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id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Electrical_resistance_standards" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical_resistance_standards"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Electrical resistance standards</span> </div> </a> <ul id="toc-Electrical_resistance_standards-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Research_status" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Research_status"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Research status</span> </div> </a> <ul id="toc-Research_status-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integer_quantum_Hall_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integer_quantum_Hall_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Integer quantum Hall effect</span> </div> </a> <button aria-controls="toc-Integer_quantum_Hall_effect-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 Integer quantum Hall effect subsection</span> </button> <ul id="toc-Integer_quantum_Hall_effect-sublist" class="vector-toc-list"> <li id="toc-Landau_levels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Landau_levels"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Landau levels</span> </div> </a> <ul id="toc-Landau_levels-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Density_of_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Density_of_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Density of states</span> </div> </a> <ul id="toc-Density_of_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Longitudinal_resistivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Longitudinal_resistivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Longitudinal resistivity</span> </div> </a> <ul id="toc-Longitudinal_resistivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transverse_resistivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transverse_resistivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Transverse resistivity</span> </div> </a> <ul id="toc-Transverse_resistivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Photonic_quantum_Hall_effect" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Photonic_quantum_Hall_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Photonic quantum Hall effect</span> </div> </a> <ul id="toc-Photonic_quantum_Hall_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_classification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topological_classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Topological classification</span> </div> </a> <ul id="toc-Topological_classification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bohr_atom_interpretation_of_the_von_Klitzing_constant" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bohr_atom_interpretation_of_the_von_Klitzing_constant"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bohr atom interpretation of the von Klitzing constant</span> </div> </a> <ul id="toc-Bohr_atom_interpretation_of_the_von_Klitzing_constant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relativistic_analogs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relativistic_analogs"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Relativistic analogs</span> </div> </a> <ul id="toc-Relativistic_analogs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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 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</div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quantum Hall effect</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. 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interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%B9%E0%A6%B2_%E0%A6%85%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="কোয়ান্টাম হল অবস্থা – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম হল অবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D1%8B_%D1%8D%D1%84%D0%B5%D0%BA%D1%82_%D0%A5%D0%BE%D0%BB%D0%B0" title="Квантавы эфект Хола – Belarusian" lang="be" hreflang="be" data-title="Квантавы эфект Хола" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Efecte_Hall_qu%C3%A0ntic" title="Efecte Hall quàntic – Catalan" lang="ca" hreflang="ca" data-title="Efecte Hall quàntic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%BD_Hall%C5%AFv_jev" title="Kvantový Hallův jev – Czech" lang="cs" hreflang="cs" data-title="Kvantový Hallův jev" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvante-Hall-effekten" title="Kvante-Hall-effekten – Danish" lang="da" hreflang="da" data-title="Kvante-Hall-effekten" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quanten-Hall-Effekt" title="Quanten-Hall-Effekt – German" lang="de" hreflang="de" data-title="Quanten-Hall-Effekt" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Efecto_Hall_cu%C3%A1ntico" title="Efecto Hall cuántico – Spanish" lang="es" hreflang="es" data-title="Efecto Hall cuántico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%B1_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C_%D9%87%D8%A7%D9%84" title="اثر کوانتومی هال – Persian" lang="fa" hreflang="fa" data-title="اثر کوانتومی هال" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Effet_Hall_quantique_entier" title="Effet Hall quantique entier – French" lang="fr" hreflang="fr" data-title="Effet Hall quantique entier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Iarmhairt_chandamach_Hall" title="Iarmhairt chandamach Hall – Irish" lang="ga" hreflang="ga" data-title="Iarmhairt chandamach Hall" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90_%ED%99%80_%ED%9A%A8%EA%B3%BC" title="양자 홀 효과 – Korean" lang="ko" hreflang="ko" data-title="양자 홀 효과" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvantni_Hallov_u%C4%8Dinak" title="Kvantni Hallov učinak – Croatian" lang="hr" hreflang="hr" data-title="Kvantni Hallov učinak" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_Hall_quantistico" title="Effetto Hall quantistico – Italian" lang="it" hreflang="it" data-title="Effetto Hall quantistico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A4%D7%A7%D7%98_%D7%94%D7%95%D7%9C_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99" title="אפקט הול הקוונטי – Hebrew" lang="he" hreflang="he" data-title="אפקט הול הקוונטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kvantu_Holla_efekts" title="Kvantu Holla efekts – Latvian" lang="lv" hreflang="lv" data-title="Kvantu Holla efekts" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantum-hall-effect" title="Kwantum-hall-effect – Dutch" lang="nl" hreflang="nl" data-title="Kwantum-hall-effect" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E3%83%9B%E3%83%BC%E3%83%AB%E5%8A%B9%E6%9E%9C" title="量子ホール効果 – Japanese" lang="ja" hreflang="ja" data-title="量子ホール効果" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvantisert_Hall-effekt" title="Kvantisert Hall-effekt – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantisert Hall-effekt" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvantehalleffekt" title="Kvantehalleffekt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvantehalleffekt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kwantowe_zjawisko_Halla" title="Kwantowe zjawisko Halla – Polish" lang="pl" hreflang="pl" data-title="Kwantowe zjawisko Halla" 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/Efeito_Hall_qu%C3%A2ntico" title="Efeito Hall quântico – Portuguese" lang="pt" hreflang="pt" data-title="Efeito Hall quântico" 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%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D1%8B%D0%B9_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82_%D0%A5%D0%BE%D0%BB%D0%BB%D0%B0" title="Квантовый эффект Холла – Russian" lang="ru" hreflang="ru" data-title="Квантовый эффект Холла" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvantittunut_Hallin_ilmi%C3%B6" title="Kvantittunut Hallin ilmiö – Finnish" lang="fi" hreflang="fi" data-title="Kvantittunut Hallin ilmiö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_Hall_etkisi" title="Kuantum Hall etkisi – Turkish" lang="tr" hreflang="tr" data-title="Kuantum Hall etkisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B5%D1%84%D0%B5%D0%BA%D1%82_%D0%A5%D0%BE%D0%BB%D0%BB%D0%B0" title="Квантовий ефект Холла – Ukrainian" lang="uk" hreflang="uk" data-title="Квантовий ефект Холла" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Hi%E1%BB%87u_%E1%BB%A9ng_Hall_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD" title="Hiệu ứng Hall lượng tử – Vietnamese" lang="vi" hreflang="vi" data-title="Hiệu ứng Hall lượng tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Electromagnetic effect in physics</div> <p>The <b>quantum Hall effect</b> (or <b>integer quantum Hall effect</b>) is a <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantized</a> version of the <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a> which is observed in <a href="/wiki/2DEG" class="mw-redirect" title="2DEG">two-dimensional electron systems</a> subjected to low <a href="/wiki/Temperature" title="Temperature">temperatures</a> and strong <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a>, in which the Hall <a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">resistance</a> <span class="texhtml"><i>R</i><sub>xy</sub></span> exhibits steps that take on the quantized values </p> <dl><dd><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_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Hall</mtext> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>channel</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ν</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd86a2343ffdfff421dafed9c25cf66d541e104f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.376ex; height:5.843ex;" alt="{\displaystyle R_{xy}={\frac {V_{\text{Hall}}}{I_{\text{channel}}}}={\frac {h}{e^{2}\nu }},}"></span></dd></dl> <p>where <span class="texhtml"><i>V</i><sub>Hall</sub></span> is the <a href="/wiki/Hall_effect" title="Hall effect">Hall voltage</a>, <span class="texhtml"><i>I</i><sub>channel</sub></span> is the channel <a href="/wiki/Electric_current" title="Electric current">current</a>, <span class="texhtml"><i>e</i></span> is the <a href="/wiki/Elementary_charge" title="Elementary charge">elementary charge</a> and <span class="texhtml"><i>h</i></span> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. The divisor <span class="texhtml"><a href="/wiki/Nu_(letter)" title="Nu (letter)"><i>ν</i></a></span> can take on either integer (<span class="texhtml"><i>ν</i> = 1, 2, 3,...