Scanning tunneling microscope

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<span>Rectangular barrier model</span> </div> </a> <ul id="toc-Rectangular_barrier_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tunneling_between_two_conductors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tunneling_between_two_conductors"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Tunneling between two conductors</span> </div> </a> <ul id="toc-Tunneling_between_two_conductors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bardeen's_formalism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bardeen's_formalism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Bardeen's formalism</span> </div> </a> <ul id="toc-Bardeen's_formalism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Gallery_of_STM_images" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gallery_of_STM_images"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Gallery of STM images</span> </div> </a> <ul id="toc-Gallery_of_STM_images-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Early_invention" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Early_invention"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Early invention</span> </div> </a> <ul id="toc-Early_invention-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_related_techniques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_related_techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other related techniques</span> </div> </a> <ul id="toc-Other_related_techniques-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">7</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">8</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">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Scanning tunneling microscope</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 42 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-42" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D9%87%D8%B1_%D9%85%D8%B3%D8%AD_%D9%86%D9%81%D9%82%D9%8A" title="مجهر مسح نفقي – Arabic" lang="ar" hreflang="ar" data-title="مجهر مسح نفقي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BD%D0%B8%D1%80%D0%B0%D1%89_%D1%82%D1%83%D0%BD%D0%B5%D0%BB%D0%B5%D0%BD_%D0%BC%D0%B8%D0%BA%D1%80%D0%BE%D1%81%D0%BA%D0%BE%D0%BF" title="Сканиращ тунелен микроскоп – Bulgarian" lang="bg" hreflang="bg" data-title="Сканиращ тунелен микроскоп" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Skeniraju%C4%87i_tunelski_mikroskop" title="Skenirajući tunelski mikroskop – Bosnian" lang="bs" hreflang="bs" data-title="Skenirajući tunelski mikroskop" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Microscopi_d%27efecte_t%C3%BAnel" title="Microscopi d'efecte túnel – Catalan" lang="ca" hreflang="ca" data-title="Microscopi d'efecte túnel" 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/%C5%98%C3%A1dkovac%C3%AD_tunelov%C3%BD_mikroskop" title="Řádkovací tunelový mikroskop – Czech" lang="cs" hreflang="cs" data-title="Řádkovací tunelový mikroskop" 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/Scanning-tunnelmikroskopi" title="Scanning-tunnelmikroskopi – Danish" lang="da" hreflang="da" data-title="Scanning-tunnelmikroskopi" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Rastertunnelmikroskop" title="Rastertunnelmikroskop – German" lang="de" hreflang="de" data-title="Rastertunnelmikroskop" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Skaneeriv_tunnelmikroskoopia" title="Skaneeriv tunnelmikroskoopia – Estonian" lang="et" hreflang="et" data-title="Skaneeriv tunnelmikroskoopia" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B9%CE%BA%CF%81%CE%BF%CF%83%CE%BA%CF%8C%CF%80%CE%B9%CE%BF_%CF%83%CE%AC%CF%81%CF%89%CF%83%CE%B7%CF%82_%CF%83%CE%AE%CF%81%CE%B1%CE%B3%CE%B3%CE%B1%CF%82" title="Μικροσκόπιο σάρωσης σήραγγας – Greek" lang="el" hreflang="el" data-title="Μικροσκόπιο σάρωσης σήραγγας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Microscopio_de_efecto_t%C3%BAnel" title="Microscopio de efecto túnel – Spanish" lang="es" hreflang="es" data-title="Microscopio de efecto túnel" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tunel-efika_mikroskopo" title="Tunel-efika mikroskopo – Esperanto" lang="eo" hreflang="eo" data-title="Tunel-efika mikroskopo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%DA%A9%D8%B1%D9%88%D8%B3%DA%A9%D9%88%D9%BE_%D8%AA%D9%88%D9%86%D9%84%DB%8C_%D8%B1%D9%88%D8%A8%D8%B4%DB%8C" title="میکروسکوپ تونلی روبشی – Persian" lang="fa" hreflang="fa" data-title="میکروسکوپ تونلی روبشی" 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/Microscope_%C3%A0_effet_tunnel" title="Microscope à effet tunnel – French" lang="fr" hreflang="fr" data-title="Microscope à effet tunnel" 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/Toll%C3%A1nmhicreasc%C3%B3p_scanach%C3%A1in" title="Tollánmhicreascóp scanacháin – Irish" lang="ga" hreflang="ga" data-title="Tollánmhicreascóp scanacháin" 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%A3%BC%EC%82%AC_%ED%84%B0%EB%84%90%EB%A7%81_%ED%98%84%EB%AF%B8%EA%B2%BD" title="주사 터널링 현미경 – Korean" lang="ko" hreflang="ko" data-title="주사 터널링 현미경" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%B5%E0%A4%B2%E0%A5%8B%E0%A4%95%E0%A4%A8_%E0%A4%9F%E0%A4%A8%E0%A4%B2%E0%A4%BF%E0%A4%82%E0%A4%97_%E0%A4%B8%E0%A5%82%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%A6%E0%A4%B0%E0%A5%8D%E0%A4%B6%E0%A5%80_%E0%A4%AF%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="अवलोकन टनलिंग सूक्ष्मदर्शी यंत्र – Hindi" lang="hi" hreflang="hi" data-title="अवलोकन टनलिंग सूक्ष्मदर्शी यंत्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pretra%C5%BEni_mikroskop_s_tuneliranjem" title="Pretražni mikroskop s tuneliranjem – Croatian" lang="hr" hreflang="hr" data-title="Pretražni mikroskop s tuneliranjem" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mikroskop_penerowongan_payaran" title="Mikroskop penerowongan payaran – Indonesian" lang="id" hreflang="id" data-title="Mikroskop penerowongan payaran" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Microscopio_a_effetto_tunnel" title="Microscopio a effetto tunnel – Italian" lang="it" hreflang="it" data-title="Microscopio a effetto tunnel" 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%9E%D7%99%D7%A7%D7%A8%D7%95%D7%A1%D7%A7%D7%95%D7%A4_%D7%9E%D7%A0%D7%94%D7%95%D7%A8_%D7%A1%D7%95%D7%A8%D7%A7" title="מיקרוסקופ מנהור סורק – Hebrew" lang="he" hreflang="he" data-title="מיקרוסקופ מנהור סורק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/P%C3%A1szt%C3%A1z%C3%B3_alag%C3%BAtmikroszk%C3%B3p" title="Pásztázó alagútmikroszkóp – Hungarian" lang="hu" hreflang="hu" data-title="Pásztázó alagútmikroszkóp" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B5%8D%E0%B4%95%E0%B4%BE%E0%B4%A8%E0%B4%BF%E0%B4%99%E0%B5%8D_%E0%B4%9F%E0%B4%A3%E0%B4%B2%E0%B4%BF%E0%B4%99%E0%B5%8D_%E0%B4%B8%E0%B5%82%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B5%8D%E0%B4%AE%E0%B4%A6%E0%B5%BC%E0%B4%B6%E0%B4%BF%E0%B4%A8%E0%B4%BF" title="സ്കാനിങ് ടണലിങ് സൂക്ഷ്മദർശിനി – Malayalam" lang="ml" hreflang="ml" data-title="സ്കാനിങ് ടണലിങ് സൂക്ഷ്മദർശിനി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Scanning_tunneling_microscopy" title="Scanning tunneling microscopy – Dutch" lang="nl" hreflang="nl" data-title="Scanning tunneling microscopy" 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/%E8%B5%B0%E6%9F%BB%E5%9E%8B%E3%83%88%E3%83%B3%E3%83%8D%E3%83%AB%E9%A1%95%E5%BE%AE%E9%8F%A1" title="走査型トンネル顕微鏡 – Japanese" lang="ja" hreflang="ja" data-title="走査型トンネル顕微鏡" 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/Scanning_tunneling_mikroskop" title="Scanning tunneling mikroskop – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Scanning tunneling mikroskop" 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/Tunnelmikroskop" title="Tunnelmikroskop – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Tunnelmikroskop" 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/Skaningowy_mikroskop_tunelowy" title="Skaningowy mikroskop tunelowy – Polish" lang="pl" hreflang="pl" data-title="Skaningowy mikroskop tunelowy" 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/Microsc%C3%B3pio_de_corrente_de_tunelamento" title="Microscópio de corrente de tunelamento – Portuguese" lang="pt" hreflang="pt" data-title="Microscópio de corrente de tunelamento" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Microscop_cu_efect_tunel" title="Microscop cu efect tunel – Romanian" lang="ro" hreflang="ro" data-title="Microscop cu efect tunel" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B0%D0%BD%D0%B8%D1%80%D1%83%D1%8E%D1%89%D0%B8%D0%B9_%D1%82%D1%83%D0%BD%D0%BD%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BC%D0%B8%D0%BA%D1%80%D0%BE%D1%81%D0%BA%D0%BE%D0%BF" title="Сканирующий туннельный микроскоп – Russian" lang="ru" hreflang="ru" data-title="Сканирующий туннельный микроскоп" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Scanning_tunneling_microscope" title="Scanning tunneling microscope – Simple English" lang="en-simple" hreflang="en-simple" data-title="Scanning tunneling microscope" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vrsti%C4%8Dni_tunelski_mikroskop" title="Vrstični tunelski mikroskop – Slovenian" lang="sl" hreflang="sl" data-title="Vrstični tunelski mikroskop" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BA%D0%B5%D0%BD%D0%B8%D1%80%D0%B0%D1%98%D1%83%D1%9B%D0%B8_%D1%82%D1%83%D0%BD%D0%B5%D0%BB%D1%81%D0%BA%D0%B8_%D0%BC%D0%B8%D0%BA%D1%80%D0%BE%D1%81%D0%BA%D0%BE%D0%BF" title="Скенирајући тунелски микроскоп – Serbian" lang="sr" hreflang="sr" data-title="Скенирајући тунелски микроскоп" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Skeniraju%C4%87i_tunelski_mikroskop" title="Skenirajući tunelski mikroskop – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Skenirajući tunelski mikroskop" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a 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searchaux" style="display:none">Instrument able to image surfaces at the atomic level by exploiting quantum tunneling effects</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">Not to be confused with <a href="/wiki/Scanning_electron_microscope" title="Scanning electron microscope">scanning electron microscope</a> (SEM).</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic_resolution_Au100.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/220px-Atomic_resolution_Au100.JPG" decoding="async" width="220" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/330px-Atomic_resolution_Au100.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/440px-Atomic_resolution_Au100.JPG 2x" data-file-width="500" data-file-height="480" /></a><figcaption>Image of <a href="/wiki/Surface_reconstruction" title="Surface reconstruction">reconstruction</a> on a clean <a href="/wiki/Miller_index" title="Miller index">(100)</a> surface of <a href="/wiki/Gold" title="Gold">gold</a></figcaption></figure> <p>A <b>scanning tunneling microscope</b> (<b>STM</b>) is a type of <a href="/wiki/Scanning_probe_microscopy" title="Scanning probe microscopy">scanning probe microscope</a> used for imaging <a href="/wiki/Surface" title="Surface">surfaces</a> at the <a href="/wiki/Atom" title="Atom">atomic</a> level. Its development in 1981 earned its inventors, <a href="/wiki/Gerd_Binnig" title="Gerd Binnig">Gerd Binnig</a> and <a href="/wiki/Heinrich_Rohrer" title="Heinrich Rohrer">Heinrich Rohrer</a>, then at <a href="/wiki/IBM_Research_%E2%80%93_Zurich" class="mw-redirect" title="IBM Research – Zurich">IBM Zürich</a>, the <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> in 1986.<sup id="cite_ref-Binnig_1-0" class="reference"><a href="#cite_note-Binnig-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> STM senses the surface by using an extremely sharp <a href="/wiki/Electrical_conductor" title="Electrical conductor">conducting</a> tip that can distinguish features smaller than 0.1 <a href="/wiki/Nanometre" title="Nanometre">nm</a> with a 0.01 nm (10 <a href="/wiki/Picometre" title="Picometre">pm</a>) depth resolution.<sup id="cite_ref-Bai_4-0" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This means that individual atoms can routinely be imaged and manipulated. Most scanning tunneling microscopes are built for use in <a href="/wiki/Ultra-high_vacuum" title="Ultra-high vacuum">ultra-high vacuum</a> at temperatures approaching <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, but variants exist for studies in air, water and other environments, and for temperatures over 1000 °C.<sup id="cite_ref-Chen_5-0" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><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> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Scanning_Tunneling_Microscope.ogv/220px--Scanning_Tunneling_Microscope.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="151" data-mwtitle="Scanning_Tunneling_Microscope.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Scanning_Tunneling_Microscope.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/1/14/Scanning_Tunneling_Microscope.ogv" type="video/ogg; codecs="theora, vorbis"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Scanning_Tunneling_Microscope.ogv/Scanning_Tunneling_Microscope.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Scanning tunneling microscope operating principle</figcaption></figure> <p>STM is based on the concept of <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a>. When the tip is brought very near to the surface to be examined, a <a href="/wiki/Biasing_(electronics)" class="mw-redirect" title="Biasing (electronics)">bias</a> voltage applied between the two allows <a href="/wiki/Electron" title="Electron">electrons</a> to tunnel through the <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> separating them. The resulting <i>tunneling <a href="/wiki/Electric_current" title="Electric current">current</a></i> is a function of the tip position, applied voltage, and the <a href="/wiki/Density_of_states#Local_density_of_states" title="Density of states">local density of states</a> (LDOS) of the sample. Information is acquired by monitoring the current as the tip scans across the surface, and is usually displayed in image form.<sup id="cite_ref-Chen_5-1" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>A refinement of the technique known as <a href="/wiki/Scanning_tunneling_spectroscopy" title="Scanning tunneling spectroscopy">scanning tunneling spectroscopy</a> consists of keeping the tip in a constant position above the surface, varying the bias voltage and recording the resultant change in current. Using this technique, the local density of the electronic states can be reconstructed.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This is sometimes performed in high magnetic fields and in presence of impurities to infer the properties and interactions of electrons in the studied material. </p><p>Scanning tunneling microscopy can be a challenging technique, as it requires extremely clean and stable surfaces, sharp tips, excellent <a href="/wiki/Vibration_isolation" title="Vibration isolation">vibration isolation</a>, and sophisticated electronics. Nonetheless, many hobbyists build their own microscopes.