Electron mobility - Wikipedia

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<li id="toc-Drift_velocity_in_an_electric_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Drift_velocity_in_an_electric_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Drift velocity in an electric field</span> </div> </a> <ul id="toc-Drift_velocity_in_an_electric_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_and_units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_and_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definition and units</span> </div> </a> <ul id="toc-Definition_and_units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_current_density" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_current_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Relation to current density</span> </div> </a> <ul id="toc-Relation_to_current_density-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_conductivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_conductivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Relation to conductivity</span> </div> </a> <ul id="toc-Relation_to_conductivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_electron_diffusion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_electron_diffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Relation to electron diffusion</span> </div> </a> <ul id="toc-Relation_to_electron_diffusion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electric_field_dependence_and_velocity_saturation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Electric_field_dependence_and_velocity_saturation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Electric field dependence and velocity saturation</span> </div> </a> <ul id="toc-Electric_field_dependence_and_velocity_saturation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_scattering_and_mobility" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_between_scattering_and_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relation between scattering and mobility</span> </div> </a> <button aria-controls="toc-Relation_between_scattering_and_mobility-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation between scattering and mobility subsection</span> </button> <ul id="toc-Relation_between_scattering_and_mobility-sublist" class="vector-toc-list"> <li id="toc-Ionized_impurity_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ionized_impurity_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Ionized impurity scattering</span> </div> </a> <ul id="toc-Ionized_impurity_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_(phonon)_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lattice_(phonon)_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Lattice (phonon) scattering</span> </div> </a> <ul id="toc-Lattice_(phonon)_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Piezoelectric_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Piezoelectric_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Piezoelectric scattering</span> </div> </a> <ul id="toc-Piezoelectric_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_roughness_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_roughness_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Surface roughness scattering</span> </div> </a> <ul id="toc-Surface_roughness_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alloy_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alloy_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Alloy scattering</span> </div> </a> <ul id="toc-Alloy_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inelastic_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inelastic_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Inelastic scattering</span> </div> </a> <ul id="toc-Inelastic_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electron–electron_scattering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electron–electron_scattering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Electron–electron scattering</span> </div> </a> <ul id="toc-Electron–electron_scattering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_mobility_and_scattering_time" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_between_mobility_and_scattering_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Relation between mobility and scattering time</span> </div> </a> <ul id="toc-Relation_between_mobility_and_scattering_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matthiessen's_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matthiessen's_rule"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Matthiessen's rule</span> </div> </a> <ul id="toc-Matthiessen's_rule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Temperature_dependence_of_mobility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Temperature_dependence_of_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.10</span> <span>Temperature dependence of mobility</span> </div> </a> <ul id="toc-Temperature_dependence_of_mobility-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Disordered_Semiconductors" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Disordered_Semiconductors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Disordered Semiconductors</span> </div> </a> <button aria-controls="toc-Disordered_Semiconductors-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Disordered Semiconductors subsection</span> </button> <ul id="toc-Disordered_Semiconductors-sublist" class="vector-toc-list"> <li id="toc-Multiple_trapping_and_release" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_trapping_and_release"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Multiple trapping and release</span> </div> </a> <ul id="toc-Multiple_trapping_and_release-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variable_Range_Hopping" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variable_Range_Hopping"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Variable Range Hopping</span> </div> </a> <ul id="toc-Variable_Range_Hopping-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measurement_of_semiconductor_mobility" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Measurement_of_semiconductor_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Measurement of semiconductor mobility</span> </div> </a> <button aria-controls="toc-Measurement_of_semiconductor_mobility-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Measurement of semiconductor mobility subsection</span> </button> <ul id="toc-Measurement_of_semiconductor_mobility-sublist" class="vector-toc-list"> <li id="toc-Hall_mobility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hall_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Hall mobility</span> </div> </a> <ul id="toc-Hall_mobility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Field-effect_mobility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Field-effect_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Field-effect mobility</span> </div> </a> <ul id="toc-Field-effect_mobility-sublist" class="vector-toc-list"> <li id="toc-Using_saturation_mode" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Using_saturation_mode"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Using saturation mode</span> </div> </a> <ul id="toc-Using_saturation_mode-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_the_linear_region" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Using_the_linear_region"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Using the linear region</span> </div> </a> <ul id="toc-Using_the_linear_region-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Optical_mobility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Optical mobility</span> </div> </a> <ul id="toc-Optical_mobility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Terahertz_mobility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Terahertz_mobility"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Terahertz mobility</span> </div> </a> <ul id="toc-Terahertz_mobility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_resolved_microwave_conductivity_(TRMC)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_resolved_microwave_conductivity_(TRMC)"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Time resolved microwave conductivity (TRMC)</span> </div> </a> <ul id="toc-Time_resolved_microwave_conductivity_(TRMC)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Doping_concentration_dependence_in_heavily-doped_silicon" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Doping_concentration_dependence_in_heavily-doped_silicon"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Doping concentration dependence in heavily-doped silicon</span> </div> </a> <ul id="toc-Doping_concentration_dependence_in_heavily-doped_silicon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Electron mobility</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 17 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B1%D9%83%D9%8A%D8%A9_%D8%A7%D9%84%D8%A5%D9%84%D9%83%D8%AA%D8%B1%D9%88%D9%86" title="حركية الإلكترون – Arabic" lang="ar" hreflang="ar" data-title="حركية الإلكترون" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mobilitat_dels_electrons" title="Mobilitat dels electrons – Catalan" lang="ca" hreflang="ca" data-title="Mobilitat dels electrons" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D8%B1%DA%A9%E2%80%8C%D9%BE%D8%B0%DB%8C%D8%B1%DB%8C" title="تحرک‌پذیری – Persian" lang="fa" hreflang="fa" data-title="تحرک‌پذیری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EC%9E%90_%EC%9D%B4%EB%8F%99%EB%8F%84" title="전자 이동도 – Korean" lang="ko" hreflang="ko" data-title="전자 이동도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%AC%D5%A5%D5%AF%D5%BF%D6%80%D5%B8%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D6%87_%D5%AB%D5%B8%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%B7%D5%A1%D6%80%D5%AA%D5%B8%D6%82%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Էլեկտրոնների և իոնների շարժունություն – Armenian" lang="hy" hreflang="hy" data-title="Էլեկտրոնների և իոնների շարժունություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%99%D7%99%D7%93%D7%95%D7%AA_%D7%97%D7%A9%D7%9E%D7%9C%D7%99%D7%AA" title="ניידות חשמלית – Hebrew" lang="he" hreflang="he" data-title="ניידות חשמלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elektronmobilit%C3%A1s" title="Elektronmobilitás – Hungarian" lang="hu" hreflang="hu" data-title="Elektronmobilitás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%BB%E5%AD%90%E7%A7%BB%E5%8B%95%E5%BA%A6" title="電子移動度 – Japanese" lang="ja" hreflang="ja" data-title="電子移動度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ruchliwo%C5%9B%C4%87" title="Ruchliwość – Polish" lang="pl" hreflang="pl" data-title="Ruchliwość" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mobilidade_(F%C3%ADsica)" title="Mobilidade (Física) – Portuguese" lang="pt" hreflang="pt" data-title="Mobilidade (Física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B2%D0%B8%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%BD%D0%BE%D1%81%D0%B8%D1%82%D0%B5%D0%BB%D0%B5%D0%B9_%D0%B7%D0%B0%D1%80%D1%8F%D0%B4%D0%B0" title="Подвижность носителей заряда – Russian" lang="ru" hreflang="ru" data-title="Подвижность носителей заряда" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Gibljivost" title="Gibljivost – Slovenian" lang="sl" hreflang="sl" data-title="Gibljivost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B1%D0%B8%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Мобилност – Serbian" lang="sr" hreflang="sr" data-title="Мобилност" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AE%A3%E0%AF%81_%E0%AE%A8%E0%AE%95%E0%AE%B0%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B1%E0%AE%A9%E0%AF%8D" title="மின்னணு நகர்திறன் – Tamil" lang="ta" hreflang="ta" data-title="மின்னணு நகர்திறன்" 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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">Quantity in solid-state physics</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">This article is about the mobility for electrons and holes in metals and semiconductors. For the general concept, see <a href="/wiki/Electrical_mobility" title="Electrical mobility">Electrical mobility</a>.</div> <p>In <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, the <b>electron mobility</b> characterises how quickly an <a href="/wiki/Electron" title="Electron">electron</a> can move through a <a href="/wiki/Metal" title="Metal">metal</a> or <a href="/wiki/Semiconductor" title="Semiconductor">semiconductor</a> when pushed or pulled by an <a href="/wiki/Electric_field" title="Electric field">electric field</a>. There is an analogous quantity for <a href="/wiki/Electron_hole" title="Electron hole">holes</a>, called <b>hole mobility</b>. The term <b>carrier mobility</b> refers in general to both electron and hole mobility. </p><p>Electron and hole mobility are <a href="/wiki/Special_case" title="Special case">special cases</a> of <a href="/wiki/Electrical_mobility" title="Electrical mobility">electrical mobility</a> of charged particles in a fluid under an applied electric field. </p><p>When an electric field <i>E</i> is applied across a piece of material, the electrons respond by moving with an average velocity called the <a href="/wiki/Drift_velocity" title="Drift velocity">drift velocity</a>, <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_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac4aa20f05fda74ee30f9f9842229d087f33133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.22ex; height:2.009ex;" alt="{\displaystyle v_{d}}" /></span>. Then the electron mobility <i>μ</i> is defined as <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 v_{d}=\mu E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=\mu E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7eedaedcd6fe91aa295e11f32d5851033070e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.142ex; height:2.676ex;" alt="{\displaystyle v_{d}=\mu E.}" /></span> </p><p>Electron mobility is almost always specified in units of <a href="/wiki/Square_centimetre" class="mw-redirect" title="Square centimetre">cm<sup>2</sup></a>/(<a href="/wiki/Volt" title="Volt">V</a>⋅<a href="/wiki/Second" title="Second">s</a>). This is different from the <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> unit of mobility, <a href="/wiki/Square_metre" title="Square metre">m<sup>2</sup></a>/(<a href="/wiki/Volt" title="Volt">V</a>⋅<a href="/wiki/Second" title="Second">s</a>). They are related by 1 m<sup>2</sup>/(V⋅s) = 10<sup>4</sup> cm<sup>2</sup>/(V⋅s). </p><p><a href="/wiki/Electrical_resistivity_and_conductivity" title="Electrical resistivity and conductivity">Conductivity</a> is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For semiconductors, the behavior of <a href="/wiki/Transistor" title="Transistor">transistors</a> and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal. </p><p>Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when <a href="/wiki/Velocity_saturation" class="mw-redirect" title="Velocity saturation">velocity saturation</a> occurs. It can be determined by the <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a>, or inferred from transistor behavior. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Electron_mobility" title="Special:EditPage/Electron mobility">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Drift_velocity_in_an_electric_field">Drift velocity in an electric field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=2" title="Edit section: Drift velocity in an electric field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Drift_velocity" title="Drift velocity">Drift velocity</a></div> <p>Without any applied electric field, in a solid, <a href="/wiki/Electron" title="Electron">electrons</a> and <a href="/wiki/Electron_hole" title="Electron hole">holes</a> <a href="/wiki/Brownian_motion" title="Brownian motion">move around randomly</a>. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time. </p><p>However, when an electric field is applied, each electron or hole is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to ever-increasing velocity (called <a href="/wiki/Ballistic_transport" class="mw-redirect" title="Ballistic transport">ballistic transport</a>). However, in a solid, the electron repeatedly scatters off <a href="/wiki/Crystallographic_defect" title="Crystallographic defect">crystal defects</a>, <a href="/wiki/Phonons" class="mw-redirect" title="Phonons">phonons</a>, impurities, etc., so that it loses some energy and changes direction. The final result is that the electron moves with a finite average velocity, called the <a href="/wiki/Drift_velocity" title="Drift velocity">drift velocity</a>. This net electron motion is usually much slower than the normally occurring random motion. </p><p>The two charge carriers, electrons and holes, will typically have different drift velocities for the same electric field. </p><p>Quasi-<a href="/wiki/Ballistic_transport" class="mw-redirect" title="Ballistic transport">ballistic transport</a> is possible in solids if the electrons are accelerated across a very small distance (as small as the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a>), or for a very short time (as short as the <a href="/wiki/Mean_free_time" title="Mean free time">mean free time</a>). In these cases, drift velocity and mobility are not meaningful. </p> <div class="mw-heading mw-heading3"><h3 id="Definition_and_units">Definition and units</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=3" title="Edit section: Definition and units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Electrical_mobility" title="Electrical mobility">Electrical mobility</a></div> <p>The electron mobility is defined by the equation: <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 v_{d}=\mu _{e}E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=\mu _{e}E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1210d27a81af9a9eda4474bf67589a496d469655" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.141ex; height:2.676ex;" alt="{\displaystyle v_{d}=\mu _{e}E.}" /></span> where: </p> <ul><li><i>E</i> is the <a href="/wiki/Euclidean_vector" title="Euclidean vector">magnitude</a> of the <a href="/wiki/Electric_field" title="Electric field">electric field</a> applied to a material,</li> <li><i>v<sub>d</sub></i> is the <a href="/wiki/Euclidean_vector" title="Euclidean vector">magnitude</a> of the electron drift velocity (in other words, the electron drift <a href="/wiki/Speed" title="Speed">speed</a>) caused by the electric field, and</li> <li><i>μ</i><sub>e</sub> is the electron mobility.</li></ul> <p>The hole mobility is defined by a similar equation: <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 v_{d}=\mu _{h}E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=\mu _{h}E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/528365aae266464035ef0821fb0b80ffe82388cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.321ex; height:2.676ex;" alt="{\displaystyle v_{d}=\mu _{h}E.}" /></span> Both electron and hole mobilities are positive by definition. </p><p>Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of the electric field). When this is not true (for example, in very large electric fields), mobility depends on the electric field. </p><p>The SI unit of velocity is <a href="/wiki/Metre_per_second" title="Metre per second">m/s</a>, and the SI unit of electric field is <a href="/wiki/Volt" title="Volt">V</a>/<a href="/wiki/Metre" title="Metre">m</a>. Therefore the SI unit of mobility is (m/s)/(V/m) = <a href="/wiki/Square_metre" title="Square metre">m<sup>2</sup></a>/(<a href="/wiki/Volt" title="Volt">V</a>⋅<a href="/wiki/Second" title="Second">s</a>). However, mobility is much more commonly expressed in cm<sup>2</sup>/(V⋅s) = 10<sup>−4</sup> m<sup>2</sup>/(V⋅s). </p><p>Mobility is usually a strong function of material impurities and temperature, and is determined empirically. Mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given material. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=4" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Starting with <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a>: <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 a=F/m_{e}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=F/m_{e}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/920e0181512d267668d7d5e5b7d3afabab1710f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.326ex; height:2.843ex;" alt="{\displaystyle a=F/m_{e}^{*}}" /></span> where: </p> <ul><li><i>a</i> is the acceleration between collisions.</li> <li><i>F</i> is the electric force exerted by the electric field, and</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 m_{e}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415a6f006ad0adf43f44dcb9a3eb72d7df0e8fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.509ex;" alt="{\displaystyle m_{e}^{*}}" /></span> is the <a href="/wiki/Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a> of an electron.</li></ul> <p>Since the force on the electron is −<i>eE</i>: <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 a=-{\frac {eE}{m_{e}^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>E</mi> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-{\frac {eE}{m_{e}^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590f69c83b717bdea2cf2ef086a8a00349e64ff3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.067ex; height:5.509ex;" alt="{\displaystyle a=-{\frac {eE}{m_{e}^{*}}}}" /></span> </p><p>This is the acceleration on the electron between collisions. The drift velocity is therefore: <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 v_{d}=a\tau _{c}=-{\frac {e\tau _{c}}{m_{e}^{*}}}E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=a\tau _{c}=-{\frac {e\tau _{c}}{m_{e}^{*}}}E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf405cf79e48bc0127eb180b8e45c87cef968b89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.768ex; height:5.009ex;" alt="{\displaystyle v_{d}=a\tau _{c}=-{\frac {e\tau _{c}}{m_{e}^{*}}}E,}" /></span> where <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 \tau _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea61c9e95de7b5b55eb1ef4b3e03290002e91089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.96ex; height:2.009ex;" alt="{\displaystyle \tau _{c}}" /></span> is the <a href="/wiki/Mean_free_time" title="Mean free time">mean free time</a> </p><p>Since we only care about how the drift velocity changes with the electric field, we lump the loose terms together to get <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 v_{d}=-\mu _{e}E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=-\mu _{e}E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef48846d9c072ffa7309dd3ff50baa64b8268ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.949ex; height:2.676ex;" alt="{\displaystyle v_{d}=-\mu _{e}E,}" /></span> where <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 _{e}={\frac {e\tau _{c}}{m_{e}^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{e}={\frac {e\tau _{c}}{m_{e}^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896b036c4d8bd33ac1ab6845cace7a90c6156ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.429ex; height:5.009ex;" alt="{\displaystyle \mu _{e}={\frac {e\tau _{c}}{m_{e}^{*}}}}" /></span> </p><p>Similarly, for holes we have <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 v_{d}=\mu _{h}E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}=\mu _{h}E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1a34d2dd249403aec332bae864d11c960ea9ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.321ex; height:2.676ex;" alt="{\displaystyle v_{d}=\mu _{h}E,}" /></span> where <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 _{h}={\frac {e\tau _{c}}{m_{h}^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{h}={\frac {e\tau _{c}}{m_{h}^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b9ce0662cdddd1f6b5bc610f6ea9daaf01a2b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.735ex; height:5.509ex;" alt="{\displaystyle \mu _{h}={\frac {e\tau _{c}}{m_{h}^{*}}}}" /></span> Note that both electron mobility and hole mobility are positive. A minus sign is added for electron drift velocity to account for the minus charge. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_current_density">Relation to current density</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=5" title="Edit section: Relation to current density"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The drift current density resulting from an electric field can be calculated from the drift velocity. Consider a sample with cross-sectional area A, length l and an electron concentration of n. The current carried by each electron must be <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 -ev_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>e</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -ev_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1e8441a950e2cfee3c55b045204a17289cd5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.111ex; height:2.343ex;" alt="{\displaystyle -ev_{d}}" /></span>, so that the total current density due to electrons is given by: <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 J_{e}={\frac {I_{n}}{A}}=-env_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>e</mi> <mi>n</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{e}={\frac {I_{n}}{A}}=-env_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a41edf7efbe7aff0a44e64e64a4bec4c1c34b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.069ex; height:5.343ex;" alt="{\displaystyle J_{e}={\frac {I_{n}}{A}}=-env_{d}}" /></span> Using the expression for <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_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac4aa20f05fda74ee30f9f9842229d087f33133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.22ex; height:2.009ex;" alt="{\displaystyle v_{d}}" /></span> gives <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 J_{e}=en\mu _{e}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{e}=en\mu _{e}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e117f6e09463fa766b40b08612e725ff34d3db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.041ex; height:2.676ex;" alt="{\displaystyle J_{e}=en\mu _{e}E}" /></span> A similar set of equations applies to the holes, (noting that the charge on a hole is positive). Therefore the current density due to holes is given by <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 J_{h}=ep\mu _{h}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mi>p</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{h}=ep\mu _{h}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8c70483e8757d8ee770c499f752d93a5e012f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.177ex; height:2.676ex;" alt="{\displaystyle J_{h}=ep\mu _{h}E}" /></span> where p is the hole concentration and <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 _{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aa30d1505e2ca4947219f340aae30b6e015bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.581ex; height:2.176ex;" alt="{\displaystyle \mu _{h}}" /></span> the hole mobility. </p><p>The total current density is the sum of the electron and hole components: <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 J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <mi>e</mi> <mi>p</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1722d5e179f372d7c1256815e4ab54efedc82a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.404ex; height:2.843ex;" alt="{\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_conductivity">Relation to conductivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=6" title="Edit section: Relation to conductivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We have previously derived the relationship between electron mobility and current density <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 J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <mi>e</mi> <mi>p</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1722d5e179f372d7c1256815e4ab54efedc82a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.404ex; height:2.843ex;" alt="{\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}" /></span> Now <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> can be written in the form <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 J=\sigma E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mi>σ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\sigma E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd9654398e639c58026b9d045060ace3ae0e554" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.675ex; height:2.176ex;" alt="{\displaystyle J=\sigma E}" /></span> where <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span> is defined as the conductivity. Therefore we can write down: <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 \sigma =en\mu _{e}+ep\mu _{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>e</mi> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <mi>e</mi> <mi>p</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =en\mu _{e}+ep\mu _{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0c92e9dd2105776516db1035a578c4436d5483" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.98ex; height:2.509ex;" alt="{\displaystyle \sigma =en\mu _{e}+ep\mu _{h}}" /></span> which can be factorised to <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 \sigma =e(n\mu _{e}+p\mu _{h})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <mi>p</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =e(n\mu _{e}+p\mu _{h})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9d1de0681dccd98ff560d5084c50560fd13c5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.706ex; height:2.843ex;" alt="{\displaystyle \sigma =e(n\mu _{e}+p\mu _{h})}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_electron_diffusion">Relation to electron diffusion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=7" title="Edit section: Relation to electron diffusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a region where n and p vary with distance, a diffusion current is superimposed on that due to conductivity. This diffusion current is governed by <a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a>: <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 F=-D_{\text{e}}\nabla n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=-D_{\text{e}}\nabla n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315e519df9e4cb8f46b65716078e227eefb1e4e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.865ex; height:2.509ex;" alt="{\displaystyle F=-D_{\text{e}}\nabla n}" /></span> where: </p> <ul><li><i>F</i> is flux.</li> <li><i>D</i><sub>e</sub> is the <a href="/wiki/Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a> or diffusivity</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 \nabla n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1b2c200324e35f520841cf24c6bb2f9ecc0e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.331ex; height:2.176ex;" alt="{\displaystyle \nabla n}" /></span> is the concentration gradient of electrons</li></ul> <p>The diffusion coefficient for a charge carrier is related to its mobility by the <a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation</a>. For a classical system (e.g. Boltzmann gas), it reads: <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 D_{\text{e}}={\frac {\mu _{\text{e}}k_{\mathrm {B} }T}{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> <mi>e</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{e}}={\frac {\mu _{\text{e}}k_{\mathrm {B} }T}{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c95662fe64d479f0bef7467ca7ff2486c3b944" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.428ex; height:5.343ex;" alt="{\displaystyle D_{\text{e}}={\frac {\mu _{\text{e}}k_{\mathrm {B} }T}{e}}}" /></span> where: </p> <ul><li><i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a></li> <li><i>T</i> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a></li> <li><i>e</i> is the electric charge of an electron</li></ul> <p>For a metal, described by a Fermi gas (Fermi liquid), quantum version of the Einstein relation should be used. Typically, temperature is much smaller than the Fermi energy, in this case one should use the following formula: <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 D_{\text{e}}={\frac {\mu _{\text{e}}E_{F}}{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mrow> <mi>e</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{e}}={\frac {\mu _{\text{e}}E_{F}}{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7df6df276f9a4d011891b9729f1723fe043bb58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.363ex; height:5.343ex;" alt="{\displaystyle D_{\text{e}}={\frac {\mu _{\text{e}}E_{F}}{e}}}" /></span> where: </p> <ul><li><i>E</i><sub>F</sub> is the Fermi energy</li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Typical electron mobility at room temperature (300 K) in metals like <a href="/wiki/Gold" title="Gold">gold</a>, <a href="/wiki/Copper" title="Copper">copper</a> and <a href="/wiki/Silver" title="Silver">silver</a> is 30–50 cm<sup>2</sup>/(V⋅s). Carrier mobility in semiconductors is doping dependent. In <a href="/wiki/Silicon" title="Silicon">silicon</a> (Si) the electron mobility is of the order of 1,000, in germanium around 4,000, and in gallium arsenide up to 10,000 cm<sup>2</sup>/(V⋅s). Hole mobilities are generally lower and range from around 100 cm<sup>2</sup>/(V⋅s) in gallium arsenide, to 450 in silicon, and 2,000 in germanium.