</span>) or fractional (<span class="texhtml"><i>ν</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">13</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">12</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>,...</span>) values. Here, <span class="texhtml"><a href="/wiki/Nu_(letter)" title="Nu (letter)"><i>ν</i></a></span> is roughly but not exactly equal to the filling factor of <a href="/wiki/Landau_quantization" class="mw-redirect" title="Landau quantization">Landau levels</a>. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether <span class="texhtml"><i>ν</i></span> is an integer or fraction, respectively. </p><p>The striking feature of the integer quantum Hall effect is the persistence of the quantization (i.e. the Hall plateau) as the electron density is varied. Since the electron density remains constant when the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> is in a clean spectral gap, this situation corresponds to one where the Fermi level is an energy with a finite density of states, though these states are localized (see <a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> is more complicated and still considered an open research problem.<sup id="cite_ref-Hansson_025005_2-0" class="reference"><a href="#cite_note-Hansson_025005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without <a href="/wiki/Landau_quantization" class="mw-redirect" title="Landau quantization">Landau levels</a>.<sup id="cite_ref-Haldane:1988_3-0" class="reference"><a href="#cite_note-Haldane:1988-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the <a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">quantum spin Hall effect</a> which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Electrical_resistance_standards">Electrical resistance standards</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=2" title="Edit section: Electrical resistance standards"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quantization of the Hall conductance (<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 G_{xy}=1/R_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{xy}=1/R_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd596c63be02edd849552fdc9e5fbfd8f25e625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.993ex; height:3.009ex;" alt="{\displaystyle G_{xy}=1/R_{xy}}"></span>) has the important property of being exceedingly precise.<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> Actual measurements of the Hall conductance have been found to be integer or fractional multiples of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>e</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>h</i></span></span>⁠</span></span> to better than one part in a billion.<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> It has allowed for the definition of a new practical <a href="/wiki/Physical_unit" class="mw-redirect" title="Physical unit">standard</a> for <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a>, based on the resistance quantum given by the <b>von Klitzing constant</b> <span class="texhtml"><i>R</i><sub>K</sub></span>. This is named after <a href="/wiki/Klaus_von_Klitzing" title="Klaus von Klitzing">Klaus von Klitzing</a>, the discoverer of exact quantization. The quantum Hall effect also provides an extremely precise independent determination of the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>, a quantity of fundamental importance in <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. </p><p><span class="anchor" id="RK-1990"></span>In 1990, a fixed <a href="/wiki/Conventional_electrical_unit" title="Conventional electrical unit">conventional value</a> <span class="texhtml"><i>R</i><sub>K-90</sub> = <span class="nowrap"><span data-sort-value="7004258128070000000♠"></span>25<span style="margin-left:.25em;">812</span>.807 Ω</span></span> was defined for use in resistance calibrations worldwide.<sup id="cite_ref-physconst-RK90_7-0" class="reference"><a href="#cite_note-physconst-RK90-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Later, the <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision of the SI</a> fixed exact values of <span class="texhtml"><i>h</i></span> and <span class="texhtml"><i>e</i></span>, resulting in an exact <span class="texhtml"><i>R</i><sub>K</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>h</i></span><span class="sr-only">/</span><span class="den"><i>e</i><sup>2</sup></span></span>⁠</span> = <span class="nowrap"><span data-sort-value="7004258128074500000♠"></span>25<span style="margin-left:.25em;">812</span>.807<span style="margin-left:.25em;">45</span>... Ω</span>.<sup id="cite_ref-physconst-RK_8-0" class="reference"><a href="#cite_note-physconst-RK-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></span> </p> <div class="mw-heading mw-heading2"><h2 id="Research_status">Research status</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=3" title="Edit section: Research status"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fractional quantum Hall effect is considered part of <i>exact quantization</i>.<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> Exact quantization in full generality is not completely understood but it has been explained as a very subtle manifestation of the combination of the principle of <a href="/wiki/Gauge_invariance" class="mw-redirect" title="Gauge invariance">gauge invariance</a> together with another symmetry (see <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a>). The integer quantum Hall effect instead is considered a solved research problem<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><sup id="cite_ref-Laughlin:1981_11-0" class="reference"><a href="#cite_note-Laughlin:1981-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and understood in the scope of <a href="/w/index.php?title=TKNN_formula&action=edit&redlink=1" class="new" title="TKNN formula (page does not exist)">TKNN formula</a> and <a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons Lagrangians</a>. </p><p>The <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> is still considered an open research problem.<sup id="cite_ref-Hansson_025005_2-1" class="reference"><a href="#cite_note-Hansson_025005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The fractional quantum Hall effect can be also understood as an integer quantum Hall effect, although not of electrons but of charge–flux composites known as <a href="/wiki/Composite_fermions" class="mw-redirect" title="Composite fermions">composite fermions</a>.<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> Other models to explain the fractional quantum Hall effect also exists.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Currently it is considered an open research problem because no single, confirmed and agreed list of fractional quantum numbers exists, neither a single agreed model to explain all of them, although there are such claims in the scope of <a href="/wiki/Composite_fermions" class="mw-redirect" title="Composite fermions">composite fermions</a> and Non Abelian <a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons Lagrangians</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1957, <a href="/wiki/Carl_Frosch" title="Carl Frosch">Carl Frosch</a> and Lincoln Derick were able to manufacture the first silicon dioxide field effect transistors at Bell Labs, the first transistors in which drain and source were adjacent at the surface.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Subsequently, a team demonstrated a working <a href="/wiki/MOSFET" title="MOSFET">MOSFET</a> at Bell Labs 1960.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This enabled physicists to study <a href="/wiki/Two-dimensional_electron_gas" title="Two-dimensional electron gas">electron behavior in a nearly ideal two-dimensional gas</a>.<sup id="cite_ref-Lindley_17-0" class="reference"><a href="#cite_note-Lindley-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>In a MOSFET, conduction electrons travel in a thin surface layer, and a "<a href="/wiki/Metal_gate" title="Metal gate">gate</a>" voltage controls the number of charge carriers in this layer. This allows researchers to explore <a href="/wiki/Quantum_effects" class="mw-redirect" title="Quantum effects">quantum effects</a> by operating high-purity MOSFETs at <a href="/wiki/Liquid_helium" title="Liquid helium">liquid helium</a> temperatures.