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Procedure">Procedure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=1" title="Edit section: Procedure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Scanning_Tunneling_Microscope_schematic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Scanning_Tunneling_Microscope_schematic.svg/300px-Scanning_Tunneling_Microscope_schematic.svg.png" decoding="async" width="300" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Scanning_Tunneling_Microscope_schematic.svg/450px-Scanning_Tunneling_Microscope_schematic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Scanning_Tunneling_Microscope_schematic.svg/600px-Scanning_Tunneling_Microscope_schematic.svg.png 2x" data-file-width="561" data-file-height="451" /></a><figcaption>Schematic view of an STM</figcaption></figure> <p>The tip is brought close to the sample by a coarse positioning mechanism that is usually monitored visually. At close range, fine control of the tip position with respect to the sample surface is achieved by <a href="/wiki/Piezoelectricity" title="Piezoelectricity">piezoelectric</a> scanner tubes whose length can be altered by a control voltage. A bias <a href="/wiki/Electric_tension" class="mw-redirect" title="Electric tension">voltage</a> is applied between the sample and the tip, and the scanner is gradually elongated until the tip starts receiving the tunneling current. The tip–sample separation <i>w</i> is then kept somewhere in the 4–7 <a href="/wiki/Angstrom" title="Angstrom">Å</a> (0.4–0.7 <a href="/wiki/Nanometre" title="Nanometre">nm</a>) range, slightly above the height where the tip would experience repulsive interaction <span class="nowrap">(<i>w</i> < 3 Å),</span> but still in the region where attractive interaction exists <span class="nowrap">(3 < <i>w</i> < 10 Å).<sup id="cite_ref-Chen_5-2" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></span> The tunneling current, being in the sub-<a href="/wiki/Nanoampere" class="mw-redirect" title="Nanoampere">nanoampere</a> range, is amplified as close to the scanner as possible. Once tunneling is established, the sample bias and tip position with respect to the sample are varied according to the requirements of the experiment. </p><p>As the tip is moved across the surface in a discrete <i>x</i>–<i>y</i> matrix, the changes in surface height and population of the electronic states cause changes in the tunneling current. Digital images of the surface are formed in one of the two ways: in the <i>constant-height mode</i> changes of the tunneling current are mapped directly, while in the <i>constant-current mode</i> the voltage that controls the height (<i>z</i>) of the tip is recorded while the tunneling current is kept at a predetermined level.<sup id="cite_ref-Chen_5-3" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In constant-current mode, feedback electronics adjust the height by a voltage to the piezoelectric height-control mechanism. If at some point the tunneling current is below the set level, the tip is moved towards the sample, and conversely. This mode is relatively slow, as the electronics need to check the tunneling current and adjust the height in a feedback loop at each measured point of the surface. When the surface is atomically flat, the voltage applied to the <i>z</i>-scanner mainly reflects variations in local charge density. But when an atomic step is encountered, or when the surface is buckled due to <a href="/wiki/Surface_reconstruction" title="Surface reconstruction">reconstruction</a>, the height of the scanner also have to change because of the overall topography. The image formed of the <i>z</i>-scanner voltages that were needed to keep the tunneling current constant as the tip scanned the surface thus contain both topographical and electron density data. In some cases it may not be clear whether height changes came as a result of one or the other. </p><p>In constant-height mode, the <i>z</i>-scanner voltage is kept constant as the scanner swings back and forth across the surface, and the tunneling current, exponentially dependent on the distance, is mapped. This mode of operation is faster, but on rough surfaces, where there may be large adsorbed molecules present, or ridges and groves, the tip will be in danger of crashing. </p><p>The <a href="/wiki/Raster_scan" title="Raster scan">raster scan</a> of the tip is anything from a 128×128 to a 1024×1024 (or more) matrix, and for each point of the raster a single value is obtained. The images produced by STM are therefore <a href="/wiki/Grayscale" title="Grayscale">grayscale</a>, and color is only added in post-processing in order to visually emphasize important features. </p><p>In addition to scanning across the sample, information on the electronic structure at a given location in the sample can be obtained by sweeping the bias voltage (along with a small AC modulation to directly measure the derivative) and measuring current change at a specific location.<sup id="cite_ref-Bai_4-1" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This type of measurement is called <a href="/wiki/Scanning_tunneling_spectroscopy" title="Scanning tunneling spectroscopy">scanning tunneling spectroscopy</a> (STS) and typically results in a plot of the local <a href="/wiki/Density_of_states" title="Density of states">density of states</a> as a function of the electrons' energy within the sample. The advantage of STM over other measurements of the density of states lies in its ability to make extremely local measurements. This is how, for example, the density of states at an <a href="/wiki/Impurity" class="mw-redirect" title="Impurity">impurity</a> site can be compared to the density of states around the impurity and elsewhere on the surface.<sup id="cite_ref-Pan_9-0" class="reference"><a href="#cite_note-Pan-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Instrumentation">Instrumentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=2" title="Edit section: Instrumentation"><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:Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG/220px-Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG/330px-Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG/440px-Scanning_tunneling_microscope-MHS_2237-IMG_3819.JPG 2x" data-file-width="5616" data-file-height="3744" /></a><figcaption>A 1986 STM from the collection of <a href="/wiki/Mus%C3%A9e_d%27histoire_des_sciences_de_la_Ville_de_Gen%C3%A8ve" title="Musée d'histoire des sciences de la Ville de Genève">Musée d'histoire des sciences de la Ville de Genève</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:STM_at_the_London_Centre_for_Nanotechnology.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/STM_at_the_London_Centre_for_Nanotechnology.jpg/220px-STM_at_the_London_Centre_for_Nanotechnology.jpg" decoding="async" width="220" height="329" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/STM_at_the_London_Centre_for_Nanotechnology.jpg/330px-STM_at_the_London_Centre_for_Nanotechnology.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/STM_at_the_London_Centre_for_Nanotechnology.jpg/440px-STM_at_the_London_Centre_for_Nanotechnology.jpg 2x" data-file-width="2592" data-file-height="3872" /></a><figcaption>A large STM setup at the <a href="/wiki/London_Centre_for_Nanotechnology" title="London Centre for Nanotechnology">London Centre for Nanotechnology</a></figcaption></figure> <p>The main components of a scanning tunneling microscope are the scanning tip, piezoelectrically controlled height (<i>z</i> axis) and lateral (<i>x</i> and <i>y</i> axes) scanner, and coarse sample-to-tip approach mechanism. The microscope is controlled by dedicated electronics and a computer. The system is supported on a vibration isolation system.<sup id="cite_ref-Chen_5-4" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The tip is often made of <a href="/wiki/Tungsten" title="Tungsten">tungsten</a> or <a href="/wiki/Platinum%E2%80%93iridium_alloy" title="Platinum–iridium alloy">platinum–iridium</a> wire, though <a href="/wiki/Gold" title="Gold">gold</a> is also used.<sup id="cite_ref-Bai_4-2" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Tungsten tips are usually made by electrochemical etching, and platinum–iridium tips by mechanical shearing. The <a href="/wiki/Image_resolution" title="Image resolution">resolution</a> of an image is limited by the <a href="/wiki/Radius_of_curvature" title="Radius of curvature">radius of curvature</a> of the scanning tip. Sometimes, image artefacts occur if the tip has more than one apex at the end; most frequently <i>double-tip imaging</i> is observed, a situation in which two apices contribute equally to the tunneling.<sup id="cite_ref-Bai_4-3" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> While several processes for obtaining sharp, usable tips are known, the ultimate test of quality of the tip is only possible when it is tunneling in the vacuum. Every so often the tips can be conditioned by applying high voltages when they are already in the tunneling range, or by making them pick up an atom or a molecule from the surface. </p><p>In most modern designs the scanner is a hollow tube of a radially polarized piezoelectric with metallized surfaces. The outer surface is divided into four long quadrants to serve as <i>x</i> and <i>y</i> motion electrodes with deflection voltages of two polarities applied on the opposing sides. The tube material is a <a href="/wiki/Lead_zirconate_titanate" title="Lead zirconate titanate">lead zirconate titanate</a> ceramic with a piezoelectric constant of about 5 nanometres per volt. The tip is mounted at the center of the tube. Because of some crosstalk between the electrodes and inherent nonlinearities, the motion is <a href="/wiki/Calibration" title="Calibration">calibrated</a>, and voltages needed for independent <i>x</i>, <i>y</i> and <i>z</i> motion applied according to calibration tables.<sup id="cite_ref-Chen_5-5" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Due to the extreme sensitivity of the tunneling current to the separation of the electrodes, proper vibration isolation or a rigid STM body is imperative for obtaining usable results. In the first STM by Binnig and Rohrer, <a href="/wiki/Magnetic_levitation" title="Magnetic levitation">magnetic levitation</a> was used to keep the STM free from vibrations; now mechanical spring or <a href="/wiki/Gas_spring" title="Gas spring">gas spring</a> systems are often employed.<sup id="cite_ref-Chen_5-6" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Additionally, mechanisms for vibration damping using <a href="/wiki/Eddy_currents" class="mw-redirect" title="Eddy currents">eddy currents</a> are sometimes implemented. Microscopes designed for long scans in scanning tunneling spectroscopy need extreme stability and are built in <a href="/wiki/Anechoic_chamber" title="Anechoic chamber">anechoic chambers</a>—dedicated concrete rooms with acoustic and electromagnetic isolation that are themselves floated on vibration isolation devices inside the laboratory. </p><p>Maintaining the tip position with respect to the sample, scanning the sample and acquiring the data is computer-controlled. Dedicated <a href="/wiki/Scanning_probe_microscopy#Visualization_and_analysis_software" title="Scanning probe microscopy">software for scanning probe microscopies</a> is used for <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a> as well as performing quantitative measurements.<sup id="cite_ref-fospm2011_10-0" class="reference"><a href="#cite_note-fospm2011-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Some scanning tunneling microscopes are capable of recording images at high frame rates.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><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> Videos made of such images can show surface <a href="/wiki/Diffusion" title="Diffusion">diffusion</a><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> or track adsorption and reactions on the surface. In video-rate microscopes, frame rates of 80 Hz have been achieved with fully working feedback that adjusts the height of the tip.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Principle_of_operation">Principle of operation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=3" title="Edit section: Principle of operation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum tunneling of electrons is a functioning concept of STM that arises from <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. Classically, a particle hitting an impenetrable barrier will not pass through. If the barrier is described by a potential acting along <i>z</i> direction, in which an electron of mass <i>m</i><sub>e</sub> acquires the potential energy <i>U</i>(<i>z</i>), the electron's trajectory will be deterministic and such that the sum <i>E</i> of its kinetic and potential energies is at all times conserved: </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 E={\frac {p^{2}}{2m_{\text{e}}}}+U(z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {p^{2}}{2m_{\text{e}}}}+U(z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc335c471920e384091a15444c166120839bb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.043ex; height:6.009ex;" alt="{\displaystyle E={\frac {p^{2}}{2m_{\text{e}}}}+U(z).}"></span></dd></dl> <p>The electron will have a defined, non-zero momentum <i>p</i> only in regions where the initial energy <i>E</i> is greater than <i>U</i>(<i>z</i>). In quantum physics, however, the electron can pass through classically forbidden regions. This is referred to as <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">tunneling</a>.<sup id="cite_ref-Chen_5-7" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rectangular_barrier_model">Rectangular barrier model</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=4" title="Edit section: Rectangular barrier model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg/300px-Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg.png" decoding="async" width="300" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg/450px-Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg/600px-Scanning_tunneling_microscope_-_rectangular_potential_barrier_model.svg.png 2x" data-file-width="484" data-file-height="304" /></a><figcaption>The real and imaginary parts of the wave function in a rectangular potential barrier model of the scanning tunneling microscope</figcaption></figure> <p>The simplest model of tunneling between the sample and the tip of a scanning tunneling microscope is that of a <a href="/wiki/Rectangular_potential_barrier" title="Rectangular potential barrier">rectangular potential barrier</a>.