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Very high mobility has been found in several ultrapure low-dimensional systems, such as two-dimensional electron gases (<a href="/wiki/2DEG" class="mw-redirect" title="2DEG">2DEG</a>) (35,000,000 cm<sup>2</sup>/(V⋅s) at low temperature),<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Carbon_nanotubes" class="mw-redirect" title="Carbon nanotubes">carbon nanotubes</a> (100,000 cm<sup>2</sup>/(V⋅s) at room temperature)<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> and freestanding <a href="/wiki/Graphene" title="Graphene">graphene</a> (200,000 cm<sup>2</sup>/(V⋅s) at low temperature).<sup id="cite_ref-:2_4-0" class="reference"><a href="#cite_note-:2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Organic_semiconductor" title="Organic semiconductor">Organic semiconductors</a> (<a href="/wiki/Polymer" title="Polymer">polymer</a>, <a href="/wiki/Oligomer" title="Oligomer">oligomer</a>) developed thus far have carrier mobilities below 50 cm<sup>2</sup>/(V⋅s), and typically below 1, with well performing materials measured below 10.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable sortable"> <caption>List of highest measured mobilities [cm<sup>2</sup>/(V⋅s)] </caption> <tbody><tr> <th>Material </th> <th>Electron mobility </th> <th>Hole mobility </th></tr> <tr> <td>AlGaAs/GaAs heterostructures </td> <td>35,000,000<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td> <td>5,800,000<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> </td></tr> <tr> <td>Freestanding graphene </td> <td>200,000<sup id="cite_ref-:2_4-1" class="reference"><a href="#cite_note-:2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </td> <td> </td></tr> <tr> <td>Carbon nanotubes </td> <td>79,000<sup id="cite_ref-DurkopGetty2004_7-0" class="reference"><a href="#cite_note-DurkopGetty2004-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SnowCampbell2005_8-0" class="reference"><a href="#cite_note-SnowCampbell2005-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </td> <td> </td></tr> <tr> <td>Cubic boron arsenide (c-BAs) </td> <td>1,600<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> </td> <td> </td></tr> <tr> <td>Crystalline silicon </td> <td>1,400<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </td> <td>450<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Polycrystalline silicon </td> <td>100 </td> <td> </td></tr> <tr> <td>Metals (Al, Au, Cu, Ag) </td> <td>10–50 </td> <td> </td></tr> <tr> <td>2D material (MoS<sub>2</sub>) </td> <td>10–50 </td> <td> </td></tr> <tr> <td>Organics </td> <td>8.6<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td> <td>43<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Amorphous silicon </td> <td>~1<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> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Electric_field_dependence_and_velocity_saturation">Electric field dependence and velocity saturation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=9" title="Edit section: Electric field dependence and velocity saturation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Velocity_saturation" class="mw-redirect" title="Velocity saturation">Velocity saturation</a></div> <p>At low fields, the drift velocity <i>v</i><sub><i>d</i></sub> is proportional to the electric field <i>E</i>, so mobility <i>μ</i> is constant. This value of <i>μ</i> is called the <i>low-field mobility</i>. </p><p>As the electric field is increased, however, the carrier velocity increases sublinearly and asymptotically towards a maximum possible value, called the <i>saturation velocity</i> <i>v</i><sub>sat</sub>. For example, the value of <i>v</i><sub>sat</sub> is on the order of 1×10<sup>7</sup> cm/s for both electrons and holes in Si. It is on the order of 6×10<sup>6</sup> cm/s for Ge. This velocity is a characteristic of the material and a strong function of <a href="/wiki/Doping_(semiconductor)" title="Doping (semiconductor)">doping</a> or impurity levels and temperature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency. </p><p>This velocity saturation phenomenon results from a process called <i><a href="/wiki/Optical_phonon" class="mw-redirect" title="Optical phonon">optical phonon</a> scattering</i>. At high fields, carriers are accelerated enough to gain sufficient <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is: <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 {\frac {m^{*}v_{\text{emit}}^{2}}{2}}\approx \hbar \omega _{\text{phonon (opt.)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emit</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≈</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>phonon (opt.)</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m^{*}v_{\text{emit}}^{2}}{2}}\approx \hbar \omega _{\text{phonon (opt.)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5d1e8039f1a201540ae31017aeeeb25c324a76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.562ex; height:6.009ex;" alt="{\displaystyle {\frac {m^{*}v_{\text{emit}}^{2}}{2}}\approx \hbar \omega _{\text{phonon (opt.)}}}" /></span> where <i>ω</i><sub>phonon(opt.)</sub> is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value of <i>E</i><sub>phonon (opt.)</sub> is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of <i>v</i><sub>emit</sub>, because the electron starts at zero velocity and accelerates up to <i>v</i><sub>emit</sub> in each cycle.<sup id="cite_ref-Mitin_13-0" class="reference"><a href="#cite_note-Mitin-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> (This is a somewhat oversimplified description.<sup id="cite_ref-Mitin_13-1" class="reference"><a href="#cite_note-Mitin-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>) </p><p>Velocity saturation is not the only possible high-field behavior. Another is the <a href="/wiki/Gunn_effect" class="mw-redirect" title="Gunn effect">Gunn effect</a>, where a sufficiently high electric field can cause intervalley electron transfer, which reduces drift velocity. This is unusual; increasing the electric field almost always <i>increases</i> the drift velocity, or else leaves it unchanged. The result is <a href="/wiki/Negative_differential_resistance" class="mw-redirect" title="Negative differential resistance">negative differential resistance</a>. </p><p>In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_between_scattering_and_mobility">Relation between scattering and mobility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=10" title="Edit section: Relation between scattering and mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other than <a href="/wiki/Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a>) is <a href="/wiki/Scattering" title="Scattering">scattering</a> time, i.e. how long the carrier is <a href="/wiki/Ballistic_transport" class="mw-redirect" title="Ballistic transport">ballistically accelerated</a> by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and <a href="/wiki/Crystallographic_defect" title="Crystallographic defect">defect</a> scattering.<sup id="cite_ref-Singh_14-0" class="reference"><a href="#cite_note-Singh-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state <b>k</b> to<b> k'</b>, while emitting or absorbing a phonon of wave vector <b>q</b>. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.<sup id="cite_ref-sct_15-0" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ionized_impurity_scattering">Ionized impurity scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=11" title="Edit section: Ionized impurity scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Semiconductors are doped with donors and/or acceptors, which are typically ionized, and are thus charged. The Coulombic forces will deflect an electron or hole approaching the ionized impurity. This is known as <i><a href="/wiki/Ionized_impurity_scattering" title="Ionized impurity scattering">ionized impurity scattering</a></i>. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the <a href="/wiki/Mean_free_time" title="Mean free time">mean free time</a> between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction. </p><p>If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi-two-dimensional electrons at the interface.<sup id="cite_ref-sct_15-1" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Lattice_(phonon)_scattering"><span id="Lattice_.28phonon.29_scattering"></span>Lattice (phonon) scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=12" title="Edit section: Lattice (phonon) scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At any temperature above <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed <a href="/wiki/Phonon" title="Phonon">phonons</a>. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, and thus increased electron scattering, which tends to reduce mobility. </p> <div class="mw-heading mw-heading3"><h3 id="Piezoelectric_scattering">Piezoelectric scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=13" title="Edit section: Piezoelectric scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Piezoelectric effect can occur only in compound semiconductor due to their polar nature. It is small in most semiconductors but may lead to local electric fields that cause scattering of carriers by deflecting them, this effect is important mainly at low temperatures where other scattering mechanisms are weak. These electric fields arise from the distortion of the basic unit cell as strain is applied in certain directions in the lattice.<sup id="cite_ref-sct_15-2" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Surface_roughness_scattering">Surface roughness scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=14" title="Edit section: Surface roughness scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface roughness scattering caused by interfacial disorder is short range scattering limiting the mobility of quasi-two-dimensional electrons at the interface. From high-resolution transmission electron micrographs, it has been determined that the interface is not abrupt on the atomic level, but actual position of the interfacial plane varies one or two atomic layers along the surface. These variations are random and cause fluctuations of the energy levels at the interface, which then causes scattering.<sup id="cite_ref-sct_15-3" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Alloy_scattering">Alloy scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=15" title="Edit section: Alloy scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In compound (alloy) semiconductors, which many thermoelectric materials are, scattering caused by the perturbation of crystal potential due to the random positioning of substituting atom species in a relevant sublattice is known as alloy scattering. This can only happen in ternary or higher alloys as their crystal structure forms by randomly replacing some atoms in one of the sublattices (sublattice) of the crystal structure. Generally, this phenomenon is quite weak but in certain materials or circumstances, it can become dominant effect limiting conductivity. In bulk materials, interface scattering is usually ignored.<sup id="cite_ref-sct_15-4" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ssp_16-0" class="reference"><a href="#cite_note-ssp-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bulusu_17-0" class="reference"><a href="#cite_note-bulusu-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pallab_18-0" class="reference"><a href="#cite_note-pallab-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Takeda_19-0" class="reference"><a href="#cite_note-Takeda-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Inelastic_scattering">Inelastic scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=16" title="Edit section: Inelastic scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During inelastic scattering processes, significant energy exchange happens. As with elastic phonon scattering also in the inelastic case, the potential arises from energy band deformations caused by atomic vibrations. Optical phonons causing inelastic scattering usually have the energy in the range 30-50 meV, for comparison energies of acoustic phonon are typically less than 1 meV but some might have energy in order of 10 meV. There is significant change in carrier energy during the scattering process. Optical or high-energy acoustic phonons can also cause intervalley or interband scattering, which means that scattering is not limited within single valley.<sup id="cite_ref-sct_15-5" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electron–electron_scattering"><span id="Electron.E2.80.93electron_scattering"></span>Electron–electron scattering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=17" title="Edit section: Electron–electron scattering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 10<sup>16</sup>~10<sup>17</sup> cm<sup>−3</sup> or electric field value 10<sup>3</sup> V/cm. However, significantly above these limits electron–electron scattering starts to dominate. Long range and nonlinearity of the Coulomb potential governing interactions between electrons make these interactions difficult to deal with.<sup id="cite_ref-sct_15-6" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ssp_16-1" class="reference"><a href="#cite_note-ssp-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-bulusu_17-1" class="reference"><a href="#cite_note-bulusu-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_between_mobility_and_scattering_time">Relation between mobility and scattering time</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=18" title="Edit section: Relation between mobility and scattering time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift mobility is:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <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 \mu ={\frac {q}{m^{*}}}{\overline {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {q}{m^{*}}}{\overline {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276074ca1e446cf2d386785829d6651d3675a8fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.869ex; height:4.843ex;" alt="{\displaystyle \mu ={\frac {q}{m^{*}}}{\overline {\tau }}}" /></span> where <i>q</i> is the <a href="/wiki/Elementary_charge" title="Elementary charge">elementary charge</a>, <i>m</i>* is the carrier <a href="/wiki/Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a>, and <span style="text-decoration:overline;"><i>τ</i></span> is the average scattering time. </p><p>If the effective mass is anisotropic (direction-dependent), <i>m</i>* is the effective mass in the direction of the electric field. </p> <div class="mw-heading mw-heading3"><h3 id="Matthiessen's_rule"><span id="Matthiessen.27s_rule"></span>Matthiessen's rule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=19" title="Edit section: Matthiessen's rule"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Normally, more than one source of scattering is present, for example both impurities and lattice phonons. It is normally a very good approximation to combine their influences using "Matthiessen's Rule" (developed from work by <a href="/wiki/Augustus_Matthiessen" title="Augustus Matthiessen">Augustus Matthiessen</a> in 1864): </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 {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>μ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9da8e88cdf763637a59306393abac7668f98209" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.471ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}.}" /></span> where <i>μ</i> is the actual mobility, <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 _{\rm {impurities}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\rm {impurities}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a2aefc8e40c8a1c4475ecc883f7c59b4e70ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.865ex; height:2.343ex;" alt="{\displaystyle \mu _{\rm {impurities}}}" /></span> is the mobility that the material would have if there was impurity scattering but no other source of scattering, and <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 _{\rm {lattice}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\rm {lattice}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256c45e1b4a4e2935046a0b6339891b2fe76fe71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.11ex; height:2.176ex;" alt="{\displaystyle \mu _{\rm {lattice}}}" /></span> is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example <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 {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}+{\frac {1}{\mu _{\rm {defects}}}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>μ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}+{\frac {1}{\mu _{\rm {defects}}}}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e4e640fecdca11a8a638d2e44d18995ee6b016" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.735ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}+{\frac {1}{\mu _{\rm {defects}}}}+\cdots .