<sup id="cite_ref-Lindley_17-1" class="reference"><a href="#cite_note-Lindley-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The integer <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantization</a> of the Hall conductance was originally predicted by <a href="/wiki/University_of_Tokyo" title="University of Tokyo">University of Tokyo</a> researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true.<sup id="cite_ref-Ando:1975_18-0" class="reference"><a href="#cite_note-Ando:1975-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1978, the <a href="/wiki/Gakushuin_University" title="Gakushuin University">Gakushuin University</a> researchers Jun-ichi Wakabayashi and Shinji Kawaji subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs.<sup id="cite_ref-Wakabayashi:1978_19-0" class="reference"><a href="#cite_note-Wakabayashi:1978-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>In 1980, <a href="/wiki/Klaus_von_Klitzing" title="Klaus von Klitzing">Klaus von Klitzing</a>, working at the high magnetic field laboratory in Grenoble with <a href="/wiki/Silicon" title="Silicon">silicon</a>-based MOSFET samples developed by <a href="/wiki/Michael_Pepper" title="Michael Pepper">Michael Pepper</a> and Gerhard Dorda, made the unexpected discovery that the Hall resistance was <i>exactly</i> quantized.<sup id="cite_ref-vonKlitzing:1980_20-0" class="reference"><a href="#cite_note-vonKlitzing:1980-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lindley_17-2" class="reference"><a href="#cite_note-Lindley-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> For this finding, von Klitzing was awarded the 1985 <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a>. A link between exact quantization and gauge invariance was subsequently proposed by <a href="/wiki/Robert_B._Laughlin" title="Robert B. Laughlin">Robert Laughlin</a>, who connected the quantized conductivity to the quantized charge transport in a Thouless charge pump.<sup id="cite_ref-Laughlin:1981_11-1" class="reference"><a href="#cite_note-Laughlin:1981-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Thouless:1983_21-0" class="reference"><a href="#cite_note-Thouless:1983-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Most integer quantum Hall experiments are now performed on <a href="/wiki/Gallium_arsenide" title="Gallium arsenide">gallium arsenide</a> <a href="/wiki/Heterostructure" class="mw-redirect" title="Heterostructure">heterostructures</a>, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in <a href="/wiki/Graphene" title="Graphene">graphene</a> at temperatures as high as room temperature,<sup id="cite_ref-Novoselov:2007_22-0" class="reference"><a href="#cite_note-Novoselov:2007-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and in the <a href="/wiki/Magnesium" title="Magnesium">magnesium</a> <a href="/wiki/Zinc" title="Zinc">zinc</a> <a href="/wiki/Oxide" title="Oxide">oxide</a> ZnO–Mg<sub><i>x</i></sub>Zn<sub>1−<i>x</i></sub>O.<sup id="cite_ref-Tsukazaki:2007_23-0" class="reference"><a href="#cite_note-Tsukazaki:2007-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Integer_quantum_Hall_effect">Integer quantum Hall effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=5" title="Edit section: Integer quantum Hall effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/360px--QuantumHallEffectExplanationWithLandauLevels.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="360" height="270" data-durationhint="20" data-mwtitle="QuantumHallEffectExplanationWithLandauLevels.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:QuantumHallEffectExplanationWithLandauLevels.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/QuantumHallEffectExplanationWithLandauLevels.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv" type="video/ogg; codecs="theora"" data-width="800" data-height="600" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/QuantumHallEffectExplanationWithLandauLevels.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/QuantumHallEffectExplanationWithLandauLevels.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/QuantumHallEffectExplanationWithLandauLevels.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="480" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/QuantumHallEffectExplanationWithLandauLevels.ogv/QuantumHallEffectExplanationWithLandauLevels.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="480" data-height="360" /></video></span><figcaption>Animated graph showing filling of Landau levels as <i>B</i> changes and the corresponding position on a graph of hall coefficient and magnetic field|Illustrative only. The levels spread out with increasing field. Between the levels the quantum hall effect is seen. DOS is the density of states.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Landau_levels">Landau levels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=6" title="Edit section: Landau levels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Landau_levels" title="Landau levels">Landau levels</a></div> <p>In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. To determine the values of the energy levels the Schrödinger equation must be solved. </p><p>Since the system is subjected to a magnetic field, it has to be introduced as an electromagnetic vector potential in the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. The system considered is an electron gas that is free to move in the x and y directions, but is tightly confined in the z direction. Then, a magnetic field is applied in the z direction and according to the <a href="/wiki/Landau_gauge" class="mw-redirect" title="Landau gauge">Landau gauge</a> the electromagnetic vector potential is <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 \mathbf {A} =(0,Bx,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>B</mi> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =(0,Bx,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8637de124dee340910bbd0b462b966ccecdcaaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.414ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} =(0,Bx,0)}"></span> and the scalar potential is <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 =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c96cdab103e6c884877c86d6e5db6e471a167d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.646ex; height:2.509ex;" alt="{\displaystyle \phi =0}"></span>. Thus the Schrödinger equation for a particle of charge <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> and effective mass <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^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2650f1055b63acf24bf275e50b4f59c3e53685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.343ex;" alt="{\displaystyle m^{*}}"></span> in this system is: </p> <dl><dd><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\{{\frac {1}{2m^{*}}}\left[\mathbf {p} -q\mathbf {A} \right]^{2}+V(z)\right\}\psi (x,y,z)=\varepsilon \psi (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ε</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\frac {1}{2m^{*}}}\left[\mathbf {p} -q\mathbf {A} \right]^{2}+V(z)\right\}\psi (x,y,z)=\varepsilon \psi (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e35220a1ebb9bed3d0c8127c74b3467d8afdea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.363ex; height:6.176ex;" alt="{\displaystyle \left\{{\frac {1}{2m^{*}}}\left[\mathbf {p} -q\mathbf {A} \right]^{2}+V(z)\right\}\psi (x,y,z)=\varepsilon \psi (x,y,z)}"></span></dd></dl> <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 \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> is the canonical momentum, which is replaced by the operator <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\hbar \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7282d9d33646ecea0e5441adc5b48e535a703db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.853ex; height:2.343ex;" alt="{\displaystyle -i\hbar \nabla }"></span> 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 \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> is the total energy. </p><p>To solve this equation it is possible to separate it into two equations since the magnetic field just affects the movement along x and y axes. The total energy becomes then, the sum of two contributions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b91d8b7676cf5fa22de45e8f6d894cec7585924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.181ex; height:2.676ex;" alt="{\displaystyle \varepsilon =\varepsilon _{z}+\varepsilon _{xy}}"></span>. The corresponding equations in z axis is: </p> <dl><dd><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[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial z^{2}}+V(z)\right]u(z)=\varepsilon _{z}u(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial z^{2}}+V(z)\right]u(z)=\varepsilon _{z}u(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b298c0ff7c211851461c359f2e20ccb8940135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.205ex; height:6.343ex;" alt="{\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial z^{2}}+V(z)\right]u(z)=\varepsilon _{z}u(z)}"></span></dd></dl> <p>To simplify things, the solution <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(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6465282979c100114e011fa5f7b749606246527a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.685ex; height:2.843ex;" alt="{\displaystyle V(z)}"></span> is considered as an infinite well. Thus the solutions for the z direction are the energies <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 \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9884da3fc74c34a98a450ac9f16a3b83f729748" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.379ex; height:4.843ex;" alt="{\textstyle \varepsilon _{z}={\frac {n_{z}^{2}\pi ^{2}\hbar ^{2}}{2m^{*}L^{2}}}}"></span>, <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 n_{z}=1,2,3...