<sup id="cite_ref-Lounis_15-0" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chen_5-8" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> An electron of energy <i>E</i> is incident upon an energy barrier of height <i>U</i>, in the region of space of width <i>w</i>. An electron's behavior in the presence of a potential <i>U</i>(<i>z</i>), assuming one-dimensional case, is described by <a href="/wiki/Wave_function" title="Wave function">wave functions</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 \psi (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle \psi (z)}"></span> that satisfy <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a> </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 -{\frac {\hbar ^{2}}{2m_{\text{e}}}}{\frac {\partial ^{2}\psi (z)}{\partial z^{2}}}+U(z)\,\psi (z)=E\,\psi (z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <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>U</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m_{\text{e}}}}{\frac {\partial ^{2}\psi (z)}{\partial z^{2}}}+U(z)\,\psi (z)=E\,\psi (z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e2d47416e45a69fd6cc412241a2a70ac4c6c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.09ex; height:6.176ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m_{\text{e}}}}{\frac {\partial ^{2}\psi (z)}{\partial z^{2}}}+U(z)\,\psi (z)=E\,\psi (z),}"></span></dd></dl> <p>where <i>ħ</i> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, <i>z</i> is the position, and <i>m</i><sub>e</sub> is the <a href="/wiki/Electron_mass" title="Electron mass">electron mass</a>. In the zero-potential regions on two sides of the barrier, the wave function takes on the forms </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 \psi _{L}(z)=e^{ikz}+r\,e^{-ikz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>z</mi> </mrow> </msup> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{L}(z)=e^{ikz}+r\,e^{-ikz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21c7c53c86f55000c4393bcf25c870cdb782bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.433ex; height:3.176ex;" alt="{\displaystyle \psi _{L}(z)=e^{ikz}+r\,e^{-ikz}}"></span> for <i>z</i> < 0,</dd> <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 \psi _{R}(z)=t\,e^{ikz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{R}(z)=t\,e^{ikz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9be45753ae6f306407422d5f767a4663965e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.724ex; height:3.176ex;" alt="{\displaystyle \psi _{R}(z)=t\,e^{ikz}}"></span> for <i>z</i> > <i>w</i>,</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 k={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mi>E</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d8bf6692b73ba83f18e3bf8c5d8611998f6e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.334ex; height:3.676ex;" alt="{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}}}"></span>. Inside the barrier, where <i>E</i> < <i>U</i>, the wave function is a superposition of two terms, each decaying from one side of the barrier: </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 \psi _{B}(z)=\xi e^{-\kappa z}+\zeta e^{\kappa z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ξ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>κ</mi> <mi>z</mi> </mrow> </msup> <mo>+</mo> <mi>ζ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{B}(z)=\xi e^{-\kappa z}+\zeta e^{\kappa z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eeff41f089a2bbf804d6873f2992f6d89ea45b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.297ex; height:3.009ex;" alt="{\displaystyle \psi _{B}(z)=\xi e^{-\kappa z}+\zeta e^{\kappa z}}"></span> for 0 < <i>z</i> < <i>w</i>,</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 \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b59c62dedf77cccdd7be95042f51b08e0fab9540" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.894ex; height:4.843ex;" alt="{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}}"></span>. </p><p>The coefficients <i>r</i> and <i>t</i> provide measure of how much of the incident electron's wave is reflected or transmitted through the barrier. Namely, of the whole impinging particle current <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_{i}=\hbar k/m_{\text{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{i}=\hbar k/m_{\text{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c659faa33dcf388610e8cf396ad6f1c26a40f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:11.566ex; height:2.843ex;" alt="{\displaystyle j_{i}=\hbar k/m_{\text{e}}}"></span> only <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_{t}=|t|^{2}\,j_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{t}=|t|^{2}\,j_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d0c78e116475034f0bf14f85ca96ee7fefe0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:10.242ex; height:3.343ex;" alt="{\displaystyle j_{t}=|t|^{2}\,j_{i}}"></span> is transmitted, as can be seen from the <a href="/wiki/Probability_current" title="Probability current">probability current</a> expression </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 j_{t}=-i{\frac {\hbar }{2m_{\text{e}}}}\left\{\psi _{R}^{*}{\frac {\partial }{\partial z}}\psi _{R}-\psi _{R}{\frac {\partial }{\partial z}}\psi _{R}^{*}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{t}=-i{\frac {\hbar }{2m_{\text{e}}}}\left\{\psi _{R}^{*}{\frac {\partial }{\partial z}}\psi _{R}-\psi _{R}{\frac {\partial }{\partial z}}\psi _{R}^{*}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c72449fe6b5705c0e356f03d258518c2c353ee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.027ex; width:38.724ex; height:6.176ex;" alt="{\displaystyle j_{t}=-i{\frac {\hbar }{2m_{\text{e}}}}\left\{\psi _{R}^{*}{\frac {\partial }{\partial z}}\psi _{R}-\psi _{R}{\frac {\partial }{\partial z}}\psi _{R}^{*}\right\},}"></span></dd></dl> <p>which evaluates to <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_{t}={\tfrac {\hbar k}{m_{\text{e}}}}\vert t\vert ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>k</mi> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">|</mo> <mi>t</mi> <msup> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{t}={\tfrac {\hbar k}{m_{\text{e}}}}\vert t\vert ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28cebf3773b096aecd5321f00ca167ae143a08c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.027ex; width:11.133ex; height:3.843ex;" alt="{\displaystyle j_{t}={\tfrac {\hbar k}{m_{\text{e}}}}\vert t\vert ^{2}}"></span>. The transmission coefficient is obtained from the continuity condition on the three parts of the wave function and their derivatives at <i>z</i> = 0 and <i>z</i> = <i>w</i> (detailed derivation is in the article <a href="/wiki/Rectangular_potential_barrier#Analysis_of_the_obtained_expressions" title="Rectangular potential barrier">Rectangular potential barrier</a>). This gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|^{2}={\big [}1+{\tfrac {1}{4}}\varepsilon ^{-1}(1-\varepsilon )^{-1}\sinh ^{2}\kappa w{\big ]}^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>κ</mi> <mi>w</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t|^{2}={\big [}1+{\tfrac {1}{4}}\varepsilon ^{-1}(1-\varepsilon )^{-1}\sinh ^{2}\kappa w{\big ]}^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cbff90eaa9f02348080c2374ce348217da21cc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.49ex; height:3.843ex;" alt="{\displaystyle |t|^{2}={\big [}1+{\tfrac {1}{4}}\varepsilon ^{-1}(1-\varepsilon )^{-1}\sinh ^{2}\kappa w{\big ]}^{-1},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon =E/U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =E/U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e321e558a0c69cc879c05ac818a380d319f820cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.903ex; height:2.843ex;" alt="{\displaystyle \varepsilon =E/U}"></span>. The expression can be further simplified, as follows: </p><p>In STM experiments, typical barrier height is of the order of the material's surface <a href="/wiki/Work_function" title="Work function">work function</a> <i>W</i>, which for most metals has a value between 4 and 6 eV.<sup id="cite_ref-Lounis_15-1" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Work_function" title="Work function">work function</a> is the minimum energy needed to bring an electron from an occupied level, the highest of which is the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> (for metals at <i>T</i> = 0 K), to <a href="/wiki/Vacuum_level" title="Vacuum level">vacuum level</a>. The electrons can tunnel between two metals only from occupied states on one side into the unoccupied states of the other side of the barrier. Without bias, Fermi energies are flush, and there is no tunneling. Bias shifts electron energies in one of the electrodes higher, and those electrons that have no match at the same energy on the other side will tunnel. In experiments, bias voltages of a fraction of 1 V are used, so <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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> is of the order of 10 to 12 nm<sup>−1</sup>, while <i>w</i> is a few tenths of a nanometre. The barrier is strongly attenuating. The expression for the transmission probability reduces to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|^{2}=16\,\varepsilon (1-\varepsilon )\,e^{-2\kappa w}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>κ</mi> <mi>w</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t|^{2}=16\,\varepsilon (1-\varepsilon )\,e^{-2\kappa w}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1591839a1e15c1a0154877b032bd5a7bd0c04c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.551ex; height:3.343ex;" alt="{\displaystyle |t|^{2}=16\,\varepsilon (1-\varepsilon )\,e^{-2\kappa w}.}"></span> The tunneling current from a single level is therefore<sup id="cite_ref-Lounis_15-2" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </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 j_{t}=\left[{\frac {4k\kappa }{k^{2}+\kappa ^{2}}}\right]^{2}\,{\frac {\hbar k}{m_{\text{e}}}}\,e^{-2\kappa w},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>k</mi> <mi>κ</mi> </mrow> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>k</mi> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>κ</mi> <mi>w</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{t}=\left[{\frac {4k\kappa }{k^{2}+\kappa ^{2}}}\right]^{2}\,{\frac {\hbar k}{m_{\text{e}}}}\,e^{-2\kappa w},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd1adb83338539122a10cd5d224709a4b2235e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.027ex; width:27.553ex; height:6.509ex;" alt="{\displaystyle j_{t}=\left[{\frac {4k\kappa }{k^{2}+\kappa ^{2}}}\right]^{2}\,{\frac {\hbar k}{m_{\text{e}}}}\,e^{-2\kappa w},}"></span></dd></dl> <p>where both wave vectors depend on the level's energy <i>E</i>, <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={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mi>E</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a63ae4b94bc0230c2f802a95c61f01f24f54ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.981ex; height:3.676ex;" alt="{\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}E}},}"></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 \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/428d7c80cbc82fa3901a51bae97ac7983c0c48c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.541ex; height:4.843ex;" alt="{\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{\text{e}}(U-E)}}.}"></span> </p><p>Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation is increased by 1 Å (0.1 nm).<sup id="cite_ref-Chen_5-9" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital.<sup id="cite_ref-Lounis_15-3" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tunneling_between_two_conductors">Tunneling between two conductors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=5" title="Edit section: Tunneling between two conductors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg/300px-Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg.png" decoding="async" width="300" height="190" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg/450px-Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg/600px-Scanning_tunneling_microscope_-_tunneling_-_Density_of_states.svg.png 2x" data-file-width="476" data-file-height="302" /></a><figcaption>Negative sample bias <i>V</i> raises its electronic levels by <i>e⋅V</i>. Only electrons that populate states between the Fermi levels of the sample and the tip are allowed to tunnel.</figcaption></figure> <p>As a result of the restriction that the tunneling from an occupied energy level on one side of the barrier requires an empty level of the same energy on the other side of the barrier, tunneling occurs mainly with electrons near the Fermi level. The tunneling current can be related to the density of available or filled states in the sample. The current due to an applied voltage <i>V</i> (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level <i>E</i><sub>F</sub> and <i>E</i><sub>F</sub> − <i>eV</i> in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip.<sup id="cite_ref-Chen_5-10" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The higher the density of available states in the tunneling region the greater the tunneling current. By convention, a positive <i>V</i> means that electrons in the tip tunnel into empty states in the sample; for a negative bias, electrons tunnel out of occupied states in the sample into the tip.<sup id="cite_ref-Chen_5-11" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>For small biases and temperatures near absolute zero, the number of electrons in a given volume (the electron concentration) that are available for tunneling is the product of the density of the electronic states <i>ρ</i>(<i>E</i><sub>F</sub>) and the energy interval between the two Fermi levels, <i>eV</i>.<sup id="cite_ref-Chen_5-12" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Half of these electrons will be travelling away from the barrier. The other half will represent the <a href="/wiki/Electric_current#Drift_speed" title="Electric current">electric current</a> impinging on the barrier, which is given by the product of the electron concentration, charge, and velocity <i>v</i> (<i>I</i><sub>i</sub> = <i>nev</i>),<sup id="cite_ref-Chen_5-13" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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_{i}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{i}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25cd6647a14bea759f477a1287c6f6a629ad2270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.085ex; height:3.509ex;" alt="{\displaystyle I_{i}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V.}"></span></dd></dl> <p>The tunneling electric current will be a small fraction of the impinging current. The proportion is determined by the transmission probability <i>T</i>,<sup id="cite_ref-Chen_5-14" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> so\ </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_{t}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V\,T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>v</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>V</mi> <mspace width="thinmathspace" /> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{t}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V\,T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42361e27bcfc28a4fe815b6cfad5bab16eaa88a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.135ex; height:3.509ex;" alt="{\displaystyle I_{t}={\tfrac {1}{2}}e^{2}v\,\rho (E_{\text{F}})\,V\,T.}"></span></dd></dl> <p>In the simplest model of a rectangular potential barrier the transmission probability coefficient <i>T</i> equals |<i>t</i>|<sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Bardeen's_formalism"><span id="Bardeen.27s_formalism"></span>Bardeen's formalism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=6" title="Edit section: Bardeen's formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Scanning_tunneling_microscope_-_tip,_barrier_and_sample_wave_functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg/300px-Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg.png" decoding="async" width="300" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg/450px-Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg/600px-Scanning_tunneling_microscope_-_tip%2C_barrier_and_sample_wave_functions.svg.png 2x" data-file-width="516" data-file-height="324" /></a><figcaption>Tip, barrier and sample wave functions in a model of the scanning tunneling microscope. Barrier width is <i>w</i>. Tip bias is <i>V</i>. Surface work functions are <i>ϕ</i>.</figcaption></figure> <p>A model that is based on more realistic wave functions for the two electrodes was devised by <a href="/wiki/John_Bardeen" title="John Bardeen">John Bardeen</a> in a study of the <a href="/wiki/Metal%E2%80%93insulator%E2%80%93metal" title="Metal–insulator–metal">metal–insulator–metal</a> junction.