}" /></span> Matthiessen's rule can also be stated in terms of the scattering time: <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 {\frac {1}{\tau }}={\frac {1}{\tau _{\rm {impurities}}}}+{\frac {1}{\tau _{\rm {lattice}}}}+{\frac {1}{\tau _{\rm {defects}}}}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\tau }}={\frac {1}{\tau _{\rm {impurities}}}}+{\frac {1}{\tau _{\rm {lattice}}}}+{\frac {1}{\tau _{\rm {defects}}}}+\cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470b1b8b0e1130a88c515ac6d8c42f06afff1b2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.379ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{\tau }}={\frac {1}{\tau _{\rm {impurities}}}}+{\frac {1}{\tau _{\rm {lattice}}}}+{\frac {1}{\tau _{\rm {defects}}}}+\cdots .}" /></span> where <i>τ</i> is the true average scattering time and τ<sub>impurities</sub> is the scattering time if there was impurity scattering but no other source of scattering, etc. </p><p>Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on each other, because individual scattering probabilities cannot be summed unless they are independent of each other.<sup id="cite_ref-Takeda_19-1" class="reference"><a href="#cite_note-Takeda-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability.<sup id="cite_ref-sct_15-7" class="reference"><a href="#cite_note-sct-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ssp_16-2" class="reference"><a href="#cite_note-ssp-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pallab_18-1" class="reference"><a href="#cite_note-pallab-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Temperature_dependence_of_mobility">Temperature dependence of mobility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=20" title="Edit section: Temperature dependence of mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="float:right; text-align:center;"> <caption>Typical temperature dependence of mobility<sup id="cite_ref-BVZ_22-0" class="reference"><a href="#cite_note-BVZ-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th> </th> <th>Si </th> <th>Ge </th> <th>GaAs </th></tr> <tr> <th>Electrons </th> <td>∝T <sup>−2.4</sup> </td> <td>∝T <sup>−1.7</sup> </td> <td>∝T <sup>−1.0</sup> </td></tr> <tr> <th>Holes </th> <td>∝T <sup>−2.2</sup> </td> <td>∝T <sup>−2.3</sup> </td> <td>∝T <sup>−2.1</sup> </td></tr></tbody></table> <p>With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in <a href="/wiki/Chemical_polarity" title="Chemical polarity">non-polar</a> semiconductors, such as silicon and germanium, is dominated by <a href="/wiki/Phonon" title="Phonon">acoustic phonon</a> interaction. The resulting mobility is expected to be proportional to <i>T</i> <sup>−3/2</sup>, while the mobility due to optical phonon scattering only is expected to be proportional to <i>T</i> <sup>−1/2</sup>. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.<sup id="cite_ref-BVZ_22-1" class="reference"><a href="#cite_note-BVZ-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>As <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="{\textstyle {\frac {1}{\tau }}\propto \left\langle v\right\rangle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> <mo>∝</mo> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\tau }}\propto \left\langle v\right\rangle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42e2a1afa19503297064c071081fd9a76ee2d5f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.787ex; height:3.343ex;" alt="{\textstyle {\frac {1}{\tau }}\propto \left\langle v\right\rangle \Sigma }" /></span>, where <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 \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" /></span> is the scattering cross section for electrons and holes at a scattering center and <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 \left\langle v\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle v\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4c5459dbec896da44d7747a340b4110b135f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \left\langle v\right\rangle }" /></span> is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors <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 \left\langle v\right\rangle \sim {\sqrt {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>T</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle v\right\rangle \sim {\sqrt {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a09cdf90930d189ebea3ae819ff1f157ffd63754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.608ex; height:3.176ex;" alt="{\displaystyle \left\langle v\right\rangle \sim {\sqrt {T}}}" /></span>. </p><p>For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σ<sub>ph</sub> is determined from the square of the average vibrational amplitude of a phonon to be proportional to <i>T</i>. The scattering from charged defects (ionized donors or acceptors) leads to the cross section <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 {\Sigma }_{\text{def}}\propto {\left\langle v\right\rangle }^{-4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </msub> <mo>∝</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Sigma }_{\text{def}}\propto {\left\langle v\right\rangle }^{-4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4dcb4c543d4dbc34f7a18e00a4cc706f1077913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.426ex; height:3.343ex;" alt="{\displaystyle {\Sigma }_{\text{def}}\propto {\left\langle v\right\rangle }^{-4}}" /></span>. This formula is the scattering cross section for "Rutherford scattering", where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction. </p><p>The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and <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 \left\langle v\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle v\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4c5459dbec896da44d7747a340b4110b135f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \left\langle v\right\rangle }" /></span>, to be for scattering from acoustic phonons <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 }_{ph}\sim T^{-3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>∼</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mu }_{ph}\sim T^{-3/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba286e5e9f4c53860b8342b6df2a665e1a56d992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.203ex; height:3.676ex;" alt="{\displaystyle {\mu }_{ph}\sim T^{-3/2}}" /></span> and from charged defects <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 }_{\text{def}}\sim T^{3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </msub> <mo>∼</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344140f660e3684046a500d1c938784b2eeab279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.298ex; height:3.343ex;" alt="{\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}" /></span>.<sup id="cite_ref-ssp_16-3" class="reference"><a href="#cite_note-ssp-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pallab_18-2" class="reference"><a href="#cite_note-pallab-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The effect of ionized impurity scattering, however, <i>decreases</i> with increasing temperature because the average thermal speeds of the carriers are increased.<sup id="cite_ref-Singh_14-1" class="reference"><a href="#cite_note-Singh-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced. </p><p>These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature. </p> <div class="mw-heading mw-heading2"><h2 id="Disordered_Semiconductors">Disordered Semiconductors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=21" title="Edit section: Disordered Semiconductors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_Edge.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Mobility_Edge.png/220px-Mobility_Edge.png" decoding="async" width="220" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Mobility_Edge.png/330px-Mobility_Edge.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Mobility_Edge.png/440px-Mobility_Edge.png 2x" data-file-width="3191" data-file-height="1934" /></a><figcaption><a href="/wiki/Density_of_states" title="Density of states">Density of states</a> of a solid possessing a mobility edge, <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 E_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f6111453eb8ad990d61ee904c781d9b9828768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.196ex; height:2.509ex;" alt="{\displaystyle E_{C}}" /></span>.</figcaption></figure> <p>While in crystalline materials electrons can be described by wavefunctions extended over the entire solid,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> this is not the case in systems with appreciable structural disorder, such as <a href="/wiki/Crystallite" title="Crystallite">polycrystalline</a> or <a href="/wiki/Amorphous_solid" title="Amorphous solid">amorphous</a> semiconductors. <a href="/wiki/Philip_W._Anderson" title="Philip W. Anderson">Anderson</a> suggested that beyond a critical value of structural disorder,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> electron states would be <i>localized</i>. Localized states are described as being confined to finite region of real space, <a href="/wiki/Normalizing_constant" title="Normalizing constant">normalizable</a>, and not contributing to transport. Extended states are spread over the extent of the material, not normalizable, and contribute to transport. Unlike crystalline semiconductors, mobility generally increases with temperature in disordered semiconductors. </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_trapping_and_release">Multiple trapping and release</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=22" title="Edit section: Multiple trapping and release"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Nevill_Francis_Mott" class="mw-redirect" title="Nevill Francis Mott">Mott</a> later developed<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> the concept of a mobility edge. This is an energy <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 E_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f6111453eb8ad990d61ee904c781d9b9828768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.196ex; height:2.509ex;" alt="{\displaystyle E_{C}}" /></span>, above which electrons undergo a transition from localized to delocalized states. In this description, termed <i>multiple trapping and release</i>, electrons are only able to travel when in extended states, and are constantly being trapped in, and re-released from, the lower energy localized states. Because the probability of an electron being released from a trap depends on its thermal energy, mobility can be described by an <a href="/wiki/Arrhenius_relationship" class="mw-redirect" title="Arrhenius relationship">Arrhenius relationship</a> in such a system: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electron_Transport_under_Multiple_Trapping_and_Release.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Electron_Transport_under_Multiple_Trapping_and_Release.png/250px-Electron_Transport_under_Multiple_Trapping_and_Release.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Electron_Transport_under_Multiple_Trapping_and_Release.png/330px-Electron_Transport_under_Multiple_Trapping_and_Release.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Electron_Transport_under_Multiple_Trapping_and_Release.png/500px-Electron_Transport_under_Multiple_Trapping_and_Release.png 2x" data-file-width="3997" data-file-height="2294" /></a><figcaption>Energy band diagram depicting electron transport under multiple trapping and release.</figcaption></figure> <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 \mu =\mu _{0}\exp \left(-{\frac {E_{\text{A}}}{k_{\text{B}}T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}\exp \left(-{\frac {E_{\text{A}}}{k_{\text{B}}T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d00587f91e6f1a1b85508b65e52993b866f6638" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.204ex; height:6.176ex;" alt="{\displaystyle \mu =\mu _{0}\exp \left(-{\frac {E_{\text{A}}}{k_{\text{B}}T}}\right)}" /></span> </p><p>where <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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" /></span> is a mobility prefactor, <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 E_{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca3e10667ff28ec068596f6cfdbfceb13517814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.18ex; height:2.509ex;" alt="{\displaystyle E_{\text{A}}}" /></span> is activation energy, <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_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9582c7c795d2def2c061f0dfa3a6f0fb3dd2de44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\text{B}}}" /></span> is the Boltzmann constant, and <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is temperature. The activation energy is typically evaluated by measuring mobility as a function of temperature. The <a href="/wiki/Urbach_energy" title="Urbach energy">Urbach Energy</a> can be used as a proxy for activation energy in some systems.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Variable_Range_Hopping">Variable Range Hopping</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=23" title="Edit section: Variable Range Hopping"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At low temperature, or in system with a large degree of structural disorder (such as fully amorphous systems), electrons cannot access delocalized states. In such a system, electrons can only travel by <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">tunnelling</a> for one site to another, in a process called <i>variable range hopping</i>. In the original theory of variable range hopping, as developed by Mott and Davis,<sup id="cite_ref-:3_27-0" class="reference"><a href="#cite_note-:3-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> the probability <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ef37c239b6d38f1e951a31eb1a3bd295271b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.969ex; height:2.843ex;" alt="{\displaystyle P_{ij}}" /></span>, of an electron hopping from one site <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span>, to another site <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /></span>, depends on their separation in space <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857845aef8b93395ad10279211c6c49180bb8791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.526ex; height:2.343ex;" alt="{\displaystyle r_{ij}}" /></span>, and their separation in energy <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 \Delta E_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2199009c56665db4ed04b25b6bcba9c0a4fd9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle \Delta E_{ij}}" /></span>. </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 P_{ij}=P_{0}\exp \left(-2\alpha r_{ij}-{\frac {\Delta E_{ij}}{k_{B}T}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mi>α</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{ij}=P_{0}\exp \left(-2\alpha r_{ij}-{\frac {\Delta E_{ij}}{k_{B}T}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb62a97d7a4aaca449b59affaa98992a77046a02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.764ex; height:6.343ex;" alt="{\displaystyle P_{ij}=P_{0}\exp \left(-2\alpha r_{ij}-{\frac {\Delta E_{ij}}{k_{B}T}}\right)}" /></span> </p><p>Here <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}" /></span> is a prefactor associated with the phonon frequency in the material,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown<sup id="cite_ref-:3_27-1" class="reference"><a href="#cite_note-:3-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> to be: </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 \mu =\mu _{0}\exp \left(-\left[{\frac {T_{0}}{T}}\right]^{-1/(d+1)}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}\exp \left(-\left[{\frac {T_{0}}{T}}\right]^{-1/(d+1)}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec6dbb01dae46490fe94780f7719b272fd5f4ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.482ex; height:7.509ex;" alt="{\displaystyle \mu =\mu _{0}\exp \left(-\left[{\frac {T_{0}}{T}}\right]^{-1/(d+1)}\right)}" /></span> </p><p>where <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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" /></span> is a mobility prefactor, <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" /></span> is a parameter (with dimensions of temperature) that quantifies the width of localized states, and <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> is the dimensionality of the system. </p> <div class="mw-heading mw-heading2"><h2 id="Measurement_of_semiconductor_mobility">Measurement of semiconductor mobility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=24" title="Edit section: Measurement of semiconductor mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Hall_mobility">Hall mobility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=25" title="Edit section: Hall mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Hall_Effect_Measurement_Setup_for_Holes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b9/Hall_Effect_Measurement_Setup_for_Holes.png" decoding="async" width="285" height="184" class="mw-file-element" data-file-width="285" data-file-height="184" /></a><figcaption>Hall effect measurement setup for holes</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Hall_Effect_Measurement_Setup_for_Electrons.