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{z}=1,2,3...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa3db306b9c89ff8898f72c246bc88e13eda15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.991ex; height:2.509ex;" alt="{\displaystyle n_{z}=1,2,3...}"></span> and the wavefunctions are sinusoidal. For the <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave in <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-direction with some unknown function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span>, 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 \psi _{xy}=u(x)e^{ik_{y}y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{xy}=u(x)e^{ik_{y}y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c24de38efd4eb721c1947e55f0ebd902573ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.454ex; height:3.343ex;" alt="{\displaystyle \psi _{xy}=u(x)e^{ik_{y}y}}"></span>. This is because the vector potential does not depend on <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and the momentum operator <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 {\hat {p}}_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57fab437f87899122e81f913d0882ab0ac3b53f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:2.498ex; height:3.009ex;" alt="{\displaystyle {\hat {p}}_{y}}"></span> therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensional <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> equation centered at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>e</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521511af8a57adb3cc08f51fd36c0641ca59258f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.788ex; height:4.343ex;" alt="{\textstyle x_{k_{y}}={\frac {\hbar k_{y}}{eB}}}"></span>. </p> <dl><dd><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[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial x^{2}}+{\frac {1}{2}}m^{*}\omega _{\rm {c}}^{2}(x-l_{B}^{2}k_{y})^{2}\right]u(x)=\varepsilon _{xy}u(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msubsup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial x^{2}}+{\frac {1}{2}}m^{*}\omega _{\rm {c}}^{2}(x-l_{B}^{2}k_{y})^{2}\right]u(x)=\varepsilon _{xy}u(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd0eac24c186b988302c8fd3cf4165677022af32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.291ex; height:6.343ex;" alt="{\displaystyle \left[-{\frac {\hbar ^{2}}{2m^{*}}}{\partial ^{2} \over \partial x^{2}}+{\frac {1}{2}}m^{*}\omega _{\rm {c}}^{2}(x-l_{B}^{2}k_{y})^{2}\right]u(x)=\varepsilon _{xy}u(x)}"></span></dd></dl> <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="{\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>B</mi> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30f97e49c8be36ac6bae1cc6e4567b0c49db0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.617ex; height:3.676ex;" alt="{\textstyle \omega _{\rm {c}}={\frac {eB}{m^{*}}}}"></span> is defined as the cyclotron frequency 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="{\textstyle l_{B}^{2}={\frac {\hbar }{eB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mi>e</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle l_{B}^{2}={\frac {\hbar }{eB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1704f32778390afbb055755bf872ea10490c1487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.121ex; height:3.843ex;" alt="{\textstyle l_{B}^{2}={\frac {\hbar }{eB}}}"></span> the magnetic length. The energies are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{xy}\equiv \varepsilon _{n_{x}}=\hbar \omega _{\rm {c}}\left(n_{x}+{\frac {1}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{xy}\equiv \varepsilon _{n_{x}}=\hbar \omega _{\rm {c}}\left(n_{x}+{\frac {1}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a906c06535d7cc9c1f39072c0e5cf6ed2fb1274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.428ex; height:6.176ex;" alt="{\displaystyle \varepsilon _{xy}\equiv \varepsilon _{n_{x}}=\hbar \omega _{\rm {c}}\left(n_{x}+{\frac {1}{2}}\right)}"></span>, <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 n_{x}=1,2,3...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}=1,2,3...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c79222b9b8c2602a4ff6657bd4d9bb5951b803c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.161ex; height:2.509ex;" alt="{\displaystyle n_{x}=1,2,3...}"></span></dd></dl> <p>And the wavefunctions for the motion in the <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 xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> plane are given by the product of a plane wave in <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> and <a href="/wiki/Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a> attenuated by the gaussian function in <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span>, which are the wavefunctions of a harmonic oscillator. </p><p>From the expression for the Landau levels one notices that the energy depends only on <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 n_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a811d9f1857b131167bd4238be8eb175ae3e85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.009ex;" alt="{\displaystyle n_{x}}"></span>, not on <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_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03d994b4735244db4b14200c7500bc3fb98f5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.261ex; height:2.843ex;" alt="{\displaystyle k_{y}}"></span>. States with the same <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 n_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a811d9f1857b131167bd4238be8eb175ae3e85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.009ex;" alt="{\displaystyle n_{x}}"></span> but different <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_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03d994b4735244db4b14200c7500bc3fb98f5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.261ex; height:2.843ex;" alt="{\displaystyle k_{y}}"></span> are degenerate. </p> <div class="mw-heading mw-heading3"><h3 id="Density_of_states">Density of states</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=7" title="Edit section: Density of states"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy </p> <dl><dd><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 n_{\rm {2D}}={\frac {m^{*}}{\pi \hbar ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mi>π</mi> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\rm {2D}}={\frac {m^{*}}{\pi \hbar ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e26f091b604216c1274824bc1c29974e450eb8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.347ex; height:5.676ex;" alt="{\displaystyle n_{\rm {2D}}={\frac {m^{*}}{\pi \hbar ^{2}}}}"></span>.</dd></dl> <p>As the field is turned on, the density of states collapses from the constant to a <a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a>, a series of Dirac <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> functions, corresponding to the Landau levels separated <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 \varepsilon _{xy}=\hbar \omega _{\rm {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \varepsilon _{xy}=\hbar \omega _{\rm {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc710a0bcb95f1175a5bcde7b97b03857fde665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.822ex; height:2.843ex;" alt="{\displaystyle \Delta \varepsilon _{xy}=\hbar \omega _{\rm {c}}}"></span>. At finite temperature, however, the Landau levels acquire a width <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 \Gamma ={\frac {\hbar }{\tau _{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Gamma ={\frac {\hbar }{\tau _{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc47bcb6d9e386a17cd45a2579ba9d1581abc9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.731ex; height:3.843ex;" alt="{\textstyle \Gamma ={\frac {\hbar }{\tau _{i}}}}"></span> being <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 _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ca8b33360ac3b7db8e9435271b5654175c853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \tau _{i}}"></span> the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a <a href="/wiki/Gaussian" class="mw-redirect" title="Gaussian">Gaussian</a> or <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Lorentzian</a> profile. </p><p>Another feature is that the wave functions form parallel strips in the <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-direction spaced equally along the <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span>-axis, along the lines of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>. Since there is nothing special about any direction in the <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 xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span>-plane if the vector potential was differently chosen one should find circular symmetry. </p><p>Given a sample of dimensions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{x}\times L_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{x}\times L_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717a559f59f9a4578d7844dd017a62f983808f7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.228ex; height:2.843ex;" alt="{\displaystyle L_{x}\times L_{y}}"></span> and applying the periodic boundary conditions in the <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-direction <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 k={\frac {2\pi }{L_{y}}}j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k={\frac {2\pi }{L_{y}}}j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8fe6a3ae1ca33903adedf720247ced3b050255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.051ex; height:4.176ex;" alt="{\textstyle k={\frac {2\pi }{L_{y}}}j}"></span> being <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> an integer, one gets that each parabolic potential is placed at a value <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 x_{k}=l_{B}^{2}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}=l_{B}^{2}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62eef27ef76028370a8a76b76047196d33ae2f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.901ex; height:3.176ex;" alt="{\displaystyle x_{k}=l_{B}^{2}k}"></span>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Potencialesparab%C3%B3licos.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Potencialesparab%C3%B3licos.jpg/270px-Potencialesparab%C3%B3licos.jpg" decoding="async" width="270" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Potencialesparab%C3%B3licos.jpg/405px-Potencialesparab%C3%B3licos.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Potencialesparab%C3%B3licos.jpg/540px-Potencialesparab%C3%B3licos.jpg 2x" data-file-width="1473" data-file-height="605" /></a><figcaption>Parabolic potentials along the <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span>-axis centered at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> with the 1st wave functions corresponding to an infinite well confinement in the <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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> direction. In the <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-direction there are travelling plane waves.</figcaption></figure> <p>The number of states for each Landau Level 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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state. </p> <dl><dd><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 N_{B}={\frac {\phi }{\phi _{0}}}={\frac {BA}{BL_{y}\Delta x_{k}}}={\frac {A}{2\pi l_{B}^{2}}}{\begin{array}{lcr}&l_{B}&\\&=&\\&&\end{array}}{\frac {AeB}{2\pi \hbar }}{\begin{array}{lcr}&\omega _{\rm {c}}&\\&=&\\&&\end{array}}{\frac {m^{*}\omega _{\rm {c}}A}{2\pi \hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϕ</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mi>A</mi> </mrow> <mrow> <mi>B</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mrow> <mn>2</mn> <mi>π</mi> <msubsup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd /> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mi>e</mi> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd /> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>A</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{B}={\frac {\phi }{\phi _{0}}}={\frac {BA}{BL_{y}\Delta x_{k}}}={\frac {A}{2\pi l_{B}^{2}}}{\begin{array}{lcr}&l_{B}&\\&=&\\&&\end{array}}{\frac {AeB}{2\pi \hbar }}{\begin{array}{lcr}&\omega _{\rm {c}}&\\&=&\\&&\end{array}}{\frac {m^{*}\omega _{\rm {c}}A}{2\pi \hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1330450855c060eacf9877cdd65eafbc30145a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:59.89ex; height:9.176ex;" alt="{\displaystyle N_{B}={\frac {\phi }{\phi _{0}}}={\frac {BA}{BL_{y}\Delta x_{k}}}={\frac {A}{2\pi l_{B}^{2}}}{\begin{array}{lcr}&l_{B}&\\&=&\\&&\end{array}}{\frac {AeB}{2\pi \hbar }}{\begin{array}{lcr}&\omega _{\rm {c}}&\\&=&\\&&\end{array}}{\frac {m^{*}\omega _{\rm {c}}A}{2\pi \hbar }}}"></span></dd></dl> <p>Thus the density of states per unit surface is </p> <dl><dd><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 n_{B}={\frac {m^{*}\omega _{\rm {c}}}{2\pi \hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{B}={\frac {m^{*}\omega _{\rm {c}}}{2\pi \hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9edaa3b55828e3efefb122cb697ad6ca257224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.312ex; height:5.509ex;" alt="{\displaystyle n_{B}={\frac {m^{*}\omega _{\rm {c}}}{2\pi \hbar }}}"></span>.</dd></dl> <p>Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since fewer energy levels are occupied. </p><p>Rewriting the last expression 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 n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mi>π</mi> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb3d26fc286b75ef24fadb98954fa33ac7b4d6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.056ex; height:4.343ex;" alt="{\textstyle n_{B}={\frac {\hbar \omega _{\rm {c}}}{2}}{\frac {m^{*}}{\pi \hbar ^{2}}}}"></span> it is clear that each Landau level contains as many states as in a <a href="/wiki/Two-dimensional_electron_gas" title="Two-dimensional electron gas">2DEG</a> in 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 \Delta \varepsilon =\hbar \omega _{\rm {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ε</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \varepsilon =\hbar \omega _{\rm {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5079fc52edda7be8e354054028ff222f87d006f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.832ex; height:2.509ex;" alt="{\displaystyle \Delta \varepsilon =\hbar \omega _{\rm {c}}}"></span>. </p><p>Given the fact that electrons are <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>, for each state available in the Landau levels it corresponds to two electrons, one electron with each value for the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</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="{\textstyle s=\pm {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s=\pm {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a0e4df440657b52a80f1350885db429e17151e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.655ex; height:3.509ex;" alt="{\textstyle s=\pm {\frac {1}{2}}}"></span>. However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies is <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 \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>g</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784cc6d110e82e242597e2311f8e96228165d032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.954ex; height:3.509ex;" alt="{\textstyle \Delta E=\pm {\frac {1}{2}}g\mu _{\rm {B}}B}"></span> being <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> a factor which depends on the material (<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 g=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de9b6986ac8bbeb3cfc90a9b420ae7391d7257c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=2}"></span> for free electrons) 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 {B}}}"> <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">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677cdbc1ede246206f4288c36a94bc92d3178684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.798ex; height:2.176ex;" alt="{\displaystyle \mu _{\rm {B}}}"></span> the <a href="/wiki/Bohr_magneton" title="Bohr magneton">Bohr magneton</a>. The sign <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 +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> is taken when the spin is parallel to the field 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 -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"></span> when it is antiparallel. This fact called spin splitting implies that the <a href="/wiki/Density_of_states" title="Density of states">density of states</a> for each level is reduced by a half. Note that <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345f0ed5ff6ec6eddc5f908d379f032c52f119c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.711ex; height:2.176ex;" alt="{\displaystyle \Delta E}"></span> is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Densidadestadossinspin.