<sup id="cite_ref-Bardeen_16-0" class="reference"><a href="#cite_note-Bardeen-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> His model takes two separate orthonormal sets of wave functions for the two electrodes and examines their time evolution as the systems are put close together.<sup id="cite_ref-Chen_5-15" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lounis_15-4" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Bardeen's novel method, ingenious in itself,<sup id="cite_ref-Chen_5-16" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> solves a time-dependent perturbative problem in which the perturbation emerges from the interaction of the two subsystems rather than an external potential of the standard <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">Rayleigh–Schrödinger perturbation theory</a>. </p><p>Each of the wave functions for the electrons of the sample (S) and the tip (T) decay into the vacuum after hitting the surface potential barrier, roughly of the size of the surface work function. The wave functions are the solutions of two separate Schrödinger's equations for electrons in potentials <i>U</i><sub>S</sub> and <i>U</i><sub>T</sub>. When the time dependence of the states of known 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="{\displaystyle E_{\mu }^{\text{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mu }^{\text{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3951b470e5d3fb45e656b965b85c8b4c25c43553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.94ex; height:3.176ex;" alt="{\displaystyle E_{\mu }^{\text{S}}}"></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 E_{\nu }^{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\nu }^{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36cfa9f97ac921732eb3eb79f389cd60f360896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.213ex; height:2.843ex;" alt="{\displaystyle E_{\nu }^{\text{T}}}"></span> is factored out, the wave functions have the following general form </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 \psi _{\mu }^{\text{S}}(t)=\psi _{\mu }^{\text{S}}\exp \left(-{\frac {i}{\hbar }}E_{\mu }^{\text{S}}t\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mu }^{\text{S}}(t)=\psi _{\mu }^{\text{S}}\exp \left(-{\frac {i}{\hbar }}E_{\mu }^{\text{S}}t\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f294291087e13b140cc5f87b60687d332ba6b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.958ex; height:6.176ex;" alt="{\displaystyle \psi _{\mu }^{\text{S}}(t)=\psi _{\mu }^{\text{S}}\exp \left(-{\frac {i}{\hbar }}E_{\mu }^{\text{S}}t\right),}"></span></dd> <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 \psi _{\nu }^{\text{T}}(t)=\psi _{\nu }^{\text{T}}\exp \left(-{\frac {i}{\hbar }}E_{\nu }^{\text{T}}t\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }^{\text{T}}(t)=\psi _{\nu }^{\text{T}}\exp \left(-{\frac {i}{\hbar }}E_{\nu }^{\text{T}}t\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9e59d31985cd6a9973b5f5d0d7eef0c394f24b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.622ex; height:6.176ex;" alt="{\displaystyle \psi _{\nu }^{\text{T}}(t)=\psi _{\nu }^{\text{T}}\exp \left(-{\frac {i}{\hbar }}E_{\nu }^{\text{T}}t\right).}"></span></dd></dl> <p>If the two systems are put closer together, but are still separated by a thin vacuum region, the potential acting on an electron in the combined system is <i>U</i><sub>T</sub> + <i>U</i><sub>S</sub>. Here, each of the potentials is spatially limited to its own side of the barrier. Only because the tail of a wave function of one electrode is in the range of the potential of the other, there is a finite probability for any state to evolve over time into the states of the other electrode.<sup id="cite_ref-Chen_5-17" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The future of the sample's state <i>μ</i> can be written as a linear combination with time-dependent coefficients 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 \psi _{\mu }^{\text{S}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mu }^{\text{S}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dac92c3bb0e13e00ac0aeea7b26a478981188e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.385ex; height:3.176ex;" alt="{\displaystyle \psi _{\mu }^{\text{S}}(t)}"></span> and all <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 _{\nu }^{\text{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }^{\text{T}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/279f41d911306ead3cd28addb59d6c5b855bb641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:3.009ex;" alt="{\displaystyle \psi _{\nu }^{\text{T}}(t)}"></span>: </p> <dl><dd><span class="mwe-math-element" id="ansatz"><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 (t)=\psi _{\mu }^{\text{S}}(t)+\sum _{\nu }c_{\nu }(t)\psi _{\nu }^{\text{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=\psi _{\mu }^{\text{S}}(t)+\sum _{\nu }c_{\nu }(t)\psi _{\nu }^{\text{T}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e9bd8a64eedff3ab21e549d405b4ac99e5d528" class="mwe-math-fallback-image-inline mw-invert skin-invert" id="ansatz" aria-hidden="true" style="vertical-align: -3.005ex; width:29.569ex; height:5.509ex;" alt="{\displaystyle \psi (t)=\psi _{\mu }^{\text{S}}(t)+\sum _{\nu }c_{\nu }(t)\psi _{\nu }^{\text{T}}(t)}"></span></dd></dl> <p>with the initial condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\nu }(0)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\nu }(0)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339a5d7142a973214096f47b1c89a9397b317286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.343ex; height:2.843ex;" alt="{\displaystyle c_{\nu }(0)=0}"></span>.<sup id="cite_ref-Chen_5-18" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> When the new wave function is inserted into the Schrödinger's equation for the potential <i>U</i><sub>T</sub> + <i>U</i><sub>S</sub>, the obtained equation is projected onto each separate <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 _{\nu }^{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }^{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669967000e4ef53487f811b40d7e21d95fa9e9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.932ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }^{\text{T}}}"></span> (that is, the equation is multiplied by 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 {\psi _{\nu }^{\text{T}}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\psi _{\nu }^{\text{T}}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767549cb4faef4b2ed08209546767e00c98549b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.986ex; height:3.009ex;" alt="{\displaystyle {\psi _{\nu }^{\text{T}}}^{*}}"></span> and integrated over the whole volume) to single out the coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\nu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7afaf12e88ab23ef9c87b7dc4af999e8562ad45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.757ex; height:2.009ex;" alt="{\displaystyle c_{\nu }.}"></span> All <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 _{\mu }^{\text{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mu }^{\text{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d6b0339ec85f5806d6bd31da6ed37f7b4f680a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:3.176ex;" alt="{\displaystyle \psi _{\mu }^{\text{S}}}"></span> are taken to be <i>nearly orthogonal</i> to all <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 _{\nu }^{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\nu }^{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669967000e4ef53487f811b40d7e21d95fa9e9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.932ex; height:2.843ex;" alt="{\displaystyle \psi _{\nu }^{\text{T}}}"></span> (their overlap is a small fraction of the total wave functions), and only first-order quantities retained. Consequently, the time evolution of the coefficients is given by </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 {\frac {\mathrm {d} }{\mathrm {d} t}}c_{\nu }(t)=-{\frac {i}{\hbar }}\int \psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,\exp \left[-{\frac {i}{\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>∫</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mspace width="thinmathspace" /> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mspace width="thinmathspace" /> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}c_{\nu }(t)=-{\frac {i}{\hbar }}\int \psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,\exp \left[-{\frac {i}{\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f51ac5025158be3ca97bbb545a819af39875c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.881ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}c_{\nu }(t)=-{\frac {i}{\hbar }}\int \psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,\exp \left[-{\frac {i}{\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t\right].}"></span></dd></dl> <p>Because the potential <i>U</i><sub>T</sub> is zero at the distance of a few atomic diameters away from the surface of the electrode, the integration over <i>z</i> can be done from a point <i>z</i><sub>0</sub> somewhere inside the barrier and into the volume of the tip (<i>z</i> > <i>z</i><sub>0</sub>). </p><p>If the tunneling matrix element is defined 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 M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mspace width="thinmathspace" /> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/166e1d4781147415d79d55551454e5bb3e9b51e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.765ex; height:6.009ex;" alt="{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}"></span></dd></dl> <p>the probability of the sample's state <i>μ</i> evolving in time <i>t</i> into the state of the tip <i>ν</i> 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 |c_{\nu }(t)|^{2}=|M_{\mu \nu }|^{2}{\frac {4\sin ^{2}{\big [}{\tfrac {1}{2\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t{\big ]}}{(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\nu }(t)|^{2}=|M_{\mu \nu }|^{2}{\frac {4\sin ^{2}{\big [}{\tfrac {1}{2\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t{\big ]}}{(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6b42283ee8f67304cc0e460cae986dac938121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.395ex; height:7.676ex;" alt="{\displaystyle |c_{\nu }(t)|^{2}=|M_{\mu \nu }|^{2}{\frac {4\sin ^{2}{\big [}{\tfrac {1}{2\hbar }}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})t{\big ]}}{(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})^{2}}}.}"></span></dd></dl> <p>In a system with many electrons impinging on the barrier, this probability will give the proportion of those that successfully tunnel. If at a time <i>t</i> this fraction was <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c_{\nu }(t)|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\nu }(t)|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bbe22fbe040eec2f99d3931e81946e66eff772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.754ex; height:3.343ex;" alt="{\displaystyle |c_{\nu }(t)|^{2},}"></span> at a later time <i>t</i> + d<i>t</i> the total fraction 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 |c_{\nu }(t+\mathrm {d} t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a4637db70471ff01f39a13711ae7fd68a7ed9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.08ex; height:3.343ex;" alt="{\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}}"></span> would have tunneled. The <i>current</i> of tunneling electrons at each instance is therefore proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}-|c_{\nu }(t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}-|c_{\nu }(t)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97c1137e88049664e575a00e483ff1cb44f8667a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.027ex; height:3.343ex;" alt="{\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}-|c_{\nu }(t)|^{2}}"></span> divided by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4331c59343cae12e47900db8866e7f9c0c89771d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} t,}"></span> which is the time derivative 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 |c_{\nu }(t)|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\nu }(t)|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bbe22fbe040eec2f99d3931e81946e66eff772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.754ex; height:3.343ex;" alt="{\displaystyle |c_{\nu }(t)|^{2},}"></span><sup id="cite_ref-Lounis_15-5" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </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 \Gamma _{\mu \to \nu }\ {\overset {\text{def}}{=}}\ {\frac {\mathrm {d} }{\mathrm {d} t}}|c_{\nu }(t)|^{2}={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}{\frac {\sin {\big [}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}){\tfrac {t}{\hbar }}{\big ]}}{\pi (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo stretchy="false">→</mo> <mi>ν</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mtext>def</mtext> </mover> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>t</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mrow> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mu \to \nu }\ {\overset {\text{def}}{=}}\ {\frac {\mathrm {d} }{\mathrm {d} t}}|c_{\nu }(t)|^{2}={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}{\frac {\sin {\big [}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}){\tfrac {t}{\hbar }}{\big ]}}{\pi (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb6f55eaaf63f600dd0cb3b3ac2a5ce64d68f148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.928ex; height:7.509ex;" alt="{\displaystyle \Gamma _{\mu \to \nu }\ {\overset {\text{def}}{=}}\ {\frac {\mathrm {d} }{\mathrm {d} t}}|c_{\nu }(t)|^{2}={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}{\frac {\sin {\big [}(E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}){\tfrac {t}{\hbar }}{\big ]}}{\pi (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}}.}"></span></dd></dl> <p>The time scale of the measurement in STM is many orders of magnitude larger than the typical <a href="/wiki/Femtosecond" title="Femtosecond">femtosecond</a> time scale of electron processes in materials, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t/\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t/\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68da3a775372f9df73032ae4f5d509e3b48c742e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\displaystyle t/\hbar }"></span> is large. The fraction part of the formula is a fast-oscillating 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 (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7a4f66d78e475ac94dc20d8e984bd1b8f1a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.802ex; height:3.176ex;" alt="{\displaystyle (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}"></span> that rapidly decays away from the central peak, 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 E_{\mu }^{\text{S}}=E_{\nu }^{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mu }^{\text{S}}=E_{\nu }^{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee77f4bc2f1cc29cc1c3ba6679aa681184be084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.251ex; height:3.176ex;" alt="{\displaystyle E_{\mu }^{\text{S}}=E_{\nu }^{\text{T}}}"></span>. In other words, the most probable tunneling process, by far, is the elastic one, in which the electron's energy is conserved. The fraction, as written above, is a representation of the <a href="/wiki/Dirac_delta_function#Oscillatory_integrals" title="Dirac delta function">delta function</a>, so </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 \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\delta (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo stretchy="false">→</mo> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>δ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\delta (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da14a4c4de632ff0bdce99e64c3e0e7eb4db10c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.814ex; height:5.343ex;" alt="{\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\delta (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}}).