4 GHz, and the power consumption per CPU at 100 W <a href="/wiki/Thermal_design_power" title="Thermal design power">TDP</a>. </p><p>The breakdown of Dennard scaling and resulting inability to increase clock frequencies significantly has caused most CPU manufacturers to focus on <a href="/wiki/Multicore_processor" class="mw-redirect" title="Multicore processor">multicore processors</a> as an alternative way to improve performance. An increased core count benefits many (though by no means all – see <a href="/wiki/Amdahl%27s_law" title="Amdahl's law">Amdahl's law</a>) workloads, but the increase in active switching elements from having multiple cores still results in increased overall power consumption and thus worsens <a href="/wiki/CPU_power_dissipation" class="mw-redirect" title="CPU power dissipation">CPU power dissipation</a> issues.<sup id="cite_ref-Dark-2012_11-0" class="reference"><a href="#cite_note-Dark-2012-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The end result is that only some fraction of an integrated circuit can actually be active at any given point in time without violating power constraints. The remaining (inactive) area is referred to as <a href="/wiki/Dark_silicon" title="Dark silicon">dark silicon</a>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dennard_scaling&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-cartesian-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-cartesian_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cartesian_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-cartesian_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMcMenamin2013" class="citation web cs1">McMenamin, Adrian (April 15, 2013). <a rel="nofollow" class="external text" href="http://cartesianproduct.wordpress.com/2013/04/15/the-end-of-dennard-scaling/">"The end of Dennard scaling"</a><span class="reference-accessdate">. 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Retrieved <span class="nowrap">March 9,</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=CMOS+Power+Consumption+and+CPD+Calculation&rft.pub=Texas+Instruments&rft.date=1997-06&rft_id=http%3A%2F%2Fwww.ti.com%2Flit%2Fan%2Fscaa035b%2Fscaa035b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADennard+scaling" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://wgropp.cs.illinois.edu/courses/cs598-s15/lectures/lecture15.pdf">https://wgropp.cs.illinois.edu/courses/cs598-s15/lectures/lecture15.pdf</a></span> </li> <li id="cite_note-retrospective-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-retrospective_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-retrospective_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohr2007" class="citation web cs1">Bohr, Mark (January 2007). <a rel="nofollow" class="external text" href="http://www.eng.auburn.edu/~agrawvd/COURSE/READING/LOWP/Boh07.pdf">"A 30 Year Retrospective on Dennard's MOSFET Scaling Paper"</a> <span class="cs1-format">(PDF)</span>. Solid-State Circuits Society<span class="reference-accessdate">. Retrieved <span class="nowrap">January 23,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+30+Year+Retrospective+on+Dennard%27s+MOSFET+Scaling+Paper&rft.pub=Solid-State+Circuits+Society&rft.date=2007-01&rft.aulast=Bohr&rft.aufirst=Mark&rft_id=http%3A%2F%2Fwww.eng.auburn.edu%2F~agrawvd%2FCOURSE%2FREADING%2FLOWP%2FBoh07.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADennard+scaling" class="Z3988"></span></span> </li> <li id="cite_note-Dark-2012-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dark-2012_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEsmaeilzadehBlemSt._AmantSankaralingam2011" class="citation book cs1">Esmaeilzadeh, Hadi; Blem, Emily; St. Amant, Renee; Sankaralingam, Karthikeyan; Burger, Doug (2011). "Dark silicon and the end of multicore scaling". <i>2011 38th Annual International Symposium on Computer Architecture (ISCA)</i>. IEEE. pp. <span class="nowrap">365–</span>376. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.222.8988">10.1.1.222.8988</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2000064.2000108">10.1145/2000064.2000108</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4503-0472-6" title="Special:BookSources/978-1-4503-0472-6"><bdi>978-1-4503-0472-6</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207188742">207188742</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dark+silicon+and+the+end+of+multicore+scaling&rft.btitle=2011+38th+Annual+International+Symposium+on+Computer+Architecture+%28ISCA%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E365-%3C%2Fspan%3E376&rft.pub=IEEE&rft.date=2011&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.222.8988%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207188742%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F2000064.2000108&rft.isbn=978-1-4503-0472-6&rft.aulast=Esmaeilzadeh&rft.aufirst=Hadi&rft.au=Blem%2C+Emily&rft.au=St.+Amant%2C+Renee&rft.au=Sankaralingam%2C+Karthikeyan&rft.au=Burger%2C+Doug&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADennard+scaling" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHruska2012" class="citation web cs1">Hruska, Joel (February 1, 2012). <a rel="nofollow" class="external text" href="http://www.extremetech.com/computing/116561-the-death-of-cpu-scaling-from-one-core-to-many-and-why-were-still-stuck">"The death of CPU scaling: From one core to many – and why we're still stuck"</a>. <a href="/wiki/ExtremeTech" title="ExtremeTech">ExtremeTech</a><span class="reference-accessdate">. Retrieved <span class="nowrap">January 23,</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+death+of+CPU+scaling%3A+From+one+core+to+many+%E2%80%93+and+why+we%27re+still+stuck&rft.pub=ExtremeTech&rft.date=2012-02-01&rft.aulast=Hruska&rft.aufirst=Joel&rft_id=http%3A%2F%2Fwww.extremetech.com%2Fcomputing%2F116561-the-death-of-cpu-scaling-from-one-core-to-many-and-why-were-still-stuck&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADennard+scaling" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeadConway1980" class="citation book cs1">Mead, Carver; Conway, Lynn (1980). <i>Introduction to VLSI systems</i>. Reading, Mass: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-04358-7" title="Special:BookSources/978-0-201-04358-7"><bdi>978-0-201-04358-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+VLSI+systems&rft.place=Reading%2C+Mass&rft.pub=Addison-Wesley&rft.date=1980&rft.isbn=978-0-201-04358-7&rft.aulast=Mead&rft.aufirst=Carver&rft.au=Conway%2C+Lynn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADennard+scaling" class="Z3988"></span>, especially Section 1.16.</li></ul> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111203053340/http://www.ieee.org/portal/cms_docs_societies/sscs/PrintEditions/200701.pdf">“The Impact of Dennard’s Scaling Theory”, SSCS Magazine, Winter 2007</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dennard_scaling&oldid=1272170848">https://en.wikipedia.org/w/index.php?title=Dennard_scaling&oldid=1272170848</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Computer_architecture_statements" title="Category:Computer architecture statements">Computer architecture statements</a></li><li><a href="/wiki/Category:Computer-related_introductions_in_1974" title="Category:Computer-related introductions in 1974">Computer-related introductions in 1974</a></li><li><a href="/wiki/Category:Computing_culture" title="Category:Computing culture">Computing culture</a></li><li><a href="/wiki/Category:Electronic_design" title="Category:Electronic design">Electronic design</a></li><li><a href="/wiki/Category:MOSFETs" title="Category:MOSFETs">MOSFETs</a></li><li><a href="/wiki/Category:Rules_of_thumb" title="Category:Rules of thumb">Rules of thumb</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 January 2025, at 12:33<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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