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Densidadestadossinspin.jpg/263px-Densidadestadossinspin.jpg" decoding="async" width="263" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Densidadestadossinspin.jpg/395px-Densidadestadossinspin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Densidadestadossinspin.jpg/526px-Densidadestadossinspin.jpg 2x" data-file-width="1600" data-file-height="625" /></a><figcaption>Density of states in a magnetic field, neglecting spin splitting. (a)The states in each range <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 \hbar \omega _{\rm {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \omega _{\rm {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b217c6e89c37a6fb4769d24670dc571ae255c534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.715ex; height:2.509ex;" alt="{\displaystyle \hbar \omega _{\rm {c}}}"></span> are squeezed into 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 \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>-function Landau level. (b) Landau levels have a non-zero width <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 \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> in a more realistic picture and overlap if <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 \hbar \omega _{\rm {c}}<\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo><</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \omega _{\rm {c}}<\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab903aaecd8db15cccd8849192e5773aabe22819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.266ex; height:2.509ex;" alt="{\displaystyle \hbar \omega _{\rm {c}}<\Gamma }"></span>. (c) The levels become distinct when <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 \hbar \omega _{\rm {c}}>\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>></mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \omega _{\rm {c}}>\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cb0e42f992262f506779ece92c9b528db3d011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.266ex; height:2.509ex;" alt="{\displaystyle \hbar \omega _{\rm {c}}>\Gamma }"></span>.</figcaption></figure> <p>In order to get the number of occupied Landau levels, one defines the so-called filling factor <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> as the ratio between the density of states in a 2DEG and the density of states in the Landau levels. </p> <dl><dd><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 \nu ={\frac {n_{\rm {2D}}}{n_{B}}}={\frac {hn_{\rm {2D}}}{eB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>e</mi> <mi>B</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {n_{\rm {2D}}}{n_{B}}}={\frac {hn_{\rm {2D}}}{eB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae768c72ea4661bba3430441bbb94995a49e1cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.849ex; height:5.676ex;" alt="{\displaystyle \nu ={\frac {n_{\rm {2D}}}{n_{B}}}={\frac {hn_{\rm {2D}}}{eB}}}"></span></dd></dl> <p>In general the filling factor <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density (and not the Fermi energy!) or varies the electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> and one cannot expect <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> to be an integer. 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 n_{B}\propto B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>∝</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{B}\propto B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7743872e36bc1fb72799db5e7997d2323d242b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.737ex; height:2.509ex;" alt="{\displaystyle n_{B}\propto B}"></span>, by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level (<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 \nu <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acd78e37ac02356984350573794735d954e9eba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu <1}"></span>) and this is called the magnetic quantum limit. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:NivelesLandausinspin.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/NivelesLandausinspin.jpg/265px-NivelesLandausinspin.jpg" decoding="async" width="265" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/NivelesLandausinspin.jpg/398px-NivelesLandausinspin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/NivelesLandausinspin.jpg/530px-NivelesLandausinspin.jpg 2x" data-file-width="1600" data-file-height="625" /></a><figcaption>Occupation of Landau levels in a magnetic field neglecting the spin splitting, showing how the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> moves to maintain a constant density of electrons. The fields are in the ratio <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 2:3:4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>:</mo> <mn>3</mn> <mo>:</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2:3:4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835b6925b0b5d1da25093effc242708b14abc67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.362ex; height:2.176ex;" alt="{\displaystyle 2:3:4}"></span> and give <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 \nu =4,{\frac {8}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =4,{\frac {8}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c3c122a2daf6ecf495b25c8c3f96312e982d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.526ex; height:5.176ex;" alt="{\displaystyle \nu =4,{\frac {8}{3}}}"></span> 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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Longitudinal_resistivity">Longitudinal resistivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=8" title="Edit section: Longitudinal resistivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is an integer, the <a href="/wiki/Fermi_energy" title="Fermi energy">Fermi energy</a> lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be approximately <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>). Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).<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> </p><p>With the conductivity <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 =\rho ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\rho ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df067284d55f9ae4779984b1868d3a6ea53043af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.963ex; height:3.176ex;" alt="{\displaystyle \sigma =\rho ^{-1}}"></span> one finds </p> <dl><dd><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 ={\frac {1}{\det \rho }}{\begin{pmatrix}\rho _{yy}&-\rho _{xy}\\-\rho _{yx}&\rho _{xx}\end{pmatrix}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {1}{\det \rho }}{\begin{pmatrix}\rho _{yy}&-\rho _{xy}\\-\rho _{yx}&\rho _{xx}\end{pmatrix}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e703c4e8daa4e2be128ccbba49e02a9d54733cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.869ex; height:6.509ex;" alt="{\displaystyle \sigma ={\frac {1}{\det \rho }}{\begin{pmatrix}\rho _{yy}&-\rho _{xy}\\-\rho _{yx}&\rho _{xx}\end{pmatrix}}\;.}"></span></dd></dl> <p>If the longitudinal resistivity is zero and transversal is finite, then <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 \det \rho \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>ρ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \rho \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d189f38f2363163dfbbcfb92c457fa04de502a8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.08ex; height:2.676ex;" alt="{\displaystyle \det \rho \neq 0}"></span>. Thus both the longitudinal conductivity and resistivity become zero. </p><p>Instead, when <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum . </p><p>This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called <i>Shubnikov–de Haas oscillations</i> which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity. It is interesting to notice that if the magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Rhoxy.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Rhoxy.jpg/263px-Rhoxy.jpg" decoding="async" width="263" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Rhoxy.jpg/395px-Rhoxy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Rhoxy.jpg/526px-Rhoxy.jpg 2x" data-file-width="707" data-file-height="492" /></a><figcaption>Longitudinal and transverse (Hall) resistivity, <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 \rho _{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906f269fe33416459c066d1e0182c8fef80e8aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.315ex; height:2.176ex;" alt="{\displaystyle \rho _{xx}}"></span> 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 \rho _{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18e4804b339fd6e032bf29ef36b8fcf7a4bf6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.191ex; height:2.343ex;" alt="{\displaystyle \rho _{xy}}"></span>, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductance <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^{2}/h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}/h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6224e6d40f7748ccebba5199a177b7d978dc93e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.639ex; height:3.176ex;" alt="{\displaystyle e^{2}/h}"></span> (units are misleading). The filling factor <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is displayed for the last 4 plateaus.