}"></span></dd></dl> <p>Solid-state systems are commonly described in terms of continuous rather than discrete energy levels. The term <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_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb371026df7f7ab2e1958c681b7b5ddffca87d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.851ex; height:3.176ex;" alt="{\displaystyle \delta (E_{\mu }^{\text{S}}-E_{\nu }^{\text{T}})}"></span> can be thought of as the <a href="/wiki/Density_of_states#Definition" title="Density of states">density of states</a> of the tip at energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mu }^{\text{S}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mu }^{\text{S}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbd99f8c73a227be993cd06433b79f8ca667be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.587ex; height:3.176ex;" alt="{\displaystyle E_{\mu }^{\text{S}},}"></span> giving </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 \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\rho _{\text{T}}(E_{\mu }^{\text{S}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo stretchy="false">→</mo> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\rho _{\text{T}}(E_{\mu }^{\text{S}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9bcaca0c662cc4e60074a8e6d3222ab2845fe41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.333ex; height:5.343ex;" alt="{\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\rho _{\text{T}}(E_{\mu }^{\text{S}}).}"></span></dd></dl> <p>The number of energy levels in the sample between 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="{\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> 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 +\mathrm {d} \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon +\mathrm {d} \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272a6595626da0fa8d5456f7e5e2b5ec83df6253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.3ex; height:2.343ex;" alt="{\displaystyle \varepsilon +\mathrm {d} \varepsilon }"></span> 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 \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696b7882a62c54a31ddd0f2ddc6d7d41a9456944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.651ex; height:2.843ex;" alt="{\displaystyle \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon .}"></span> When occupied, these levels are spin-degenerate (except in a few special classes of materials) and contain 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 2e\cdot \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>e</mi> <mo>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2e\cdot \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1983d343bdc1a8522f2e6b01039c2056c719670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.929ex; height:2.843ex;" alt="{\displaystyle 2e\cdot \rho _{\text{S}}(\varepsilon )\,\mathrm {d} \varepsilon }"></span> of either spin. With the sample biased to voltage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace9595e3ce66fdec7e9d30202626accd676b11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.434ex; height:2.509ex;" alt="{\displaystyle V,}"></span> tunneling can occur only between states whose occupancies, given for each electrode by the <a href="/wiki/Fermi%E2%80%93Dirac_distribution" class="mw-redirect" title="Fermi–Dirac distribution">Fermi–Dirac distribution</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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, are not the same, that is, when either one or the other is occupied, but not both. That will be for all 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="{\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> for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf6d8d1d72dff9fdda55833e3adf7fe7fc83765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.617ex; height:2.843ex;" alt="{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}"></span> is not zero. For example, an electron will tunnel from energy 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 E_{\text{F}}-eV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{F}}-eV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771e299e88cc7c44fa283f5d54a7363badbd4a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.732ex; height:2.509ex;" alt="{\displaystyle E_{\text{F}}-eV}"></span> in the sample into energy 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 E_{\text{F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df54e5da3da6ebc2c30a44235e448a3bebed687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.021ex; height:2.509ex;" alt="{\displaystyle E_{\text{F}}}"></span> in the tip (<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 =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbe653d369abd71fc0be98efbf59e9edc1bdfb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon =0}"></span>), an electron 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 E_{\text{F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df54e5da3da6ebc2c30a44235e448a3bebed687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.021ex; height:2.509ex;" alt="{\displaystyle E_{\text{F}}}"></span> in the sample will find unoccupied states in the tip 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 E_{\text{F}}+eV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>e</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{F}}+eV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e7a813573f5fdd3374f5e84342cdf3009de56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.732ex; height:2.509ex;" alt="{\displaystyle E_{\text{F}}+eV}"></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 \varepsilon =eV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mi>e</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =eV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773a303881073700447d62ac3a26ebb534f1e095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.053ex; height:2.176ex;" alt="{\displaystyle \varepsilon =eV}"></span>), and so will be for all energies in between. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: <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 2e\cdot \rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\mathrm {d} \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>e</mi> <mo>⋅</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2e\cdot \rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\mathrm {d} \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf40c90cdf102b0e3ecddb6ab48d9ddc5408b485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.501ex; height:2.843ex;" alt="{\displaystyle 2e\cdot \rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\mathrm {d} \varepsilon }"></span> representing available electrons, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf6d8d1d72dff9fdda55833e3adf7fe7fc83765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.617ex; height:2.843ex;" alt="{\displaystyle f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )}"></span> for those that are allowed to tunnel, and the probability 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 \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> for those that will actually tunnel: </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_{t}={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )]\,\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>e</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )]\,\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a8486ad191dde95bc814bf16fe7734c83dd060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:82.75ex; height:6.176ex;" alt="{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{\text{F}}-eV+\varepsilon )-f(E_{\text{F}}+\varepsilon )]\,\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}"></span></dd></dl> <p>Typical experiments are run at a liquid-helium temperature (around 4 K), at which the Fermi-level cut-off of the electron population is less than a millielectronvolt wide. The allowed energies are only those between the two step-like Fermi levels, and the integral becomes </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_{t}={\frac {4\pi e}{\hbar }}\int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mi>e</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>V</mi> </mrow> </msubsup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>M</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a2f2efe6f604415780a7b66ab59b7131017d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.948ex; height:6.176ex;" alt="{\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,|M|^{2}\,d\varepsilon .}"></span></dd></dl> <p>When the bias is small, it is reasonable to assume that the electron wave functions and, consequently, the tunneling matrix element do not change significantly in the narrow range of energies. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip: </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_{t}\propto \int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,d\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∝</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>V</mi> </mrow> </msubsup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>−</mo> <mi>e</mi> <mi>V</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{t}\propto \int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,d\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e59194698b78a921afaea8231fccc4b56f344ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.969ex; height:6.176ex;" alt="{\displaystyle I_{t}\propto \int _{0}^{eV}\rho _{\text{S}}(E_{\text{F}}-eV+\varepsilon )\,\rho _{\text{T}}(E_{\text{F}}+\varepsilon )\,d\varepsilon .}"></span></dd></dl> <p>How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element </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 M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mspace width="thinmathspace" /> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44930389dc36366e2be6d543e6eb55f4f82749c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.765ex; height:6.009ex;" alt="{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{\text{S}}\,U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}"></span></dd></dl> <p>This formula can be transformed so that no explicit dependence on the potential remains. First, 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 U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662285b63698caf5cfdf8f4efca2234dd16faf5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.38ex; height:3.009ex;" alt="{\displaystyle U_{\text{T}}\,{\psi _{\nu }^{\text{T}}}^{*}}"></span> part is taken out from the Schrödinger equation for the tip, and the elastic tunneling condition is used so that </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 M_{\mu \nu }=\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}E_{\mu }\psi _{\mu }^{\text{S}}+\psi _{\mu }^{\text{S}}{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <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> <mi>m</mi> </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> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}E_{\mu }\psi _{\mu }^{\text{S}}+\psi _{\mu }^{\text{S}}{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0d67865de072da61991e11c6acf7b24d00a2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.633ex; height:6.509ex;" alt="{\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}E_{\mu }\psi _{\mu }^{\text{S}}+\psi _{\mu }^{\text{S}}{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}"></span></dd></dl> <p>Now <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_{\mu }\,{\psi _{\mu }^{\text{S}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mu }\,{\psi _{\mu }^{\text{S}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2fa3164f1a0f4609e57750e4d2d650db9c055d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.062ex; height:3.176ex;" alt="{\displaystyle E_{\mu }\,{\psi _{\mu }^{\text{S}}}}"></span> is present in the Schrödinger equation for the sample and equals the kinetic plus the potential operator acting 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 \psi _{\mu }^{\text{S}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mu }^{\text{S}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace332a45883a4464cd6cba2a4a2d24b58c4fcde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.383ex; height:3.176ex;" alt="{\displaystyle \psi _{\mu }^{\text{S}}.}"></span> However, the potential part containing <i>U</i><sub>S</sub> is on the tip side of the barrier nearly zero. What remains, </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 M_{\mu \nu }=-{\frac {\hbar ^{2}}{2m}}\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\mu }^{\text{S}}}-{\psi _{\mu }^{\text{S}}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <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> <mi>m</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>></mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <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> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </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> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }=-{\frac {\hbar ^{2}}{2m}}\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\mu }^{\text{S}}}-{\psi _{\mu }^{\text{S}}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afdb71cb7f16e9b5e66e2deb0c246af079e86c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.188ex; height:6.509ex;" alt="{\displaystyle M_{\mu \nu }=-{\frac {\hbar ^{2}}{2m}}\int _{z>z_{0}}\left({\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\mu }^{\text{S}}}-{\psi _{\mu }^{\text{S}}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{\text{T}}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}"></span></dd></dl> <p>can be integrated over <i>z</i> because the integrand in the parentheses equals <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 \partial _{z}\left({\psi _{\nu }^{\text{T}}}^{*}\,\partial _{z}\psi _{\mu }^{\text{S}}-{\psi _{\mu }^{\text{S}}}\,\partial _{z}{\psi _{\nu }^{\text{T}}}^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mrow> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{z}\left({\psi _{\nu }^{\text{T}}}^{*}\,\partial _{z}\psi _{\mu }^{\text{S}}-{\psi _{\mu }^{\text{S}}}\,\partial _{z}{\psi _{\nu }^{\text{T}}}^{*}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f96056b79e7d78d4eef86971c06752277ecdab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.319ex; height:3.343ex;" alt="{\displaystyle \partial _{z}\left({\psi _{\nu }^{\text{T}}}^{*}\,\partial _{z}\psi _{\mu }^{\text{S}}-{\psi _{\mu }^{\text{S}}}\,\partial _{z}{\psi _{\nu }^{\text{T}}}^{*}\right).