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Hall_Effect_Measurement_Setup_for_Electrons.png" decoding="async" width="285" height="184" class="mw-file-element" data-file-width="285" data-file-height="184" /></a><figcaption>Hall effect measurement setup for electrons</figcaption></figure> <p>Carrier mobility is most commonly measured using the <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a>. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement"). </p><p>Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the <i>x</i>-direction and a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> is applied in the <i>z</i>-direction. The resulting Lorentz force will accelerate the electrons (<i>n</i>-type materials) or holes (<i>p</i>-type materials) in the (−<i>y</i>) direction, according to the <a href="/wiki/Right_hand_rule" class="mw-redirect" title="Right hand rule">right hand rule</a> and set up an electric field <i>ξ<sub>y</sub></i>. As a result there is a voltage across the sample, which can be measured with a <a href="/wiki/High_impedance" title="High impedance">high-impedance</a> voltmeter. This voltage, <i>V<sub>H</sub></i>, is called the <a href="/wiki/Hall_effect" title="Hall effect">Hall voltage</a>. <i>V<sub>H</sub></i> is negative for <i>n</i>-type material and positive for <i>p</i>-type material. </p><p>Mathematically, the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a> acting on a charge <i>q</i> is given by </p><p>For electrons: <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 \mathbf {F} _{Hn}=-q(\mathbf {v} _{n}\times \mathbf {B} _{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{Hn}=-q(\mathbf {v} _{n}\times \mathbf {B} _{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90163080d6f82ed803c383dfec6b578f65566dfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.518ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{Hn}=-q(\mathbf {v} _{n}\times \mathbf {B} _{z})}" /></span> </p><p>For holes: <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 \mathbf {F} _{Hp}=+q(\mathbf {v} _{p}\times \mathbf {B} _{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{Hp}=+q(\mathbf {v} _{p}\times \mathbf {B} _{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9cc170216b79d85e778613b61c68e2bd8fe991" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.2ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{Hp}=+q(\mathbf {v} _{p}\times \mathbf {B} _{z})}" /></span> </p><p>In steady state this force is balanced by the force set up by the Hall voltage, so that there is no <a href="/wiki/Net_force" title="Net force">net force</a> on the carriers in the <i>y</i> direction. For electrons, </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 \mathbf {F} _{y}=(-q)\xi _{y}+(-q)[\mathbf {v} _{n}\times \mathbf {B} _{z}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{y}=(-q)\xi _{y}+(-q)[\mathbf {v} _{n}\times \mathbf {B} _{z}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf982828aabe02c406b8e0f377e23d99e065626" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.04ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{y}=(-q)\xi _{y}+(-q)[\mathbf {v} _{n}\times \mathbf {B} _{z}]=0}" /></span> </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 \Rightarrow -q\xi _{y}+qv_{x}B_{z}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mo>−</mo> <mi>q</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>q</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow -q\xi _{y}+qv_{x}B_{z}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/394f60ff06c152002ddb7ca125adccc322470f7e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.151ex; height:2.843ex;" alt="{\displaystyle \Rightarrow -q\xi _{y}+qv_{x}B_{z}=0}" /></span> </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 \xi _{y}=v_{x}B_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{y}=v_{x}B_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944d628bc222fb88b8fe349f1cc4fc61271ceb0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.232ex; height:2.843ex;" alt="{\displaystyle \xi _{y}=v_{x}B_{z}}" /></span> </p><p>For electrons, the field points in the −<i>y</i> direction, and for holes, it points in the +<i>y</i> direction. </p><p>The <a href="/wiki/Electric_current" title="Electric current">electron current</a> <i>I</i> is given by <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=-qnv_{x}tW}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mi>n</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>t</mi> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=-qnv_{x}tW}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1dec5f642e67cb8224af0378dd4aa95eaa3991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.118ex; height:2.509ex;" alt="{\displaystyle I=-qnv_{x}tW}" /></span>. Sub <i>v</i><sub><i>x</i></sub> into the expression for <i>ξ</i><sub><i>y</i></sub>, </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 \xi _{y}=-{\frac {IB}{nqtW}}=+{\frac {R_{Hn}IB}{tW}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mi>B</mi> </mrow> <mrow> <mi>n</mi> <mi>q</mi> <mi>t</mi> <mi>W</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mi>I</mi> <mi>B</mi> </mrow> <mrow> <mi>t</mi> <mi>W</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{y}=-{\frac {IB}{nqtW}}=+{\frac {R_{Hn}IB}{tW}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79a6f7b6bc1f715e1f64e73f126ff5309ab3a2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.67ex; height:5.676ex;" alt="{\displaystyle \xi _{y}=-{\frac {IB}{nqtW}}=+{\frac {R_{Hn}IB}{tW}}}" /></span> </p><p>where <i>R<sub>Hn</sub></i> is the Hall coefficient for electron, and is defined as <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 R_{Hn}=-{\frac {1}{nq}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{Hn}=-{\frac {1}{nq}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d029af861e76eeeabce7cba55320c6a2b87f70" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.649ex; height:5.676ex;" alt="{\displaystyle R_{Hn}=-{\frac {1}{nq}}}" /></span> </p><p>Since <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 \xi _{y}={\frac {V_{H}}{W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mi>W</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{y}={\frac {V_{H}}{W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b2972552235a1b114ab3f2be37fc40eec64496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.049ex; height:5.343ex;" alt="{\displaystyle \xi _{y}={\frac {V_{H}}{W}}}" /></span> <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 R_{Hn}=-{\frac {1}{nq}}={\frac {V_{Hn}t}{IB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> <mrow> <mi>I</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{Hn}=-{\frac {1}{nq}}={\frac {V_{Hn}t}{IB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc25c8c8b319d5afb13384fc8613ef1d24fdcf2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.456ex; height:5.676ex;" alt="{\displaystyle R_{Hn}=-{\frac {1}{nq}}={\frac {V_{Hn}t}{IB}}}" /></span> </p><p>Similarly, for holes <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 R_{Hp}={\frac {1}{pq}}={\frac {V_{Hp}t}{IB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>p</mi> </mrow> </msub> <mi>t</mi> </mrow> <mrow> <mi>I</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{Hp}={\frac {1}{pq}}={\frac {V_{Hp}t}{IB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ba9c7cd663b509a69f40edf8299de7c1d7789d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.104ex; height:6.009ex;" alt="{\displaystyle R_{Hp}={\frac {1}{pq}}={\frac {V_{Hp}t}{IB}}}" /></span> </p><p>From the Hall coefficient, we can obtain the carrier mobility as follows: <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 {\begin{aligned}\mu _{n}&=\left(-nq\right)\mu _{n}\left(-{\frac {1}{nq}}\right)\\&=-\sigma _{n}R_{Hn}\\&=-{\frac {\sigma _{n}V_{Hn}t}{IB}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>n</mi> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mi>q</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> <mrow> <mi>I</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{n}&=\left(-nq\right)\mu _{n}\left(-{\frac {1}{nq}}\right)\\&=-\sigma _{n}R_{Hn}\\&=-{\frac {\sigma _{n}V_{Hn}t}{IB}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8d63561cd12710a39d666a70dc1eb55115fc0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.465ex; margin-bottom: -0.206ex; width:24.476ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\mu _{n}&=\left(-nq\right)\mu _{n}\left(-{\frac {1}{nq}}\right)\\&=-\sigma _{n}R_{Hn}\\&=-{\frac {\sigma _{n}V_{Hn}t}{IB}}\end{aligned}}}" /></span> </p><p>Similarly, <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 \mu _{p}={\frac {\sigma _{p}V_{Hp}t}{IB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mi>p</mi> </mrow> </msub> <mi>t</mi> </mrow> <mrow> <mi>I</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{p}={\frac {\sigma _{p}V_{Hp}t}{IB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cdbb7bbb4a0489b1bd3bc001d76a0e82ff6d38a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.495ex; height:5.509ex;" alt="{\displaystyle \mu _{p}={\frac {\sigma _{p}V_{Hp}t}{IB}}}" /></span> </p><p>Here the value of <i>V<sub>Hp</sub></i> (Hall voltage), <i>t</i> (sample thickness), <i>I</i> (current) and <i>B</i> (magnetic field) can be measured directly, and the conductivities <i>σ</i><sub>n</sub> or <i>σ</i><sub>p</sub> are either known or can be obtained from measuring the resistivity. </p> <div class="mw-heading mw-heading3"><h3 id="Field-effect_mobility">Field-effect mobility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=26" title="Edit section: Field-effect mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/MOSFET" title="MOSFET">MOSFET</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Wien_effect" title="Wien effect">Wien effect</a>.</div> <p>The mobility can also be measured using a <a href="/wiki/Field-effect_transistor" title="Field-effect transistor">field-effect transistor</a> (FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement"). </p><p>The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements.<sup id="cite_ref-Rost_29-0" class="reference"><a href="#cite_note-Rost-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> (See <a href="/wiki/MOSFET" title="MOSFET">MOSFET</a> for a description of the different modes or regions of operation.) </p> <div class="mw-heading mw-heading4"><h4 id="Using_saturation_mode">Using saturation mode</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=27" title="Edit section: Using saturation mode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this technique,<sup id="cite_ref-Rost_29-1" class="reference"><a href="#cite_note-Rost-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> for each fixed gate voltage V<sub>GS</sub>, the drain-source voltage V<sub>DS</sub> is increased until the current I<sub>D</sub> saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope <i>m</i><sub>sat</sub> is measured. Then the mobility is: <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 \mu =m_{\text{sat}}^{2}{\frac {2L}{W}}{\frac {1}{C_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sat</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mi>W</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =m_{\text{sat}}^{2}{\frac {2L}{W}}{\frac {1}{C_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd8ce5fa6246b20daf7d4ddbb7ff1f299417e21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.761ex; height:5.676ex;" alt="{\displaystyle \mu =m_{\text{sat}}^{2}{\frac {2L}{W}}{\frac {1}{C_{i}}}}" /></span> where <i>L</i> and <i>W</i> are the length and width of the channel and <i>C</i><sub><i>i</i></sub> is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode: <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 I_{D}={\frac {\mu C_{i}}{2}}{\frac {W}{L}}(V_{GS}-V_{th})^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mi>L</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>S</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{D}={\frac {\mu C_{i}}{2}}{\frac {W}{L}}(V_{GS}-V_{th})^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720dbb97996a503cdea56c9d77772697cda12b86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.103ex; height:5.509ex;" alt="{\displaystyle I_{D}={\frac {\mu C_{i}}{2}}{\frac {W}{L}}(V_{GS}-V_{th})^{2}.}" /></span> where <i>V</i><sub>th</sub> is the threshold voltage. This approximation ignores the <a href="/wiki/Early_effect" title="Early effect">Early effect</a> (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.<sup id="cite_ref-Rost2_30-0" class="reference"><a href="#cite_note-Rost2-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Using_the_linear_region">Using the linear region</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=28" title="Edit section: Using the linear region"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this technique,<sup id="cite_ref-Rost_29-2" class="reference"><a href="#cite_note-Rost-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> the transistor is operated in the linear region (or "ohmic mode"), where V<sub>DS</sub> is small and <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_{D}\propto V_{GS}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>∝</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{D}\propto V_{GS}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb03b8045a0584dafd484c244e5344447c5891e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.654ex; height:2.509ex;" alt="{\displaystyle I_{D}\propto V_{GS}}" /></span> with slope <i>m</i><sub>lin</sub>. Then the mobility is: <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 \mu =m_{\text{lin}}{\frac {L}{W}}{\frac {1}{V_{DS}}}{\frac {1}{C_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>lin</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>W</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>S</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =m_{\text{lin}}{\frac {L}{W}}{\frac {1}{V_{DS}}}{\frac {1}{C_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad22b077e5484696df5d05cd2250a48196647a9e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.662ex; height:5.676ex;" alt="{\displaystyle \mu =m_{\text{lin}}{\frac {L}{W}}{\frac {1}{V_{DS}}}{\frac {1}{C_{i}}}.}" /></span> This equation comes from the approximate equation for a MOSFET in the linear region: <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 I_{D}=\mu C_{i}{\frac {W}{L}}\left((V_{GS}-V_{th})V_{DS}-{\frac {V_{DS}^{2}}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mi>L</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>S</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>S</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{D}=\mu C_{i}{\frac {W}{L}}\left((V_{GS}-V_{th})V_{DS}-{\frac {V_{DS}^{2}}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f191ad5756741fc3174b7906204f1f4838fa8105" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.327ex; height:7.509ex;" alt="{\displaystyle I_{D}=\mu C_{i}{\frac {W}{L}}\left((V_{GS}-V_{th})V_{DS}-{\frac {V_{DS}^{2}}{2}}\right)}" /></span> In practice, this technique may overestimate the true mobility, because if V<sub>DS</sub> is not small enough and V<sub>G</sub> is not large enough, the MOSFET may not stay in the linear region.<sup id="cite_ref-Rost2_30-1" class="reference"><a href="#cite_note-Rost2-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Optical_mobility">Optical mobility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=29" title="Edit section: Optical mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Electron mobility may be determined from non-contact laser <a href="/w/index.php?title=Photo-reflectance_technique&action=edit&redlink=1" class="new" title="Photo-reflectance technique (page does not exist)">photo-reflectance technique</a> measurements. A series of photo-reflectance measurements are made as the sample is stepped through focus. The electron diffusion length and recombination time are determined by a regressive fit to the data. Then the Einstein relation is used to calculate the mobility.