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Transverse_resistivity">Transverse resistivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=9" title="Edit section: Transverse resistivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the classical relation of the transverse resistivity <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 \rho _{xy}={\frac {B}{en_{\rm {2D}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mrow> <mi>e</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho _{xy}={\frac {B}{en_{\rm {2D}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8fcb09925f81a7df5e881421aaad480a7269f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.729ex; height:3.676ex;" alt="{\textstyle \rho _{xy}={\frac {B}{en_{\rm {2D}}}}}"></span> and substituting <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 n_{\rm {2D}}=\nu {\frac {eB}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>B</mi> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n_{\rm {2D}}=\nu {\frac {eB}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef051cfac473bed20a8593d8711ba64665b90e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.885ex; height:3.843ex;" alt="{\textstyle n_{\rm {2D}}=\nu {\frac {eB}{h}}}"></span> one finds out the quantization of the transverse resistivity and conductivity: </p> <dl><dd><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 \rho _{xy}={\frac {h}{\nu e^{2}}}\Rightarrow \sigma =\nu {\frac {e^{2}}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mi>ν</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <mi>σ</mi> <mo>=</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xy}={\frac {h}{\nu e^{2}}}\Rightarrow \sigma =\nu {\frac {e^{2}}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9eec4652369b85fb2be8ca13d7c325d8f38ad89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.744ex; height:6.009ex;" alt="{\displaystyle \rho _{xy}={\frac {h}{\nu e^{2}}}\Rightarrow \sigma =\nu {\frac {e^{2}}{h}}}"></span></dd></dl> <p>One concludes then, that the transverse resistivity is a multiple of the inverse of the so-called conductance quantum <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^{2}/h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}/h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6224e6d40f7748ccebba5199a177b7d978dc93e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.639ex; height:3.176ex;" alt="{\displaystyle e^{2}/h}"></span> if the filling factor is an integer. In experiments, however, plateaus are observed for whole plateaus of filling values <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>, which indicates that there are in fact electron states between the Landau levels. These states are localized in, for example, impurities of the material where they are trapped in orbits so they can not contribute to the conductivity. That is why the resistivity remains constant in between Landau levels. Again if the magnetic field decreases, one gets the classical result in which the resistivity is proportional to the magnetic field. </p> <div class="mw-heading mw-heading2"><h2 id="Photonic_quantum_Hall_effect">Photonic quantum Hall effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=10" title="Edit section: Photonic quantum Hall effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quantum Hall effect, in addition to being observed in <a href="/wiki/2DEG" class="mw-redirect" title="2DEG">two-dimensional electron systems</a>, can be observed in photons. <a href="/wiki/Photon" title="Photon">Photons</a> do not possess inherent <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>, but through the manipulation of discrete <a href="/wiki/Optical_cavity" title="Optical cavity">optical resonators</a> and coupling phases or on-site phases, an artificial <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> can be created.<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><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><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><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><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> This process can be expressed through a metaphor of photons bouncing between multiple mirrors. By shooting the light across multiple mirrors, the photons are routed and gain additional phase proportional to their <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum</a>. This creates an effect like they are in a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Topological_classification">Topological classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=11" title="Edit section: Topological classification"><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:Hofstadter%27s_butterfly.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Hofstadter%27s_butterfly.png/220px-Hofstadter%27s_butterfly.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Hofstadter%27s_butterfly.png/330px-Hofstadter%27s_butterfly.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Hofstadter%27s_butterfly.png/440px-Hofstadter%27s_butterfly.png 2x" data-file-width="3000" data-file-height="2000" /></a><figcaption><a href="/wiki/Hofstadter%27s_butterfly" title="Hofstadter's butterfly">Hofstadter's butterfly</a></figcaption></figure> <p>The integers that appear in the Hall effect are examples of <a href="/wiki/Topological_quantum_number" title="Topological quantum number">topological quantum numbers</a>. They are known in mathematics as the first <a href="/wiki/Chern_class#Chern_numbers" title="Chern class">Chern numbers</a> and are closely related to <a href="/wiki/Geometric_phase" title="Geometric phase">Berry's phase</a>. A striking model of much interest in this context is the Azbel–Harper–Hofstadter model whose quantum phase diagram is the <a href="/wiki/Hofstadter_butterfly" class="mw-redirect" title="Hofstadter butterfly">Hofstadter butterfly</a> shown in the figure. The vertical axis is the strength of the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> and the horizontal axis is the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a>, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. Note, however, that the density of states in these regions of quantized Hall conductance is zero; hence, they cannot produce the plateaus observed in the experiments. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious <a href="/wiki/Self-similarity" title="Self-similarity">self-similarity</a>. In the presence of disorder, which is the source of the plateaus seen in the experiments, this diagram is very different and the fractal structure is mostly washed away. Also, the experiments control the filling factor and not the Fermi energy. If this diagram is plotted as a function of filling factor, all the features are completely washed away, hence, it has very little to do with the actual Hall physics. </p><p>Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a>. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called <i><a href="/wiki/Composite_fermions" class="mw-redirect" title="Composite fermions">composite fermions</a></i>. </p> <div class="mw-heading mw-heading2"><h2 id="Bohr_atom_interpretation_of_the_von_Klitzing_constant">Bohr atom interpretation of the von Klitzing constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=12" title="Edit section: Bohr atom interpretation of the von Klitzing constant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value of the von Klitzing constant may be obtained already on the level of a single atom within the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> while looking at it as a single-electron Hall effect. While during the <a href="/wiki/Cyclotron" title="Cyclotron">cyclotron motion</a> on a circular orbit the centrifugal force is balanced by the <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a> responsible for the transverse induced voltage and the Hall effect, one may look at the Coulomb potential difference in the Bohr atom as the induced single atom Hall voltage and the periodic electron motion on a circle as a Hall current. Defining the single atom Hall current as a rate a single electron charge <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> is making Kepler revolutions with angular frequency <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> </p> <dl><dd><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={\frac {\omega e}{2\pi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ω</mi> <mi>e</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I={\frac {\omega e}{2\pi }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d858f4b548ec333c0a12746296261678edd7e541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.282ex; height:4.676ex;" alt="{\displaystyle I={\frac {\omega e}{2\pi }},}"></span></dd></dl> <p>and the induced Hall voltage as a difference between the hydrogen nucleus Coulomb potential at the electron orbital point and at infinity: </p> <dl><dd><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 U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232a05a6a132c4bc151ad7aba341d7da07fad796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.47ex; height:5.176ex;" alt="{\displaystyle U=V_{\text{C}}(\infty )-V_{\text{C}}(r)=0-V_{\text{C}}(r)={\frac {e}{4\pi \epsilon _{0}r}}}"></span></dd></dl> <p>One obtains the quantization of the defined Bohr orbit Hall resistance in steps of the von Klitzing constant as </p> <dl><dd><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_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Bohr</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>I</mi> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab37a602fb77bc8967c8bd1796d4e9997f3af0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.929ex; height:5.676ex;" alt="{\displaystyle R_{\text{Bohr}}(n)={\frac {U}{I}}=n{\frac {h}{e^{2}}}}"></span></dd></dl> <p>which for the Bohr atom is linear but not inverse in the integer <i>n</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Relativistic_analogs">Relativistic analogs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_Hall_effect&action=edit&section=13" title="Edit section: Relativistic analogs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Relativistic examples of the integer quantum Hall effect and <a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">quantum spin Hall effect</a> arise in the context of <a href="/wiki/Lattice_gauge_theory" title="Lattice gauge theory">lattice gauge theory</a>.