}"></span> </p><p>Bardeen's tunneling matrix element is an integral of the wave functions and their gradients over a surface separating the two planar electrodes: </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 M_{\mu \nu }={\frac {\hbar ^{2}}{2m}}\int _{z=z_{0}}\left({\psi _{\mu }^{\text{S}}}{\frac {\partial }{\partial z}}{\psi _{\nu }^{\text{T}}}^{*}-{\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial }{\partial z}}{\psi _{\mu }^{\text{S}}}\right)\,\mathrm {d} x\,\mathrm {d} y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <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> <mi>m</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msubsup> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\mu \nu }={\frac {\hbar ^{2}}{2m}}\int _{z=z_{0}}\left({\psi _{\mu }^{\text{S}}}{\frac {\partial }{\partial z}}{\psi _{\nu }^{\text{T}}}^{*}-{\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial }{\partial z}}{\psi _{\mu }^{\text{S}}}\right)\,\mathrm {d} x\,\mathrm {d} y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3f5981325c5410a33a614d9de4d9c1461bc380" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.499ex; height:6.509ex;" alt="{\displaystyle M_{\mu \nu }={\frac {\hbar ^{2}}{2m}}\int _{z=z_{0}}\left({\psi _{\mu }^{\text{S}}}{\frac {\partial }{\partial z}}{\psi _{\nu }^{\text{T}}}^{*}-{\psi _{\nu }^{\text{T}}}^{*}{\frac {\partial }{\partial z}}{\psi _{\mu }^{\text{S}}}\right)\,\mathrm {d} x\,\mathrm {d} y.}"></span></dd></dl> <p>The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that <i>leak</i> through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. </p><p>The tunneling matrix elements show appreciable energy dependence, which is such that tunneling from the upper end of the <i>eV</i> interval is nearly an order of magnitude more likely than tunneling from the states at its bottom. When the sample is biased positively, its unoccupied levels are probed as if the density of states of the tip is concentrated at its Fermi level. Conversely, when the sample is biased negatively, its occupied electronic states are probed, but the spectrum of the electronic states of the tip dominates. In this case it is important that the density of states of the tip is as flat as possible.<sup id="cite_ref-Chen_5-19" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The results identical to Bardeen's can be obtained by considering adiabatic approach of the two electrodes and using the standard time-dependent perturbation theory.<sup id="cite_ref-Lounis_15-6" class="reference"><a href="#cite_note-Lounis-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This leads to <a href="/wiki/Fermi%27s_golden_rule" title="Fermi's golden rule">Fermi's golden rule</a> for the transition probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{\mu \to \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo stretchy="false">→</mo> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mu \to \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94cbf080c945b47366895b7bea5d2887c2657fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.191ex; height:2.843ex;" alt="{\displaystyle \Gamma _{\mu \to \nu }}"></span> in the form given above. </p><p>Bardeen's model is for tunneling between two planar electrodes and does not explain scanning tunneling microscope's lateral resolution. Tersoff and Hamann<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> used Bardeen's theory and modeled the tip as a structureless geometric point.<sup id="cite_ref-Chen_5-20" class="reference"><a href="#cite_note-Chen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This helped them disentangle the properties of the tip—which are hard to model—from the properties of the sample surface. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically symmetric tip (<i>s</i>-wave tip model). With such a simplification, their model proved valuable for interpreting images of surface features bigger than a nanometre, even though it predicted atomic-scale corrugations of less than a picometre. These are well below the microscope's detection limit and below the values actually observed in experiments. </p><p>In sub-nanometre-resolution experiments, the convolution of the tip and sample surface states will always be important, to the extent of the apparent inversion of the atomic corrugations that may be observed within the same scan. Such effects can only be explained by modeling of the surface and tip electronic states and the ways the two electrodes interact from <a href="/wiki/Ab_initio_quantum_chemistry_methods" title="Ab initio quantum chemistry methods">first principles</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Gallery_of_STM_images">Gallery of STM images</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=7" title="Edit section: Gallery of STM images"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Scanning_tunneling_microscope_(STM)_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_(111)_surface.png" class="mw-file-description" title="One-atom-thick silver islands grown on terraces of the (111) surface of palladium. Image size is 250 nm by 250 nm."><img alt="One-atom-thick silver islands grown on terraces of the (111) surface of palladium. Image size is 250 nm by 250 nm." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png/120px-Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png/180px-Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png/240px-Scanning_tunneling_microscope_%28STM%29_250_nm_by_250_nm_image_of_one-atom-thick_silver_islands_grown_on_palladium_%28111%29_surface.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="gallerytext">One-atom-thick silver islands grown on terraces of the (111) surface of palladium. Image size is 250 nm by 250 nm.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Atomic_resolution_Au100.JPG" class="mw-file-description" title="The characteristic reconstruction fringes on the (100) surface of gold are 1.44 nm wide[20] and consist of six atomic rows that sit on top of five rows of the crystal bulk. Image size is approximately 10 nm by 10 nm."><img alt="The characteristic reconstruction fringes on the (100) surface of gold are 1.44 nm wide[20] and consist of six atomic rows that sit on top of five rows of the crystal bulk. Image size is approximately 10 nm by 10 nm." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/120px-Atomic_resolution_Au100.JPG" decoding="async" width="120" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/180px-Atomic_resolution_Au100.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Atomic_resolution_Au100.JPG/240px-Atomic_resolution_Au100.JPG 2x" data-file-width="500" data-file-height="480" /></a></span></div> <div class="gallerytext">The characteristic reconstruction fringes on the (100) surface of gold are 1.44 <a href="/wiki/Nanometre" title="Nanometre">nm</a> wide<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and consist of six atomic rows that sit on top of five rows of the crystal bulk. Image size is approximately 10 nm by 10 nm.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Chiraltube.png" class="mw-file-description" title="A 7 nm long part of a single-walled carbon nanotube."><img alt="A 7 nm long part of a single-walled carbon nanotube." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Chiraltube.png/120px-Chiraltube.png" decoding="async" width="120" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Chiraltube.png/180px-Chiraltube.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Chiraltube.png/240px-Chiraltube.png 2x" data-file-width="486" data-file-height="155" /></a></span></div> <div class="gallerytext">A 7 nm long part of a single-walled <a href="/wiki/Carbon_nanotube" title="Carbon nanotube">carbon nanotube</a>.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Silicium-atomes.png" class="mw-file-description" title="Atoms on the surface of a crystal of silicon carbide (SiC) are arranged in a hexagonal lattice and are 0.3 nm apart."><img alt="Atoms on the surface of a crystal of silicon carbide (SiC) are arranged in a hexagonal lattice and are 0.3 nm apart." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Silicium-atomes.png/120px-Silicium-atomes.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7a/Silicium-atomes.png 1.5x" data-file-width="177" data-file-height="177" /></a></span></div> <div class="gallerytext">Atoms on the surface of a crystal of <a href="/wiki/Silicon_carbide" title="Silicon carbide">silicon carbide</a> (SiC) are arranged in a hexagonal lattice and are 0.3 nm apart.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Cens_nanomanipulation3d_Trixler.jpg" class="mw-file-description" title="STM nanomanipulation of PTCDA molecules on graphite to inscribe the logo of the Center for NanoScience (CeNS), Munich."><img alt="STM nanomanipulation of PTCDA molecules on graphite to inscribe the logo of the Center for NanoScience (CeNS), Munich." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Cens_nanomanipulation3d_Trixler.jpg/120px-Cens_nanomanipulation3d_Trixler.jpg" decoding="async" width="120" height="84" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Cens_nanomanipulation3d_Trixler.jpg/180px-Cens_nanomanipulation3d_Trixler.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Cens_nanomanipulation3d_Trixler.jpg/240px-Cens_nanomanipulation3d_Trixler.jpg 2x" data-file-width="638" data-file-height="445" /></a></span></div> <div class="gallerytext">STM nanomanipulation of PTCDA molecules on <a href="/wiki/Graphite" title="Graphite">graphite</a> to inscribe the logo of the <a href="/wiki/Center_for_NanoScience" title="Center for NanoScience">Center for NanoScience</a> (CeNS), Munich. </div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Early_invention">Early invention</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=8" title="Edit section: Early invention"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An earlier invention similar to Binnig and Rohrer's, the <i>Topografiner</i> of R. Young, J. Ward, and F. Scire from the <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>, relied on field emission.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> However, Young is credited by the Nobel Committee as the person who realized that it should be possible to achieve better resolution by using the tunnel effect.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_related_techniques">Other related techniques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=9" title="Edit section: Other related techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many other microscopy techniques have been developed based upon STM. These include <a href="/wiki/Photon_scanning_microscopy" title="Photon scanning microscopy">photon scanning microscopy</a> (PSTM), which uses an optical tip to tunnel photons;<sup id="cite_ref-Bai_4-4" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> scanning tunneling potentiometry (STP), which measures electric potential across a surface;<sup id="cite_ref-Bai_4-5" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Spin-polarized_scanning_tunneling_microscopy" title="Spin-polarized scanning tunneling microscopy">spin-polarized scanning tunneling microscopy</a> (SPSTM), which uses a <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> tip to tunnel spin-polarized electrons into a magnetic sample;<sup id="cite_ref-Wiesendanger_23-0" class="reference"><a href="#cite_note-Wiesendanger-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Multi-tip_scanning_tunneling_microscopy" title="Multi-tip scanning tunneling microscopy">multi-tip scanning tunneling microscopy</a>, which enables electrical measurements to be performed at the nanoscale; and <a href="/wiki/Atomic_force_microscopy" title="Atomic force microscopy">atomic force microscopy</a> (AFM), in which the <a href="/wiki/Force" title="Force">force</a> caused by interaction between the tip and sample is measured. </p><p>STM can be used to manipulate atoms and change the topography of the sample. This is attractive for several reasons. Firstly the STM has an atomically precise positioning system, which enables very accurate atomic-scale manipulation. Furthermore, after the surface is modified by the tip, the same instrument can be used to image the resulting structures. <a href="/wiki/IBM" title="IBM">IBM</a> researchers famously developed a way to manipulate <a href="/wiki/Xenon" title="Xenon">xenon</a> atoms adsorbed on a <a href="/wiki/Nickel" title="Nickel">nickel</a> surface.<sup id="cite_ref-Bai_4-6" class="reference"><a href="#cite_note-Bai-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This technique has been used to create electron <i>corrals</i> with a small number of adsorbed atoms and observe <a href="/wiki/Friedel_oscillations" title="Friedel oscillations">Friedel oscillations</a> in the electron density on the surface of the substrate. Aside from modifying the actual sample surface, one can also use the STM to tunnel electrons into a layer of electron-beam <a href="/wiki/Photoresist" title="Photoresist">photoresist</a> on the sample, in order to do <a href="/wiki/Lithography" title="Lithography">lithography</a>. This has the advantage of offering more control of the exposure than traditional <a href="/wiki/Electron-beam_lithography" title="Electron-beam lithography">electron-beam lithography</a>. Another practical application of STM is atomic deposition of metals (gold, silver, tungsten, etc.) with any desired (pre-programmed) pattern, which can be used as contacts to nanodevices or as nanodevices themselves.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2020)">citation needed</span></a></i>]</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=Scanning_tunneling_microscope&action=edit&section=10" 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"> <ul><li><a href="/wiki/Scanning_probe_microscopy" title="Scanning probe microscopy">Scanning probe microscopy</a></li> <li><a href="/wiki/Atomic_force_microscope" class="mw-redirect" title="Atomic force microscope">Atomic force microscope</a></li> <li><a href="/wiki/Electrochemical_scanning_tunneling_microscope" title="Electrochemical scanning tunneling microscope">Electrochemical scanning tunneling microscope</a></li> <li><a href="/wiki/Microscopy" title="Microscopy">Microscopy</a></li> <li><a href="/wiki/Electron_microscope" title="Electron microscope">Electron microscope</a></li> <li><a href="/wiki/Multi-tip_scanning_tunneling_microscopy" title="Multi-tip scanning tunneling microscopy">Multi-tip scanning tunneling microscopy</a></li> <li><a href="/wiki/IBM_(atoms)" title="IBM (atoms)">IBM (atoms)</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=Scanning_tunneling_microscope&action=edit&section=11" 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-Binnig-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Binnig_1-0">^</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 .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBinnigRohrer1986" class="citation journal cs1">Binnig G, Rohrer H (1986). 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"Recent advances in spin-polarized scanning tunneling microscopy". <i>Ultramicroscopy</i>. 42–44: 338–344. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3991%2892%2990289-V">10.1016/0304-3991(92)90289-V</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:95739038">95739038</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ultramicroscopy&rft.atitle=Recent+advances+in+spin-polarized+scanning+tunneling+microscopy&rft.volume=42%E2%80%9344&rft.pages=338-344&rft.date=1992&rft_id=info%3Adoi%2F10.1016%2F0304-3991%2892%2990289-V&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A95739038%23id-name%3DS2CID&rft.aulast=Wiesendanger&rft.aufirst=R&rft.au=Shvets%2C+IV&rft.au=B%C3%BCrgler%2C+D&rft.au=Tarrach%2C+G&rft.au=G%C3%BCntherodt%2C+HJ&rft.au=Coey%2C+JM&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Scanning_tunneling_microscope&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen1993" class="citation book cs1">Chen CJ (1993). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221218190343/http://www.columbia.edu/~jcc2161/documents/STM_book.pdf"><i>Introduction to Scanning Tunneling Microscopy</i></a> <span class="cs1-format">(PDF)</span>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-507150-4" title="Special:BookSources/978-0-19-507150-4"><bdi>978-0-19-507150-4</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.columbia.edu/~jcc2161/documents/STM_book.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2022-12-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Scanning+Tunneling+Microscopy&rft.pub=Oxford+University+Press&rft.date=1993&rft.isbn=978-0-19-507150-4&rft.aulast=Chen&rft.aufirst=CJ&rft_id=http%3A%2F%2Fwww.columbia.edu%2F~jcc2161%2Fdocuments%2FSTM_book.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWiesendanger1994" class="citation book cs1">Wiesendanger R (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EXae0pjS2vwC"><i>Scanning probe microscopy and spectroscopy: methods and applications</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42847-7" title="Special:BookSources/978-0-521-42847-7"><bdi>978-0-521-42847-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scanning+probe+microscopy+and+spectroscopy%3A+methods+and+applications&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-42847-7&rft.aulast=Wiesendanger&rft.aufirst=R&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEXae0pjS2vwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWiesendangerGüntherodt1996" class="citation book cs1">Wiesendanger R, Güntherodt HJ, eds. (1996). <i>Scanning Tunneling Microscopy III – Theory of STM and Related Scanning Probe Methods</i>. Springer Series in Surface Sciences. Vol. 29. Springer-Verlag Berlin Heidelberg. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-80118-1">10.1007/978-3-642-80118-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-60824-0" title="Special:BookSources/978-3-540-60824-0"><bdi>978-3-540-60824-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scanning+Tunneling+Microscopy+III+%E2%80%93+Theory+of+STM+and+Related+Scanning+Probe+Methods&rft.series=Springer+Series+in+Surface+Sciences&rft.pub=Springer-Verlag+Berlin+Heidelberg&rft.date=1996&rft_id=info%3Adoi%2F10.1007%2F978-3-642-80118-1&rft.isbn=978-3-540-60824-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBai2000" class="citation book cs1">Bai C (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3Q08jRmmtrkC&pg=PA345"><i>Scanning tunneling microscopy and its applications</i></a>. New York: Springer Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65715-6" title="Special:BookSources/978-3-540-65715-6"><bdi>978-3-540-65715-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scanning+tunneling+microscopy+and+its+applications&rft.place=New+York&rft.pub=Springer+Verlag&rft.date=2000&rft.isbn=978-3-540-65715-6&rft.aulast=Bai&rft.aufirst=C&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3Q08jRmmtrkC%26pg%3DPA345&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoigtländer2015" class="citation book cs1">Voigtländer B (2015). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-662-45240-0"><i>Scanning Probe Microscopy</i></a>. NanoScience and Technology. <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/2015spma.book.....V">2015spma.book.....V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-45240-0">10.1007/978-3-662-45240-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-45239-4" title="Special:BookSources/978-3-662-45239-4"><bdi>978-3-662-45239-4</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-4904">1434-4904</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:94208893">94208893</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scanning+Probe+Microscopy&rft.series=NanoScience+and+Technology&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A94208893%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015spma.book.....V&rft.issn=1434-4904&rft_id=info%3Adoi%2F10.1007%2F978-3-662-45240-0&rft.isbn=978-3-662-45239-4&rft.aulast=Voigtl%C3%A4nder&rft.aufirst=B&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-662-45240-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLounis2014" class="citation arxiv cs1">Lounis S (2014-04-03). "Theory of Scanning Tunneling Microscopy". <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/1404.0961">1404.0961</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cond-mat.mes-hall">cond-mat.mes-hall</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Theory+of+Scanning+Tunneling+Microscopy&rft.date=2014-04-03&rft_id=info%3Aarxiv%2F1404.0961&rft.aulast=Lounis&rft.aufirst=S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBinnigRohrerGerberWeibel1983" class="citation journal cs1">Binnig G, Rohrer H, Gerber C, Weibel E (1983-01-10). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.50.120">"7 × 7 Reconstruction on Si(111) Resolved in Real Space"</a>. <i>Physical Review Letters</i>. <b>50</b> (2): 120–123. <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/1983PhRvL..50..120B">1983PhRvL..50..120B</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/10.1103%2FPhysRevLett.50.120">10.1103/PhysRevLett.50.120</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=7+%C3%97+7+Reconstruction+on+Si%28111%29+Resolved+in+Real+Space&rft.volume=50&rft.issue=2&rft.pages=120-123&rft.date=1983-01-10&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.50.120&rft_id=info%3Abibcode%2F1983PhRvL..50..120B&rft.aulast=Binnig&rft.aufirst=G&rft.au=Rohrer%2C+H&rft.au=Gerber%2C+C&rft.au=Weibel%2C+E&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.50.120&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBinnigRohrerGerberWeibel1982" class="citation journal cs1">Binnig G, Rohrer H, Gerber C, Weibel E (1982-07-05). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.49.57">"Surface Studies by Scanning Tunneling Microscopy"</a>. <i>Physical Review Letters</i>. <b>49</b> (1): 57–61. <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/1982PhRvL..49...57B">1982PhRvL..49...57B</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/10.1103%2FPhysRevLett.49.57">10.1103/PhysRevLett.49.57</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Surface+Studies+by+Scanning+Tunneling+Microscopy&rft.volume=49&rft.issue=1&rft.pages=57-61&rft.date=1982-07-05&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.49.57&rft_id=info%3Abibcode%2F1982PhRvL..49...57B&rft.aulast=Binnig&rft.aufirst=G&rft.au=Rohrer%2C+H&rft.au=Gerber%2C+C&rft.au=Weibel%2C+E&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%252FPhysRevLett.49.57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBinnigRohrerGerberWeibel1982" class="citation journal cs1">Binnig G, Rohrer H, Gerber C, Weibel E (1982-01-15). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.92999">"Tunneling through a controllable vacuum gap"</a>. <i>Applied Physics Letters</i>. <b>40</b> (2): 178–180. <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/1982ApPhL..40..178B">1982ApPhL..40..178B</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%2F1.92999">10.1063/1.92999</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-6951">0003-6951</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+Letters&rft.atitle=Tunneling+through+a+controllable+vacuum+gap&rft.volume=40&rft.issue=2&rft.pages=178-180&rft.date=1982-01-15&rft.issn=0003-6951&rft_id=info%3Adoi%2F10.1063%2F1.92999&rft_id=info%3Abibcode%2F1982ApPhL..40..178B&rft.aulast=Binnig&rft.aufirst=G&rft.au=Rohrer%2C+H&rft.au=Gerber%2C+C&rft.au=Weibel%2C+E&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.92999&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBardeen1961" class="citation journal cs1">Bardeen J (1961-01-15). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.6.57">"Tunnelling from a Many-Particle Point of View"</a>. <i>Physical Review Letters</i>. <b>6</b> (2): 57–59. <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/1961PhRvL...6...57B">1961PhRvL...6...57B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.6.57">10.1103/PhysRevLett.6.57</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Tunnelling+from+a+Many-Particle+Point+of+View&rft.volume=6&rft.issue=2&rft.pages=57-59&rft.date=1961-01-15&rft.issn=0031-9007&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.6.57&rft_id=info%3Abibcode%2F1961PhRvL...6...57B&rft.aulast=Bardeen&rft.aufirst=J&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.6.57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTersoffHamann1985" class="citation journal cs1">Tersoff J, Hamann DR (January 1985). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevB.31.805">"Theory of the scanning tunneling microscope"</a>. <i>Physical Review B</i>. <b>31</b> (2): 805–813. <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/1985PhRvB..31..805T">1985PhRvB..31..805T</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.31.805">10.1103/PhysRevB.31.805</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/9935822">9935822</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Theory+of+the+scanning+tunneling+microscope&rft.volume=31&rft.issue=2&rft.pages=805-813&rft.date=1985-01&rft_id=info%3Apmid%2F9935822&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.31.805&rft_id=info%3Abibcode%2F1985PhRvB..31..805T&rft.aulast=Tersoff&rft.aufirst=J&rft.au=Hamann%2C+DR&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevB.31.805&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen1990" class="citation journal cs1">Chen CJ (July 1990). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.65.448">"Origin of atomic resolution on metal surfaces in scanning tunneling microscopy"</a>. <i>Physical Review Letters</i>. <b>65</b> (4): 448–451. <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/1990PhRvL..65..448C">1990PhRvL..65..448C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.65.448">10.1103/PhysRevLett.65.448</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/10042923">10042923</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Origin+of+atomic+resolution+on+metal+surfaces+in+scanning+tunneling+microscopy&rft.volume=65&rft.issue=4&rft.pages=448-451&rft.date=1990-07&rft_id=info%3Apmid%2F10042923&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.65.448&rft_id=info%3Abibcode%2F1990PhRvL..65..448C&rft.aulast=Chen&rft.aufirst=CJ&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.65.448&rfr_id=info%3Asid%2Fen.wikipedia.org%3AScanning+tunneling+microscope" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFujitaSagisaka2008" class="citation journal cs1">Fujita D, Sagisaka K (January 2008). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5099790">"Active nanocharacterization of nanofunctional materials by scanning tunneling microscopy"</a>. <i>Science and Technology of Advanced Materials</i>. <b>9</b> (1): 013003. <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/2008STAdM...9a3003F">2008STAdM...9a3003F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1468-6996%2F9%2F1%2F013003">10.1088/1468-6996/9/1/013003</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5099790">5099790</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/27877921">27877921</a>.</cite><span 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Scanning_tunneling_microscope" class="extiw" title="commons:Category:Scanning tunneling microscope">Scanning tunneling microscope</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.fz-juelich.de/en/pgi/pgi-3/groups/research/multi-tip-spm-instrument-development/a-scanning-tunelling-microscope-filmed-during-operation-by-an-electron-microscope">A scanning tunelling microscope filmed during operation by an electron microscope</a></li> <li><a rel="nofollow" class="external text" href="https://wecanfigurethisout.org/VL/easyScan_STM.htm">The Inner Workings of an STM - An Animated Explanation</a> WeCanFigureThisOut.org</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091028073926/http://www.geocities.com/spm_stm/Project.html">Build a simple STM with a cost of materials less than $100 excluding oscilloscope</a></li> <li><a rel="nofollow" class="external text" href="http://toutestquantique.fr/en/microscopy/">Animations and explanations on various types of microscopes including electron microscopes</a> (Université Paris Sud)</li> <li><a rel="nofollow" class="external text" href="https://hoffman.physics.harvard.edu/research/STMintro.php">Introduction to STM in plain English</a> (Harvard University)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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template">e</abbr></a></li></ul></div><div id="Scanning_probe_microscopy" style="font-size:114%;margin:0 4em"><a href="/wiki/Scanning_probe_microscopy" title="Scanning probe microscopy">Scanning probe microscopy</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Common</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomic_force_microscopy" title="Atomic force microscopy">Atomic force</a> <ul><li><a href="/wiki/Conductive_atomic_force_microscopy" title="Conductive atomic force microscopy">Conductive</a></li> <li><a href="/wiki/AFM-IR" class="mw-redirect" title="AFM-IR">Infrared</a></li> <li><a href="/wiki/Non-contact_atomic_force_microscopy" title="Non-contact atomic force microscopy">Non-contact</a></li> <li><a href="/wiki/Photoconductive_atomic_force_microscopy" title="Photoconductive atomic force microscopy">Photoconductive</a></li></ul></li> <li><a class="mw-selflink selflink">Scanning tunneling</a> <ul><li><a href="/wiki/Electrochemical_scanning_tunneling_microscope" title="Electrochemical scanning tunneling microscope">Electrochemical</a></li> <li><a href="/wiki/Spin_polarized_scanning_tunneling_microscopy" class="mw-redirect" title="Spin polarized scanning tunneling microscopy">Spin polarized</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:AFMsetup.jpg" class="mw-file-description" title="Typical atomic force microscopy set-up"><img alt="Typical atomic force microscopy set-up" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/AFMsetup.jpg/75px-AFMsetup.jpg" decoding="async" width="75" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/AFMsetup.jpg/113px-AFMsetup.