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Terahertz_mobility">Terahertz mobility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=30" title="Edit section: Terahertz mobility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Electron mobility can be calculated from time-resolved <a href="/wiki/Terahertz_time-domain_spectroscopy" title="Terahertz time-domain spectroscopy">terahertz probe</a> measurement.<sup id="cite_ref-UlbrichtHendry2011_33-0" class="reference"><a href="#cite_note-UlbrichtHendry2011-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lloyd-HughesJeon2012_34-0" class="reference"><a href="#cite_note-Lloyd-HughesJeon2012-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Femtosecond_laser" class="mw-redirect" title="Femtosecond laser">Femtosecond laser</a> pulses excite the semiconductor and the resulting <a href="/wiki/Photoconductivity" title="Photoconductivity">photoconductivity</a> is measured using a terahertz probe, which detects changes in the terahertz electric field.<sup id="cite_ref-EversSchins2015_35-0" class="reference"><a href="#cite_note-EversSchins2015-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Time_resolved_microwave_conductivity_(TRMC)"><span id="Time_resolved_microwave_conductivity_.28TRMC.29"></span>Time resolved microwave conductivity (TRMC)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=31" title="Edit section: Time resolved microwave conductivity (TRMC)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Time_resolved_microwave_conductivity" title="Time resolved microwave conductivity">Time resolved microwave conductivity</a></div> <p>A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC).<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> A pulsed optical laser is used to create electrons and holes in a semiconductor, which are then detected as an increase in photoconductance. With knowledge of the sample absorbance, dimensions, and incident laser fluence, the parameter <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 \phi \Sigma \mu =\phi (\mu _{e}+\mu _{h})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mi mathvariant="normal">Σ</mi> <mi>μ</mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \Sigma \mu =\phi (\mu _{e}+\mu _{h})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b1283909a52ce64ef037a593fda472d036a698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.58ex; height:2.843ex;" alt="{\displaystyle \phi \Sigma \mu =\phi (\mu _{e}+\mu _{h})}" /></span> can be evaluated, where <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" /></span> is the carrier generation yield (between 0 and 1), <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 _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7072c3c52f9f8228872e82dc61668e0bc7b6d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.4ex; height:2.176ex;" alt="{\displaystyle \mu _{e}}" /></span> is the electron mobility and <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 _{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aa30d1505e2ca4947219f340aae30b6e015bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.581ex; height:2.176ex;" alt="{\displaystyle \mu _{h}}" /></span> is the hole mobility. <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 \phi \Sigma \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mi mathvariant="normal">Σ</mi> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \Sigma \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d8d061a8798ebc6080ade3e8396505043acb35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.465ex; height:2.676ex;" alt="{\displaystyle \phi \Sigma \mu }" /></span> has the same dimensions as mobility, but carrier type (electron or hole) is obscured. </p> <div class="mw-heading mw-heading2"><h2 id="Doping_concentration_dependence_in_heavily-doped_silicon">Doping concentration dependence in heavily-doped silicon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=32" title="Edit section: Doping concentration dependence in heavily-doped silicon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Charge_carrier" title="Charge carrier">charge carriers</a> in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility. </p><p>While there is considerable scatter in the <a href="/wiki/Experimental_data" title="Experimental data">experimental data</a>, for noncompensated material (no counter doping) for heavily doped substrates (i.e. <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 10^{18}\mathrm {cm} ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{18}\mathrm {cm} ^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204d8ec444bf0552026bd88e36543f593d9e1b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.502ex; height:2.676ex;" alt="{\displaystyle 10^{18}\mathrm {cm} ^{-3}}" /></span> and up), the mobility in silicon is often characterized by the <a href="/wiki/Empirical_relationship" title="Empirical relationship">empirical relationship</a>:<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> <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 \mu =\mu _{o}+{\frac {\mu _{1}}{1+\left({\frac {N}{N_{\text{ref}}}}\right)^{\alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>ref</mtext> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{o}+{\frac {\mu _{1}}{1+\left({\frac {N}{N_{\text{ref}}}}\right)^{\alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76985e2fced0c8afedb25831b0d7164c20466203" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.515ex; height:7.509ex;" alt="{\displaystyle \mu =\mu _{o}+{\frac {\mu _{1}}{1+\left({\frac {N}{N_{\text{ref}}}}\right)^{\alpha }}}}" /></span> where <i>N</i> is the doping concentration (either <i>N<sub>D</sub></i> or <i>N<sub>A</sub></i>), and <i>N</i><sub>ref</sub> and α are fitting parameters. At <a href="/wiki/Room_temperature" title="Room temperature">room temperature</a>, the above equation becomes: </p><p>Majority carriers:<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <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 \mu _{n}(N_{D})=65+{\frac {1265}{1+\left({\frac {N_{D}}{8.5\times 10^{16}}}\right)^{0.72}}}}"> <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> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>65</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1265</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mn>8.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0.72</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}(N_{D})=65+{\frac {1265}{1+\left({\frac {N_{D}}{8.5\times 10^{16}}}\right)^{0.72}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbc8080c441b103672b476989e11696c0113f17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:34.281ex; height:8.343ex;" alt="{\displaystyle \mu _{n}(N_{D})=65+{\frac {1265}{1+\left({\frac {N_{D}}{8.5\times 10^{16}}}\right)^{0.72}}}}" /></span> <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 \mu _{p}(N_{A})=48+{\frac {447}{1+\left({\frac {N_{A}}{6.3\times 10^{16}}}\right)^{0.76}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>48</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>447</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <mn>6.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0.76</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{p}(N_{A})=48+{\frac {447}{1+\left({\frac {N_{A}}{6.3\times 10^{16}}}\right)^{0.76}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a593e0fc0ab9d17d608e67e7488ab2a6fe2408" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:33.994ex; height:8.343ex;" alt="{\displaystyle \mu _{p}(N_{A})=48+{\frac {447}{1+\left({\frac {N_{A}}{6.3\times 10^{16}}}\right)^{0.76}}}}" /></span> </p><p>Minority carriers:<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> <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 \mu _{n}(N_{A})=232+{\frac {1180}{1+\left({\frac {N_{A}}{8\times 10^{16}}}\right)^{0.9}}}}"> <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> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>232</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1180</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0.9</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}(N_{A})=232+{\frac {1180}{1+\left({\frac {N_{A}}{8\times 10^{16}}}\right)^{0.9}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d3bfcb9afa87aca09585c219c4f85bbf624502" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:33.214ex; height:8.343ex;" alt="{\displaystyle \mu _{n}(N_{A})=232+{\frac {1180}{1+\left({\frac {N_{A}}{8\times 10^{16}}}\right)^{0.9}}}}" /></span> <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 \mu _{p}(N_{D})=130+{\frac {370}{1+\left({\frac {N_{D}}{8\times 10^{17}}}\right)^{1.25}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>130</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>370</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1.25</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{p}(N_{D})=130+{\frac {370}{1+\left({\frac {N_{D}}{8\times 10^{17}}}\right)^{1.25}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236b369d3402256f4473c8bfc9b126b1e8be1c5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:34.005ex; height:8.343ex;" alt="{\displaystyle \mu _{p}(N_{D})=130+{\frac {370}{1+\left({\frac {N_{D}}{8\times 10^{17}}}\right)^{1.25}}}}" /></span> </p><p>These equations apply only to silicon, and only under low field. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=33" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Speed_of_electricity" title="Speed of electricity">Speed of electricity</a></li></ul> <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=Electron_mobility&action=edit&section=34" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.matprop.ru/">"NSM Archive - Physical Properties of Semiconductors"</a>. <i>www.matprop.ru</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.matprop.ru&rft.atitle=NSM+Archive+-+Physical+Properties+of+Semiconductors&rft_id=http%3A%2F%2Fwww.matprop.ru%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><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="CITEREFUmanskyHeiblumLevinsonSmet2009" class="citation journal cs1">Umansky, V.; Heiblum, M.; Levinson, Y.; Smet, J.; Nübler, J.; Dolev, M. (2009). "MBE growth of ultra-low disorder 2DEG with mobility exceeding 35×106 cm2 V−1 s−1". <i>Journal of Crystal Growth</i>. <b>311</b> (7): <span class="nowrap">1658–</span>1661. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JCrGr.311.1658U">2009JCrGr.311.1658U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcrysgro.2008.09.151">10.1016/j.jcrysgro.2008.09.151</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Crystal+Growth&rft.atitle=MBE+growth+of+ultra-low+disorder+2DEG+with+mobility+exceeding+35%C3%97106+cm2+V%E2%88%921+s%E2%88%921&rft.volume=311&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1658-%3C%2Fspan%3E1661&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.jcrysgro.2008.09.151&rft_id=info%3Abibcode%2F2009JCrGr.311.1658U&rft.aulast=Umansky&rft.aufirst=V.&rft.au=Heiblum%2C+M.&rft.au=Levinson%2C+Y.&rft.au=Smet%2C+J.&rft.au=N%C3%BCbler%2C+J.&rft.au=Dolev%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDürkopGettyCobasFuhrer2004" class="citation journal cs1">Dürkop, T.; Getty, S. A.; Cobas, Enrique; Fuhrer, M. S. (2004). "Extraordinary Mobility in Semiconducting Carbon Nanotubes". <i>Nano Letters</i>. <b>4</b> (1): 35. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004NanoL...4...35D">2004NanoL...4...35D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fnl034841q">10.1021/nl034841q</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:45010238">45010238</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nano+Letters&rft.atitle=Extraordinary+Mobility+in+Semiconducting+Carbon+Nanotubes&rft.volume=4&rft.issue=1&rft.pages=35&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45010238%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1021%2Fnl034841q&rft_id=info%3Abibcode%2F2004NanoL...4...35D&rft.aulast=D%C3%BCrkop&rft.aufirst=T.&rft.au=Getty%2C+S.+A.&rft.au=Cobas%2C+Enrique&rft.au=Fuhrer%2C+M.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-:2-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_4-1"><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="CITEREFBolotinSikesJiangKlima2008" class="citation journal cs1">Bolotin, K; Sikes, K; Jiang, Z; Klima, M; Fudenberg, G; Hone, J; Kim, P; Stormer, H (2008). "Ultrahigh electron mobility in suspended graphene". <i>Solid State Communications</i>. <b>146</b> (9): <span class="nowrap">351–</span>355. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0802.2389">0802.2389</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SSCom.146..351B">2008SSCom.146..351B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ssc.2008.02.024">10.1016/j.ssc.2008.02.024</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:118392999">118392999</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Solid+State+Communications&rft.atitle=Ultrahigh+electron+mobility+in+suspended+graphene&rft.volume=146&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E351-%3C%2Fspan%3E355&rft.date=2008&rft_id=info%3Aarxiv%2F0802.2389&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118392999%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.ssc.2008.02.024&rft_id=info%3Abibcode%2F2008SSCom.146..351B&rft.aulast=Bolotin&rft.aufirst=K&rft.au=Sikes%2C+K&rft.au=Jiang%2C+Z&rft.au=Klima%2C+M&rft.au=Fudenberg%2C+G&rft.au=Hone%2C+J&rft.au=Kim%2C+P&rft.au=Stormer%2C+H&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNawrocki2016" class="citation journal cs1">Nawrocki, Robert (2016). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Faelm.201500452">"300-nm Imperceptible, Ultraflexible, and Biocompatible e-Skin Fit with Tactile Sensors and Organic Transistors"</a>. <i>Advanced Electronic Materials</i>. <b>2</b> (4): 1500452. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Faelm.201500452">10.1002/aelm.201500452</a></span>. <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:138355533">138355533</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advanced+Electronic+Materials&rft.atitle=300-nm+Imperceptible%2C+Ultraflexible%2C+and+Biocompatible+e-Skin+Fit+with+Tactile+Sensors+and+Organic+Transistors&rft.volume=2&rft.issue=4&rft.pages=1500452&rft.date=2016&rft_id=info%3Adoi%2F10.1002%2Faelm.201500452&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A138355533%23id-name%3DS2CID&rft.aulast=Nawrocki&rft.aufirst=Robert&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Faelm.201500452&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChung,_Y._J.Wang,_C.Singh,_S._K.Gupta,_A.2022" class="citation journal cs1">Chung, Y. J., Wang, C., Singh, S. K., Gupta, A., Baldwin, K. W., West, K. W., Shayegan, M., Pfeiffer, L. N., Winkler, R. (14 March 2022). "Record-quality GaAs two-dimensional hole systems". <i>Physical Review Materials</i>. <b>6</b> (3): 034005. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2203.10713">2203.10713</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022PhRvM...6c4005C">2022PhRvM...6c4005C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevMaterials.6.034005">10.1103/PhysRevMaterials.6.034005</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2475-9953">2475-9953</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Materials&rft.atitle=Record-quality+GaAs+two-dimensional+hole+systems&rft.volume=6&rft.issue=3&rft.pages=034005&rft.date=2022-03-14&rft_id=info%3Aarxiv%2F2203.10713&rft.issn=2475-9953&rft_id=info%3Adoi%2F10.1103%2FPhysRevMaterials.6.034005&rft_id=info%3Abibcode%2F2022PhRvM...6c4005C&rft.au=Chung%2C+Y.+J.&rft.au=Wang%2C+C.&rft.au=Singh%2C+S.+K.&rft.au=Gupta%2C+A.&rft.au=Baldwin%2C+K.+W.&rft.au=West%2C+K.+W.&rft.au=Shayegan%2C+M.&rft.au=Pfeiffer%2C+L.+N.&rft.au=Winkler%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-DurkopGetty2004-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-DurkopGetty2004_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDürkopGettyCobasFuhrer2004" class="citation journal cs1">Dürkop, T.; Getty, S. A.; Cobas, Enrique; Fuhrer, M. S. (2004). "Extraordinary Mobility in Semiconducting Carbon Nanotubes". <i>Nano Letters</i>. <b>4</b> (1): <span class="nowrap">35–</span>39. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004NanoL...4...35D">2004NanoL...4...35D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fnl034841q">10.1021/nl034841q</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:45010238">45010238</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nano+Letters&rft.atitle=Extraordinary+Mobility+in+Semiconducting+Carbon+Nanotubes&rft.volume=4&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E35-%3C%2Fspan%3E39&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A45010238%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1021%2Fnl034841q&rft_id=info%3Abibcode%2F2004NanoL...4...35D&rft.aulast=D%C3%BCrkop&rft.aufirst=T.