<sup id="cite_ref-Kaplan:1992_30-0" class="reference"><a href="#cite_note-Kaplan:1992-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Golterman:1993_31-0" class="reference"><a href="#cite_note-Golterman:1993-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </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=Quantum_Hall_effect&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Quantum_Hall_transitions" title="Quantum Hall transitions">Quantum Hall transitions</a></li> <li><a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">Fractional quantum Hall effect</a></li> <li><a href="/wiki/Quantum_anomalous_Hall_effect" title="Quantum anomalous Hall effect">Quantum anomalous Hall effect</a></li> <li><a href="/wiki/Quantum_cellular_automata" class="mw-redirect" title="Quantum cellular automata">Quantum cellular automata</a></li> <li><a href="/wiki/Composite_fermions" class="mw-redirect" title="Composite fermions">Composite fermions</a></li> <li><a href="/wiki/Conductance_quantum" title="Conductance quantum">Conductance Quantum</a></li> <li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a href="/wiki/Hall_probe" class="mw-redirect" title="Hall probe">Hall probe</a></li> <li><a href="/wiki/Graphene" title="Graphene">Graphene</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Static_forces_and_virtual-particle_exchange#Coulomb_potential_between_two_current_loops_embedded_in_a_magnetic_field" title="Static forces and virtual-particle exchange">Coulomb potential between two current loops embedded in a magnetic field</a></li></ul> </div> <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=Quantum_Hall_effect&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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 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R. Yennie (1987). "Integral quantum Hall effect for nonspecialists". <i>Rev. Mod. Phys</i>. <b>59</b> (3): <span class="nowrap">781–</span>824. <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/1987RvMP...59..781Y">1987RvMP...59..781Y</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.59.781">10.1103/RevModPhys.59.781</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rev.+Mod.+Phys.&rft.atitle=Integral+quantum+Hall+effect+for+nonspecialists&rft.volume=59&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E781-%3C%2Fspan%3E824&rft.date=1987&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.59.781&rft_id=info%3Abibcode%2F1987RvMP...59..781Y&rft.au=D.+R.+Yennie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+Hall+effect" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._HsiehD._QianL._WrayY._Xia2008" class="citation journal cs1">D. Hsieh; D. Qian; L. Wray; Y. Xia; Y. S. Hor; R. J. Cava; M. Z. Hasan (2008). "A topological Dirac insulator in a quantum spin Hall phase". <i>Nature</i>. <b>452</b> (7190): <span class="nowrap">970–</span>974. <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/0902.1356">0902.1356</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/2008Natur.452..970H">2008Natur.452..970H</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%2Fnature06843">10.1038/nature06843</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/18432240">18432240</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:4402113">4402113</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+topological+Dirac+insulator+in+a+quantum+spin+Hall+phase&rft.volume=452&rft.issue=7190&rft.pages=%3Cspan+class%3D%22nowrap%22%3E970-%3C%2Fspan%3E974&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4402113%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2008Natur.452..970H&rft_id=info%3Aarxiv%2F0902.1356&rft_id=info%3Apmid%2F18432240&rft_id=info%3Adoi%2F10.1038%2Fnature06843&rft.au=D.+Hsieh&rft.au=D.+Qian&rft.au=L.+Wray&rft.au=Y.+Xia&rft.au=Y.+S.+Hor&rft.au=R.+J.+Cava&rft.au=M.+Z.+Hasan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+Hall+effect" class="Z3988"></span></li> <li><i>25 years of Quantum Hall Effect</i>, K. von Klitzing, Poincaré Seminar (Paris-2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070304025657/http://parthe.lpthe.jussieu.fr/poincare/textes/novembre2004.html">Postscript</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110707025318/http://hrma.physics.sjtu.edu.cn/PhysicsHorizon/25yearsQHE-lecture.pdf">Pdf</a>.</li> <li>Magnet Lab Press Release <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071222020625/http://www.magnet.fsu.edu/mediacenter/news/pressreleases/2007february15.html">Quantum Hall Effect Observed at Room Temperature</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvronOsadchy,_DanielSeiler,_Ruedi2003" class="citation journal cs1">Avron, Joseph E.; Osadchy, Daniel; Seiler, Ruedi (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1611351">"A Topological Look at the Quantum Hall Effect"</a>. <i>Physics Today</i>. <b>56</b> (8): 38. <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/2003PhT....56h..38A">2003PhT....56h..38A</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/10.1063%2F1.1611351">10.1063/1.1611351</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=A+Topological+Look+at+the+Quantum+Hall+Effect&rft.volume=56&rft.issue=8&rft.pages=38&rft.date=2003&rft_id=info%3Adoi%2F10.1063%2F1.1611351&rft_id=info%3Abibcode%2F2003PhT....56h..38A&rft.aulast=Avron&rft.aufirst=Joseph+E.&rft.au=Osadchy%2C+Daniel&rft.au=Seiler%2C+Ruedi&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.1611351&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+Hall+effect" class="Z3988"></span></li> <li>Zyun F. Ezawa: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p3JpcdbqBPoC"><i>Quantum Hall Effects - Field Theoretical Approach and Related Topics.</i></a> World Scientific, Singapore 2008, <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/978-981-270-032-2" title="Special:BookSources/978-981-270-032-2">978-981-270-032-2</a></li> <li>Sankar D. Sarma, <a href="/wiki/Aron_Pinczuk" title="Aron Pinczuk">Aron Pinczuk</a>: <i>Perspectives in Quantum Hall Effects.</i> Wiley-VCH, Weinheim 2004, <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/978-0-471-11216-7" title="Special:BookSources/978-0-471-11216-7">978-0-471-11216-7</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA._BaumgartnerT._IhnK._EnsslinK._Maranowski2007" class="citation journal cs1">A. Baumgartner; T. Ihn; K. Ensslin; K. Maranowski; A. Gossard (2007). "Quantum Hall effect transition in scanning gate experiments". <i>Phys. Rev. B</i>. <b>76</b> (8): 085316. <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/2007PhRvB..76h5316B">2007PhRvB..76h5316B</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%2FPhysRevB.76.085316">10.1103/PhysRevB.76.085316</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+B&rft.atitle=Quantum+Hall+effect+transition+in+scanning+gate+experiments&rft.volume=76&rft.issue=8&rft.pages=085316&rft.date=2007&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.76.085316&rft_id=info%3Abibcode%2F2007PhRvB..76h5316B&rft.au=A.+Baumgartner&rft.au=T.+Ihn&rft.au=K.+Ensslin&rft.au=K.+Maranowski&rft.au=A.+Gossard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+Hall+effect" class="Z3988"></span></li> <li><a href="/wiki/Emmanuel_Rashba" title="Emmanuel Rashba">E. 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href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li> <li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a class="mw-selflink selflink">Quantum Hall effect</a></li> <li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Drude_model" title="Drude model">Drude model</a></li> <li><a href="/wiki/Free_electron_model" title="Free electron model">Free electron model</a></li> <li><a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid theory</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li> <li><a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a></li> <li><a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a></li> <li><a href="/wiki/Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li> <li><a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a></li> <li><a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li> <li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li> <li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li> <li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li> <li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li> <li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li> <li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li> <li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li> <li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li> <li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li> <li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list 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