&rft.au=Getty%2C+S.+A.&rft.au=Cobas%2C+Enrique&rft.au=Fuhrer%2C+M.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-SnowCampbell2005-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-SnowCampbell2005_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSnowCampbellAnconaNovak2005" class="citation journal cs1">Snow, E. S.; Campbell, P. M.; Ancona, M. G.; Novak, J. P. (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170924194648/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA573996">"High-mobility carbon-nanotube thin-film transistors on a polymeric substrate"</a>. <i>Applied Physics Letters</i>. <b>86</b> (3): 033105. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005ApPhL..86c3105S">2005ApPhL..86c3105S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1854721">10.1063/1.1854721</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-6951">0003-6951</a>. Archived from <a rel="nofollow" class="external text" href="http://www.dtic.mil/get-tr-doc/pdf?AD=ADA573996">the original</a> on September 24, 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+Letters&rft.atitle=High-mobility+carbon-nanotube+thin-film+transistors+on+a+polymeric+substrate&rft.volume=86&rft.issue=3&rft.pages=033105&rft.date=2005&rft.issn=0003-6951&rft_id=info%3Adoi%2F10.1063%2F1.1854721&rft_id=info%3Abibcode%2F2005ApPhL..86c3105S&rft.aulast=Snow&rft.aufirst=E.+S.&rft.au=Campbell%2C+P.+M.&rft.au=Ancona%2C+M.+G.&rft.au=Novak%2C+J.+P.&rft_id=http%3A%2F%2Fwww.dtic.mil%2Fget-tr-doc%2Fpdf%3FAD%3DADA573996&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShinGamageDingChen2022" class="citation journal cs1">Shin, Jungwoo; Gamage, Geethal Amila; Ding, Zhiwei; Chen, Ke; Tian, Fei; Qian, Xin; Zhou, Jiawei; Lee, Hwijong; Zhou, Jianshi; Shi, Li; Nguyen, Thanh; Han, Fei; Li, Mingda; Broido, David; Schmidt, Aaron; Ren, Zhifeng; Chen, Gang (2022). <a rel="nofollow" class="external text" href="https://www.science.org/doi/10.1126/science.abn4290">"High ambipolar mobility in cubic boron arsenide"</a>. <i>Science</i>. <b>377</b> (6604): <span class="nowrap">437–</span>440. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022Sci...377..437S">2022Sci...377..437S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.abn4290">10.1126/science.abn4290</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35862526">35862526</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:250952849">250952849</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=High+ambipolar+mobility+in+cubic+boron+arsenide&rft.volume=377&rft.issue=6604&rft.pages=%3Cspan+class%3D%22nowrap%22%3E437-%3C%2Fspan%3E440&rft.date=2022&rft_id=info%3Adoi%2F10.1126%2Fscience.abn4290&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250952849%23id-name%3DS2CID&rft_id=info%3Apmid%2F35862526&rft_id=info%3Abibcode%2F2022Sci...377..437S&rft.aulast=Shin&rft.aufirst=Jungwoo&rft.au=Gamage%2C+Geethal+Amila&rft.au=Ding%2C+Zhiwei&rft.au=Chen%2C+Ke&rft.au=Tian%2C+Fei&rft.au=Qian%2C+Xin&rft.au=Zhou%2C+Jiawei&rft.au=Lee%2C+Hwijong&rft.au=Zhou%2C+Jianshi&rft.au=Shi%2C+Li&rft.au=Nguyen%2C+Thanh&rft.au=Han%2C+Fei&rft.au=Li%2C+Mingda&rft.au=Broido%2C+David&rft.au=Schmidt%2C+Aaron&rft.au=Ren%2C+Zhifeng&rft.au=Chen%2C+Gang&rft_id=https%3A%2F%2Fwww.science.org%2Fdoi%2F10.1126%2Fscience.abn4290&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeStolteWürthner2013" class="citation journal cs1">He, Tao; Stolte, Matthias; Würthner, Frank (2013-12-23). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fadma.201303392">"Air-Stable n-Channel Organic Single Crystal Field-Effect Transistors Based on Microribbons of Core-Chlorinated Naphthalene Diimide"</a>. <i>Advanced Materials</i>. <b>25</b> (48): <span class="nowrap">6951–</span>6955. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013AdM....25.6951H">2013AdM....25.6951H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fadma.201303392">10.1002/adma.201303392</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24105872">24105872</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advanced+Materials&rft.atitle=Air-Stable+n-Channel+Organic+Single+Crystal+Field-Effect+Transistors+Based+on+Microribbons+of+Core-Chlorinated+Naphthalene+Diimide&rft.volume=25&rft.issue=48&rft.pages=%3Cspan+class%3D%22nowrap%22%3E6951-%3C%2Fspan%3E6955&rft.date=2013-12-23&rft_id=info%3Apmid%2F24105872&rft_id=info%3Adoi%2F10.1002%2Fadma.201303392&rft_id=info%3Abibcode%2F2013AdM....25.6951H&rft.aulast=He&rft.aufirst=Tao&rft.au=Stolte%2C+Matthias&rft.au=W%C3%BCrthner%2C+Frank&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fadma.201303392&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYuan2014" class="citation journal cs1">Yuan, Yongbo (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fncomms4005">"Ultra-high mobility transparent organic thin film transistors grown by an off-centre spin-coating method"</a>. <i>Nature Communications</i>. <b>5</b>: 3005. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014NatCo...5.3005Y">2014NatCo...5.3005Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fncomms4005">10.1038/ncomms4005</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24398476">24398476</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Ultra-high+mobility+transparent+organic+thin+film+transistors+grown+by+an+off-centre+spin-coating+method&rft.volume=5&rft.pages=3005&rft.date=2014&rft_id=info%3Apmid%2F24398476&rft_id=info%3Adoi%2F10.1038%2Fncomms4005&rft_id=info%3Abibcode%2F2014NatCo...5.3005Y&rft.aulast=Yuan&rft.aufirst=Yongbo&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252Fncomms4005&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" 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="CITEREFHeremans2015" class="citation journal cs1">Heremans, Paul (2015). "Mechanical and Electronic Properties of Thin-Film Transistors on Plastic, and Their Integration in Flexible Electronic Applications". <i>Advanced Materials</i>. <b>28</b> (22): <span class="nowrap">4266–</span>4282. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fadma.201504360">10.1002/adma.201504360</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26707947">26707947</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:25457390">25457390</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advanced+Materials&rft.atitle=Mechanical+and+Electronic+Properties+of+Thin-Film+Transistors+on+Plastic%2C+and+Their+Integration+in+Flexible+Electronic+Applications&rft.volume=28&rft.issue=22&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4266-%3C%2Fspan%3E4282&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A25457390%23id-name%3DS2CID&rft_id=info%3Apmid%2F26707947&rft_id=info%3Adoi%2F10.1002%2Fadma.201504360&rft.aulast=Heremans&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-Mitin-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mitin_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mitin_13-1"><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="CITEREFVladimir_Vasilʹevich_MitinVi︠a︡cheslav_Aleksandrovich_KochelapMichael_A._Stroscio1999" class="citation book cs1">Vladimir Vasilʹevich Mitin; Vi︠a︡cheslav Aleksandrovich Kochelap; Michael A. Stroscio (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Wzo4IdxS48oC&pg=PA308"><i>Quantum heterostructures: microelectronics and optoelectronics</i></a>. Cambridge University Press. pp. <span class="nowrap">307–</span>9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-63635-3" title="Special:BookSources/978-0-521-63635-3"><bdi>978-0-521-63635-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+heterostructures%3A+microelectronics+and+optoelectronics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E307-%3C%2Fspan%3E9&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-63635-3&rft.au=Vladimir+Vasil%CA%B9evich+Mitin&rft.au=Vi%EF%B8%A0a%EF%B8%A1cheslav+Aleksandrovich+Kochelap&rft.au=Michael+A.+Stroscio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWzo4IdxS48oC%26pg%3DPA308&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-Singh-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Singh_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Singh_14-1"><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="CITEREFSingh2008" class="citation book cs1">Singh (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2aqtlybkFE0C&pg=PA77"><i>Electronic Devices And Integrated Circuits</i></a>. PHI Learning Pvt. Ltd. pp. 77–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-3192-1" title="Special:BookSources/978-81-203-3192-1"><bdi>978-81-203-3192-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Devices+And+Integrated+Circuits&rft.pages=77-&rft.pub=PHI+Learning+Pvt.+Ltd.&rft.date=2008&rft.isbn=978-81-203-3192-1&rft.au=Singh&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2aqtlybkFE0C%26pg%3DPA77&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-sct-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-sct_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sct_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sct_15-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-sct_15-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-sct_15-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-sct_15-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-sct_15-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-sct_15-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text">Ferry, David K. Semiconductor transport. London: Taylor & Francis, 2000. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7484-0865-7" title="Special:BookSources/0-7484-0865-7">0-7484-0865-7</a> (hbk.), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7484-0866-5" title="Special:BookSources/0-7484-0866-5">0-7484-0866-5</a> (pbk.)</span> </li> <li id="cite_note-ssp-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-ssp_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ssp_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ssp_16-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-ssp_16-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Ibach, Harald.; Luth, Hans. Solid-state physics : an introduction to principles of materials science / Harald Ibach, Hans Luth. New York: Springer, 2009. -(Advanced texts in physics) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-93803-3" title="Special:BookSources/978-3-540-93803-3">978-3-540-93803-3</a></span> </li> <li id="cite_note-bulusu-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-bulusu_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bulusu_17-1"><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="CITEREFBulusu2008" class="citation journal cs1">Bulusu, A. (2008). "Review of electronic transport models for thermoelectric materials". <i>Superlattices and Microstructures</i>. <b>44</b> (1): <span class="nowrap">1–</span>36. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SuMi...44....1B">2008SuMi...44....1B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spmi.2008.02.008">10.1016/j.spmi.2008.02.008</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Superlattices+and+Microstructures&rft.atitle=Review+of+electronic+transport+models+for+thermoelectric+materials&rft.volume=44&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E36&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.spmi.2008.02.008&rft_id=info%3Abibcode%2F2008SuMi...44....1B&rft.aulast=Bulusu&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span>.</span> </li> <li id="cite_note-pallab-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-pallab_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pallab_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-pallab_18-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Bhattacharya, Pallab. Semiconductor optoelectronic devices / Pallab Bhattacharya. Upper Saddle River (NJ): Prentice-Hall, 1997. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-495656-7" title="Special:BookSources/0-13-495656-7">0-13-495656-7</a> (nid.)</span> </li> <li id="cite_note-Takeda-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Takeda_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Takeda_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Y. Takeda and T.P. Pearsall, "Failure of Mattheissen's Rule in the Calculation of Carrier Mobility and Alloy Scattering Effects in Ga0.47In0.53As", Electronics Lett. 17, 573-574 (1981).</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeter_Y._YuManuel_Cardona2010" class="citation book cs1">Peter Y. Yu; Manuel Cardona (30 May 2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA205+"><i>Fundamentals of Semiconductors: Physics and Materials Properties</i></a>. Springer. pp. 205–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-00709-5" title="Special:BookSources/978-3-642-00709-5"><bdi>978-3-642-00709-5</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Semiconductors%3A+Physics+and+Materials+Properties&rft.pages=205-&rft.pub=Springer&rft.date=2010-05-30&rft.isbn=978-3-642-00709-5&rft.au=Peter+Y.+Yu&rft.au=Manuel+Cardona&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5aBuKYBT_hsC%26pg%3DPA205%2B&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAntonio_LuqueSteven_Hegedus2003" class="citation book cs1">Antonio Luque; Steven Hegedus (9 June 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u-bCMhl_JjQC"><i>Handbook of photovoltaic science and engineering</i></a>. John Wiley and Sons. p. 79, eq. 3.58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-49196-5" title="Special:BookSources/978-0-471-49196-5"><bdi>978-0-471-49196-5</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+photovoltaic+science+and+engineering&rft.pages=79%2C+eq.+3.58&rft.pub=John+Wiley+and+Sons&rft.date=2003-06-09&rft.isbn=978-0-471-49196-5&rft.au=Antonio+Luque&rft.au=Steven+Hegedus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Du-bCMhl_JjQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://www.knovel.com/web/portal/browse/display?_EXT_KNOVEL_DISPLAY_bookid=1081">weblink (subscription only)</a></span> </li> <li id="cite_note-BVZ-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-BVZ_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BVZ_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_7.htm">Chapter 2: Semiconductor Fundamentals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090121102148/http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_7.htm">Archived</a> 2009-01-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Online textbook by B. Van Zeghbroeck]</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHookHall1991" class="citation book cs1">Hook, J. R.; Hall, H. E. (1991-09-05). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DLHvAAAAMAAJ"><i>Solid State Physics</i></a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92804-1" title="Special:BookSources/978-0-471-92804-1"><bdi>978-0-471-92804-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solid+State+Physics&rft.pub=Wiley&rft.date=1991-09-05&rft.isbn=978-0-471-92804-1&rft.aulast=Hook&rft.aufirst=J.+R.&rft.au=Hall%2C+H.+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDLHvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAnderson1958" class="citation journal cs1">Anderson, P. W. (1958-03-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.109.1492">"Absence of Diffusion in Certain Random Lattices"</a>. <i>Physical Review</i>. <b>109</b> (5): <span class="nowrap">1492–</span>1505. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1958PhRv..109.1492A">1958PhRv..109.1492A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.109.1492">10.1103/PhysRev.109.1492</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=Absence+of+Diffusion+in+Certain+Random+Lattices&rft.volume=109&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1492-%3C%2Fspan%3E1505&rft.date=1958-03-01&rft_id=info%3Adoi%2F10.1103%2FPhysRev.109.1492&rft_id=info%3Abibcode%2F1958PhRv..109.1492A&rft.aulast=Anderson&rft.aufirst=P.+W.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.109.1492&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMott1967" class="citation journal cs1">Mott, N. F. (1967-01-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1080/00018736700101265">"Electrons in disordered structures"</a>. <i>Advances in Physics</i>. <b>16</b> (61): <span class="nowrap">49–</span>144. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967AdPhy..16...49M">1967AdPhy..16...49M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00018736700101265">10.1080/00018736700101265</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-8732">0001-8732</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Physics&rft.atitle=Electrons+in+disordered+structures&rft.volume=16&rft.issue=61&rft.pages=%3Cspan+class%3D%22nowrap%22%3E49-%3C%2Fspan%3E144&rft.date=1967-01-01&rft.issn=0001-8732&rft_id=info%3Adoi%2F10.1080%2F00018736700101265&rft_id=info%3Abibcode%2F1967AdPhy..16...49M&rft.aulast=Mott&rft.aufirst=N.+F.&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%2F00018736700101265&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrotherton2013" class="citation book cs1">Brotherton, S. D. (2013). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9783319000015"><i>Introduction to Thin Film Transistors: Physics and Technology of TFTs</i></a>. Springer International Publishing. p. 143. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-00001-5" title="Special:BookSources/978-3-319-00001-5"><bdi>978-3-319-00001-5</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+Thin+Film+Transistors%3A+Physics+and+Technology+of+TFTs&rft.pages=143&rft.pub=Springer+International+Publishing&rft.date=2013&rft.isbn=978-3-319-00001-5&rft.aulast=Brotherton&rft.aufirst=S.+D.&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9783319000015&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-:3-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_27-1"><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 class="citation book cs1"><a rel="nofollow" class="external text" href="https://global.oup.com/academic/product/electronic-processes-in-non-crystalline-materials-9780199645336?cc=us&lang=en&"><i>Electronic Processes in Non-Crystalline Materials</i></a>. Oxford Classic Texts in the Physical Sciences. Oxford, New York: Oxford University Press. 2012-03-24. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-964533-6" title="Special:BookSources/978-0-19-964533-6"><bdi>978-0-19-964533-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Processes+in+Non-Crystalline+Materials&rft.place=Oxford%2C+New+York&rft.series=Oxford+Classic+Texts+in+the+Physical+Sciences&rft.pub=Oxford+University+Press&rft.date=2012-03-24&rft.isbn=978-0-19-964533-6&rft_id=https%3A%2F%2Fglobal.oup.com%2Facademic%2Fproduct%2Felectronic-processes-in-non-crystalline-materials-9780199645336%3Fcc%3Dus%26lang%3Den%26&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEmin1974" class="citation journal cs1">Emin, David (1974-02-11). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.32.303">"Phonon-Assisted Jump Rate in Noncrystalline Solids"</a>. <i>Physical Review Letters</i>. <b>32</b> (6): <span class="nowrap">303–</span>307. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974PhRvL..32..303E">1974PhRvL..32..303E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.32.303">10.1103/PhysRevLett.32.303</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Phonon-Assisted+Jump+Rate+in+Noncrystalline+Solids&rft.volume=32&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E303-%3C%2Fspan%3E307&rft.date=1974-02-11&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.32.303&rft_id=info%3Abibcode%2F1974PhRvL..32..303E&rft.aulast=Emin&rft.aufirst=David&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.32.303&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-Rost-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rost_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rost_29-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Rost_29-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConstance_Rost-Bietsch2005" class="citation book cs1">Constance Rost-Bietsch (August 2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xxvt0CkVKaIC&pg=PA17"><i>Ambipolar and Light-Emitting Organic Field-Effect Transistors</i></a>. Cuvillier Verlag. pp. 17–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-86537-535-3" title="Special:BookSources/978-3-86537-535-3"><bdi>978-3-86537-535-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 March</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ambipolar+and+Light-Emitting+Organic+Field-Effect+Transistors&rft.pages=17-&rft.pub=Cuvillier+Verlag&rft.date=2005-08&rft.isbn=978-3-86537-535-3&rft.au=Constance+Rost-Bietsch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXxvt0CkVKaIC%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span>. This reference mistakenly leaves out a factor of 1/V<sub>DS</sub> in eqn (2.11). The correct version of that equation can be found, e.g., in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStassenDe_BoerIosadMorpurgo2004" class="citation journal cs1">Stassen, A. F.; De Boer, R. W. I.; Iosad, N. N.; Morpurgo, A. F. (2004). <a rel="nofollow" class="external text" href="http://resolver.tudelft.nl/uuid:868f9c8e-b994-47e8-b2fd-69cca21b1415">"Influence of the gate dielectric on the mobility of rubrene single-crystal field-effect transistors"</a>. <i>Applied Physics Letters</i>. <b>85</b> (17): <span class="nowrap">3899–</span>3901. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0407293">cond-mat/0407293</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004ApPhL..85.3899S">2004ApPhL..85.3899S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1812368">10.1063/1.1812368</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:119532427">119532427</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+Letters&rft.atitle=Influence+of+the+gate+dielectric+on+the+mobility+of+rubrene+single-crystal+field-effect+transistors&rft.volume=85&rft.issue=17&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3899-%3C%2Fspan%3E3901&rft.date=2004&rft_id=info%3Aarxiv%2Fcond-mat%2F0407293&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119532427%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1812368&rft_id=info%3Abibcode%2F2004ApPhL..85.3899S&rft.aulast=Stassen&rft.aufirst=A.+F.&rft.au=De+Boer%2C+R.+W.+I.&rft.au=Iosad%2C+N.+N.&rft.au=Morpurgo%2C+A.+F.&rft_id=http%3A%2F%2Fresolver.tudelft.nl%2Fuuid%3A868f9c8e-b994-47e8-b2fd-69cca21b1415&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-Rost2-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rost2_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rost2_30-1"><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="CITEREFConstance_Rost-Bietsch2005" class="citation book cs1">Constance Rost-Bietsch (August 2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xxvt0CkVKaIC&pg=PA19"><i>Ambipolar and Light-Emitting Organic Field-Effect Transistors</i></a>. Cuvillier Verlag. pp. 19–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-86537-535-3" title="Special:BookSources/978-3-86537-535-3"><bdi>978-3-86537-535-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">20 April</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ambipolar+and+Light-Emitting+Organic+Field-Effect+Transistors&rft.pages=19-&rft.pub=Cuvillier+Verlag&rft.date=2005-08&rft.isbn=978-3-86537-535-3&rft.au=Constance+Rost-Bietsch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXxvt0CkVKaIC%26pg%3DPA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span> "Extracting the field-effect mobility directly from the linear region of the output characteristic may yield larger values for the field-effect mobility than the actual one, since the drain current is linear only for very small VDS and large VG. In contrast, extracting the field-effect mobility from the saturated region might yield rather conservative values for the field-effect mobility, since the drain-current dependence from the gate-voltage becomes sub-quadratic for large VG as well as for small VDS."</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">W. Chism, "Precise Optical Measurement of Carrier Mobilities Using Z-scanning Laser Photoreflectance," <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1711.01138">arXiv:1711.01138</a> [physics:ins-det], Oct. 2017.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">W. Chism, "Z-scanning Laser Photoreflectance as a Tool for Characterization of Electronic Transport Properties," <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1808.01897">arXiv:1808.01897</a> [cond-mat:mes-hall], Aug. 2018.</span> </li> <li id="cite_note-UlbrichtHendry2011-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-UlbrichtHendry2011_33-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFUlbrichtHendryShanHeinz2011" class="citation journal cs1">Ulbricht, Ronald; Hendry, Euan; Shan, Jie; Heinz, Tony F.; Bonn, Mischa (2011). <a rel="nofollow" class="external text" href="https://ore.exeter.ac.uk/repository/bitstream/10871/15671/2/RevModPhys.83.543.pdf">"Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy"</a> <span class="cs1-format">(PDF)</span>. <i>Reviews of Modern Physics</i>. <b>83</b> (2): <span class="nowrap">543–</span>586. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011RvMP...83..543U">2011RvMP...83..543U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.83.543">10.1103/RevModPhys.83.543</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10871%2F15671">10871/15671</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Carrier+dynamics+in+semiconductors+studied+with+time-resolved+terahertz+spectroscopy&rft.volume=83&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E543-%3C%2Fspan%3E586&rft.date=2011&rft_id=info%3Ahdl%2F10871%2F15671&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.83.543&rft_id=info%3Abibcode%2F2011RvMP...83..543U&rft.aulast=Ulbricht&rft.aufirst=Ronald&rft.au=Hendry%2C+Euan&rft.au=Shan%2C+Jie&rft.au=Heinz%2C+Tony+F.&rft.au=Bonn%2C+Mischa&rft_id=https%3A%2F%2Fore.exeter.ac.uk%2Frepository%2Fbitstream%2F10871%2F15671%2F2%2FRevModPhys.83.543.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-Lloyd-HughesJeon2012-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lloyd-HughesJeon2012_34-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLloyd-HughesJeon2012" class="citation journal cs1">Lloyd-Hughes, James; Jeon, Tae-In (2012). "A Review of the Terahertz Conductivity of Bulk and Nano-Materials". <i>Journal of Infrared, Millimeter, and Terahertz Waves</i>. <b>33</b> (9): <span class="nowrap">871–</span>925. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012JIMTW..33..871L">2012JIMTW..33..871L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10762-012-9905-y">10.1007/s10762-012-9905-y</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1866-6892">1866-6892</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:13849900">13849900</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Infrared%2C+Millimeter%2C+and+Terahertz+Waves&rft.atitle=A+Review+of+the+Terahertz+Conductivity+of+Bulk+and+Nano-Materials&rft.volume=33&rft.issue=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E871-%3C%2Fspan%3E925&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2Fs10762-012-9905-y&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13849900%23id-name%3DS2CID&rft.issn=1866-6892&rft_id=info%3Abibcode%2F2012JIMTW..33..871L&rft.aulast=Lloyd-Hughes&rft.aufirst=James&rft.au=Jeon%2C+Tae-In&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-EversSchins2015-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-EversSchins2015_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEversSchinsAertsKulkarni2015" class="citation journal cs1">Evers, Wiel H.; Schins, Juleon M.; Aerts, Michiel; Kulkarni, Aditya; Capiod, Pierre; Berthe, Maxime; Grandidier, Bruno; Delerue, Christophe; van der Zant, Herre S. J.; van Overbeek, Carlo; Peters, Joep L.; Vanmaekelbergh, Daniel; Siebbeles, Laurens D. A. (2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4598357">"High charge mobility in two-dimensional percolative networks of PbSe quantum dots connected by atomic bonds"</a>. <i>Nature Communications</i>. <b>6</b>: 8195. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015NatCo...6.8195E">2015NatCo...6.8195E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fncomms9195">10.1038/ncomms9195</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2041-1723">2041-1723</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4598357">4598357</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26400049">26400049</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=High+charge+mobility+in+two-dimensional+percolative+networks+of+PbSe+quantum+dots+connected+by+atomic+bonds&rft.volume=6&rft.pages=8195&rft.date=2015&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4598357%23id-name%3DPMC&rft_id=info%3Abibcode%2F2015NatCo...6.8195E&rft_id=info%3Apmid%2F26400049&rft_id=info%3Adoi%2F10.1038%2Fncomms9195&rft.issn=2041-1723&rft.aulast=Evers&rft.aufirst=Wiel+H.&rft.au=Schins%2C+Juleon+M.&rft.au=Aerts%2C+Michiel&rft.au=Kulkarni%2C+Aditya&rft.au=Capiod%2C+Pierre&rft.au=Berthe%2C+Maxime&rft.au=Grandidier%2C+Bruno&rft.au=Delerue%2C+Christophe&rft.au=van+der+Zant%2C+Herre+S.+J.&rft.au=van+Overbeek%2C+Carlo&rft.au=Peters%2C+Joep+L.&rft.au=Vanmaekelbergh%2C+Daniel&rft.au=Siebbeles%2C+Laurens+D.+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4598357&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSavenijeFergusonKopidakisRumbles2013" class="citation journal cs1">Savenije, Tom J.; Ferguson, Andrew J.; Kopidakis, Nikos; Rumbles, Garry (2013-11-21). <a rel="nofollow" class="external text" href="https://doi.org/10.1021/jp406706u">"Revealing the Dynamics of Charge Carriers in Polymer:Fullerene Blends Using Photoinduced Time-Resolved Microwave Conductivity"</a>. <i>The Journal of Physical Chemistry C</i>. <b>117</b> (46): <span class="nowrap">24085–</span>24103. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fjp406706u">10.1021/jp406706u</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-7447">1932-7447</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Physical+Chemistry+C&rft.atitle=Revealing+the+Dynamics+of+Charge+Carriers+in+Polymer%3AFullerene+Blends+Using+Photoinduced+Time-Resolved+Microwave+Conductivity&rft.volume=117&rft.issue=46&rft.pages=%3Cspan+class%3D%22nowrap%22%3E24085-%3C%2Fspan%3E24103&rft.date=2013-11-21&rft_id=info%3Adoi%2F10.1021%2Fjp406706u&rft.issn=1932-7447&rft.aulast=Savenije&rft.aufirst=Tom+J.&rft.au=Ferguson%2C+Andrew+J.&rft.au=Kopidakis%2C+Nikos&rft.au=Rumbles%2C+Garry&rft_id=https%3A%2F%2Fdoi.org%2F10.1021%2Fjp406706u&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">B. L. Anderson and R. L. Anderson, "Fundamentals of Semiconductor Devices, " Mc Graw Hill, 2005</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCaugheyThomas1967" class="citation journal cs1">Caughey, D.M.; Thomas, R.E. (1967). "Carrier mobilities in silicon empirically related to doping and field". <i>Proceedings of the IEEE</i>. <b>55</b> (12): <span class="nowrap">2192–</span>2193. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPROC.1967.6123">10.1109/PROC.1967.6123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IEEE&rft.atitle=Carrier+mobilities+in+silicon+empirically+related+to+doping+and+field&rft.volume=55&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2192-%3C%2Fspan%3E2193&rft.date=1967&rft_id=info%3Adoi%2F10.1109%2FPROC.1967.6123&rft.aulast=Caughey&rft.aufirst=D.M.&rft.au=Thomas%2C+R.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDel_Alamo1985" class="citation journal cs1">Del Alamo, J (1985). "Measuring and modeling minority carrier transport in heavily doped silicon". <i>Solid-State Electronics</i>. <b>28</b> (1): <span class="nowrap">47–</span>54. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985SSEle..28...47D">1985SSEle..28...47D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0038-1101%2885%2990209-6">10.1016/0038-1101(85)90209-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Solid-State+Electronics&rft.atitle=Measuring+and+modeling+minority+carrier+transport+in+heavily+doped+silicon&rft.volume=28&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E54&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0038-1101%2885%2990209-6&rft_id=info%3Abibcode%2F1985SSEle..28...47D&rft.aulast=Del+Alamo&rft.aufirst=J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectron+mobility" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electron_mobility&action=edit&section=35" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://semiconductorglossary.com/default.asp?searchterm=electron+mobility">semiconductor glossary entry for electron mobility</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090104011855/http://semiconductorglossary.com/default.asp?searchterm=electron+mobility">Archived</a> 2009-01-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080723211649/http://ee.byu.edu/cleanroom/ResistivityCal.phtml">Resistivity and Mobility Calculator from the BYU Cleanroom</a></li> <li>Online lecture- <a rel="nofollow" class="external text" href="http://nanohub.org/resources/6151">Mobility</a> from an atomistic point of view</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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