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/AFMsetup.jpg/150px-AFMsetup.jpg 2x" data-file-width="721" data-file-height="569" /></a><figcaption>Typical atomic force microscopy set-up</figcaption></figure></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ballistic_electron_emission_microscopy" title="Ballistic electron emission microscopy">Ballistic electron emission</a></li> <li><a href="/wiki/Chemical_force_microscopy" title="Chemical force microscopy">Chemical force</a></li> <li><a href="/wiki/Electrostatic_force_microscope" title="Electrostatic force microscope">Electrostatic force</a></li> <li><a href="/wiki/Kelvin_probe_force_microscope" title="Kelvin probe force microscope">Kelvin probe force</a></li> <li><a href="/wiki/Magnetic_force_microscope" title="Magnetic force microscope">Magnetic force</a></li> <li><a href="/wiki/Magnetic_resonance_force_microscopy" title="Magnetic resonance force microscopy">Magnetic resonance force</a></li> <li><a href="/wiki/Near-field_scanning_optical_microscope" title="Near-field scanning optical microscope">Near-field scanning optical</a> <ul><li><a href="/wiki/Nano-FTIR" title="Nano-FTIR">Nano-FTIR</a></li></ul></li> <li><a href="/wiki/Photon_scanning_microscopy" title="Photon scanning microscopy">Photon scanning</a></li> <li><a href="/wiki/Photothermal_microspectroscopy" title="Photothermal microspectroscopy">Photothermal microspectroscopy</a></li> <li><a href="/wiki/Piezoresponse_force_microscopy" title="Piezoresponse force microscopy">Piezoresponse force</a></li> <li><a href="/wiki/Scanning_capacitance_microscopy" title="Scanning capacitance microscopy">Scanning capacitance</a></li> <li><a href="/wiki/Scanning_electrochemical_microscopy" title="Scanning electrochemical microscopy">Scanning electrochemical</a></li> <li><a href="/wiki/Scanning_gate_microscopy" title="Scanning gate microscopy">Scanning gate</a></li> <li><a href="/wiki/Scanning_Hall_probe_microscope" title="Scanning Hall probe microscope">Scanning Hall probe</a></li> <li><a href="/wiki/Scanning_ion-conductance_microscopy" title="Scanning ion-conductance microscopy">Scanning ion-conductance</a></li> <li><a href="/wiki/Scanning_joule_expansion_microscopy" title="Scanning joule expansion microscopy">Scanning joule expansion</a></li> <li><a href="/wiki/Scanning_Kelvin_probe" class="mw-redirect" title="Scanning Kelvin probe">Scanning Kelvin probe</a></li> <li><a href="/wiki/Scanning_quantum_dot_microscopy" title="Scanning quantum dot microscopy">Scanning quantum dot microscopy</a></li> <li><a href="/wiki/Scanning_SQUID_microscope" class="mw-redirect" title="Scanning SQUID microscope">Scanning SQUID microscope</a></li> <li><a href="/wiki/Scanning_SQUID_microscopy" title="Scanning SQUID microscopy">Scanning SQUID microscopy</a></li> <li><a href="/wiki/Scanning_thermal_microscopy" title="Scanning thermal microscopy">Scanning thermal</a></li> <li><a href="/wiki/Scanning_voltage_microscopy" title="Scanning voltage microscopy">Scanning voltage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scanning_probe_lithography" title="Scanning probe lithography">Scanning probe lithography</a></li> <li><a href="/wiki/Dip-pen_nanolithography" title="Dip-pen nanolithography">Dip-pen nanolithography</a></li> <li><a href="/wiki/Feature-oriented_scanning" title="Feature-oriented scanning">Feature-oriented scanning</a></li> <li><a href="/wiki/Millipede_memory" title="Millipede memory">Millipede memory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nanotechnology" title="Nanotechnology">Nanotechnology</a></li> <li><a href="/wiki/Microscope" title="Microscope">Microscope</a></li> <li><a href="/wiki/Microscopy" title="Microscopy">Microscopy</a></li> <li><a href="/wiki/Vibrational_analysis_with_scanning_probe_microscopy" title="Vibrational analysis with scanning probe microscopy">Vibrational analysis</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="IBM" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:IBM" title="Template:IBM"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:IBM" title="Template talk:IBM"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:IBM" title="Special:EditPage/Template:IBM"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="IBM" style="font-size:114%;margin:0 4em"><a href="/wiki/IBM" title="IBM">IBM</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_IBM" title="History of IBM">History</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/History_of_IBM" title="History of IBM">History</a></li> <li><a href="/wiki/List_of_mergers_and_acquisitions_by_IBM" title="List of mergers and acquisitions by IBM">Mergers and acquisitions</a> <ul><li><a href="/wiki/Acquisition_of_the_IBM_PC_business_by_Lenovo" title="Acquisition of the IBM PC business by Lenovo">PC business acquisition by Lenovo</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_IBM_products" title="List of IBM products">Products</a></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%">Hardware</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%">Current</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/IBM_mainframe" title="IBM mainframe">Mainframe</a> <ul><li><a href="/wiki/IBM_Z" title="IBM Z">IBM Z</a></li></ul></li> <li><a href="/wiki/IBM_Power_microprocessors" title="IBM Power microprocessors">Power microprocessors</a></li> <li><a href="/wiki/IBM_Power_Systems" title="IBM Power Systems">Power Systems</a></li> <li><a href="/wiki/IBM_storage" title="IBM storage">Storage</a> <ul><li><a href="/wiki/IBM_FlashSystem" title="IBM FlashSystem">FlashSystem</a></li> <li><a href="/wiki/IBM_DS8000_series" title="IBM DS8000 series">DS8000</a></li></ul></li> <li>Quantum <ul><li><a href="/wiki/IBM_Q_System_One" title="IBM Q System One">Q System One</a></li> <li><a href="/wiki/IBM_Q_System_Two" title="IBM Q System Two">Q System Two</a></li> <li><a href="/wiki/IBM_Eagle" title="IBM Eagle">Eagle</a></li> <li><a href="/wiki/IBM_Osprey" title="IBM Osprey">Osprey</a></li> <li><a href="/wiki/IBM_Heron" title="IBM Heron">Heron</a></li> <li><a href="/wiki/IBM_Condor" title="IBM Condor">Condor</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Former</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/IBM_Blue_Gene" title="IBM Blue Gene">Blue Gene</a></li> <li><a href="/wiki/Cell_(processor)" title="Cell (processor)">Cell microprocessors</a></li> <li><a href="/wiki/PowerPC" title="PowerPC">PowerPC</a></li> <li><a href="/wiki/Midrange_computer" title="Midrange computer">Midrange computer</a></li> <li><a href="/wiki/IBM_Personal_Computer" title="IBM Personal Computer">Personal Computer</a></li> <li><a href="/wiki/IBM_Selectric" title="IBM Selectric">Selectric</a></li> <li><a href="/wiki/ThinkPad" title="ThinkPad">ThinkPad</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Carbon_Design_System" title="Carbon Design System">Carbon Design System</a></li> <li><a href="/wiki/IBM_Cloud" title="IBM Cloud">Cloud</a> <ul><li><a href="/wiki/Cloudant" title="Cloudant">Cloudant</a></li></ul></li> <li><a href="/wiki/IBM_Cognos_Analytics" title="IBM Cognos Analytics">Cognos Analytics</a></li> <li><a href="/wiki/HCL_Connections" title="HCL Connections">Connections</a></li> <li><a href="/wiki/Criminal_Reduction_Utilising_Statistical_History" title="Criminal Reduction Utilising Statistical History">Criminal Reduction Utilising Statistical History</a></li> <li><a href="/wiki/Fortran" title="Fortran">Fortran</a></li> <li><a href="/wiki/ILOG" title="ILOG">ILOG</a></li> <li><a href="/wiki/IBM_Information_Management_Software" title="IBM Information Management Software">Information Management Software</a></li> <li><a href="/wiki/Lotus_Software" title="Lotus Software">Lotus Software</a></li> <li><a href="/wiki/History_of_IBM_mainframe_operating_systems" title="History of IBM mainframe operating systems">Mainframe operating systems</a></li> <li><a href="/wiki/IBM_Mashup_Center" title="IBM Mashup Center">Mashup Center</a></li> <li><a href="/wiki/IBM_Planning_Analytics" title="IBM Planning Analytics">Planning Analytics</a></li> <li><a href="/wiki/IBM_PureQuery" class="mw-redirect" title="IBM PureQuery">PureQuery</a></li> <li><a href="/wiki/IBM_Quantum_Platform" title="IBM Quantum Platform">Quantum Platform</a> <ul><li><a href="/wiki/Qiskit" title="Qiskit">Qiskit</a></li> <li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a></li></ul></li> <li><a href="/wiki/Rational_Software" title="Rational Software">Rational Software</a></li> <li><a href="/wiki/SPSS" title="SPSS">SPSS</a></li> <li><a href="/wiki/Tivoli_Software" title="Tivoli Software">Tivoli Software</a> <ul><li><a href="/wiki/Tivoli_Service_Automation_Manager" title="Tivoli Service Automation Manager">Service Automation Manager</a></li></ul></li> <li><a href="/wiki/IBM_Watson" title="IBM Watson">Watson</a></li> <li><a href="/wiki/IBM_Watsonx" title="IBM Watsonx">Watsonx</a> <ul><li><a href="/wiki/IBM_Granite" title="IBM Granite">Granite</a></li></ul></li> <li><a href="/wiki/IBM_WebSphere" title="IBM WebSphere">WebSphere</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Business<br />entities</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%">Current</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/Apptio" title="Apptio">Apptio</a></li> <li><a href="/wiki/IBM_Center_for_The_Business_of_Government" title="IBM Center for The Business of Government">Center for The Business of Government</a></li> <li><a href="/wiki/IBM_Consulting" title="IBM Consulting">Consulting</a> <ul><li><a href="/wiki/Promontory_Financial_Group" title="Promontory Financial Group">Promontory</a></li></ul></li> <li><a href="/wiki/Kenexa" title="Kenexa">Kenexa</a></li> <li><a href="/wiki/List_of_international_subsidiaries_of_IBM" title="List of international subsidiaries of IBM">International subsidiaries</a> <ul><li><a href="/wiki/IBM_India" title="IBM India">India</a></li></ul></li> <li><a href="/wiki/IBM_Press" title="IBM Press">Press</a></li> <li><a href="/wiki/Red_Hat" title="Red Hat">Red Hat</a></li> <li><a href="/wiki/IBM_Research" title="IBM Research">Research</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Former</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/AdStar" title="AdStar">AdStar</a></li> <li><a href="/wiki/AIM_alliance" title="AIM alliance">AIM alliance</a> <ul><li><a href="/wiki/Kaleida_Labs" title="Kaleida Labs">Kaleida Labs</a></li> <li><a href="/wiki/Taligent" title="Taligent">Taligent</a></li></ul></li> <li><a href="/wiki/Ambra_Computer_Corporation" title="Ambra Computer Corporation">Ambra Computer</a></li> <li><a href="/wiki/Cognos" title="Cognos">Cognos</a></li> <li><a href="/wiki/EduQuest" title="EduQuest">EduQuest</a></li> <li><a href="/wiki/Kyndryl" title="Kyndryl">Kyndryl</a></li> <li><a href="/wiki/Lexmark" title="Lexmark">Lexmark</a></li> <li><a href="/wiki/Merative" title="Merative">Merative</a></li> <li><a href="/wiki/IBM_Microelectronics" title="IBM Microelectronics">Microelectronics</a></li> <li><a href="/wiki/IBM_Product_Center" title="IBM Product Center">Product Center</a></li> <li><a href="/wiki/Science_Research_Associates" title="Science Research Associates">Science Research Associates</a></li> <li><a href="/wiki/Service_Bureau_Corporation" title="Service Bureau Corporation">Service Bureau</a></li> <li><a href="/wiki/The_Weather_Company" title="The Weather Company">The Weather Company</a> (<a href="/wiki/Weather_Underground_(weather_service)" title="Weather Underground (weather service)">Weather Underground</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Facilities</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Towers <ul><li><a href="/wiki/1250_Ren%C3%A9-L%C3%A9vesque" title="1250 René-Lévesque">1250 René-Lévesque</a>, Montreal, QC</li> <li><a href="/wiki/One_Atlantic_Center" title="One Atlantic Center">One Atlantic Center</a>, Atlanta, GA</li></ul></li> <li>Software Labs <ul><li><a href="/wiki/IBM_Rome_Software_Lab" title="IBM Rome Software Lab">Rome Software Lab</a></li> <li><a href="/wiki/IBM_Toronto_Software_Lab" title="IBM Toronto Software Lab">Toronto Software Lab</a></li></ul></li> <li>IBM Buildings <ul><li><a href="/wiki/330_North_Wabash" title="330 North Wabash">330 North Wabash</a>, Chicago, IL</li> <li><a href="/wiki/IBM_Building,_Honolulu" title="IBM Building, Honolulu">Honolulu</a></li> <li><a href="/wiki/1200_Fifth" title="1200 Fifth">Seattle</a></li></ul></li> <li>Facilities <ul><li><a href="/wiki/Thomas_J._Watson_Research_Center" title="Thomas J. Watson Research Center">Thomas J. Watson Research Center</a></li> <li><a href="/wiki/IBM_Hakozaki_Facility" title="IBM Hakozaki Facility">Hakozaki Facility</a></li> <li><a href="/wiki/IBM_Yamato_Facility" title="IBM Yamato Facility">Yamato Facility</a></li></ul></li> <li><a href="/wiki/Cambridge_Scientific_Center" title="Cambridge Scientific Center">Cambridge Scientific Center</a></li> <li><a href="/wiki/IBM_Hursley" title="IBM Hursley">IBM Hursley</a></li> <li><a href="/wiki/IBM_Canada_Head_Office_Building" title="IBM Canada Head Office Building">Canada Head Office Building</a></li> <li><a href="/wiki/IBM_Rochester" title="IBM Rochester">IBM Rochester</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Initiatives</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Academy of Technology</li> <li><a href="/wiki/IBM_Deep_Thunder" title="IBM Deep Thunder">Deep Thunder</a> <ul><li><a href="/wiki/Develothon" title="Develothon">Develothon</a></li></ul></li> <li><a href="/wiki/IBM_Fellow" title="IBM Fellow">Fellow</a></li> <li><a href="/wiki/IBM%27s_The_Great_Mind_Challenge" title="IBM's The Great Mind Challenge">The Great Mind Challenge</a></li> <li><a href="/wiki/Linux_Technology_Center" title="Linux Technology Center">Linux Technology Center</a></li> <li><a href="/wiki/IBM_SkillsBuild" title="IBM SkillsBuild">SkillsBuild</a></li> <li><a href="/wiki/Smarter_Planet" title="Smarter Planet">Smarter Planet</a></li> <li><a href="/wiki/IBM_Virtual_Universe_Community" title="IBM Virtual Universe Community">Virtual Universe Community</a></li> <li><a href="/wiki/World_Community_Grid" title="World Community Grid">World Community Grid</a></li> <li><a href="/wiki/IBM_Think_conference" title="IBM Think conference">Think conference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inventions</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/Automated_teller_machine" class="mw-redirect" title="Automated teller machine">Automated teller machine</a></li> <li><a href="/wiki/Cynefin_framework" title="Cynefin framework">Cynefin framework</a></li> <li><a href="/wiki/Dynamic_random-access_memory" title="Dynamic random-access memory">DRAM</a></li> <li><a href="/wiki/Keypunch" title="Keypunch">Electronic keypunch</a></li> <li><a href="/wiki/Floppy_disk" title="Floppy disk">Floppy disk</a></li> <li><a href="/wiki/Hard_disk_drive" title="Hard disk drive">Hard disk drive</a></li> <li><a href="/wiki/Magnetic_stripe_card" class="mw-redirect" title="Magnetic stripe card">Magnetic stripe card</a></li> <li><a href="/wiki/Relational_model" title="Relational model">Relational model</a></li> <li><a href="/wiki/Sabre_(travel_reservation_system)" title="Sabre (travel reservation system)">Sabre airline reservation system</a></li> <li><a class="mw-selflink selflink">Scanning tunneling microscope</a></li> <li><a href="/wiki/Swap_(finance)" title="Swap (finance)">Financial swaps</a></li> <li><a href="/wiki/Universal_Product_Code" title="Universal Product Code">Universal Product Code</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Terminology</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/Big_Blue" class="mw-redirect" title="Big Blue">Big Blue</a></li> <li><a href="/wiki/Commercial_Processing_Workload" title="Commercial Processing Workload">Commercial Processing Workload</a></li> <li><a href="/wiki/Customer_engineer" title="Customer engineer">Customer engineer</a></li> <li><a href="/wiki/Globally_integrated_enterprise" title="Globally integrated enterprise">Globally integrated enterprise</a></li> <li><a href="/wiki/Electronic_business" title="Electronic business">e-business</a></li> <li><a href="/wiki/Think_(slogan)" title="Think (slogan)">Think slogan</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_IBM_CEOs" title="List of IBM CEOs">CEOs</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/Thomas_J._Watson" title="Thomas J. Watson">Thomas J. Watson</a> (1914–1956)</li> <li><a href="/wiki/Thomas_J._Watson_Jr." title="Thomas J. Watson Jr.">Thomas Watson Jr.</a> (1956–1971)</li> <li><a href="/wiki/T._Vincent_Learson" title="T. Vincent Learson">T. Vincent Learson</a> (1971–1973)</li> <li><a href="/wiki/Frank_T._Cary" title="Frank T. Cary">Frank T. Cary</a> (1973–1981)</li> <li><a href="/wiki/John_R._Opel" title="John R. Opel">John R. Opel</a> (1981–1985)</li> <li><a href="/wiki/John_Fellows_Akers" title="John Fellows Akers">John Fellows Akers</a> (1985–1993)</li> <li><a href="/wiki/Lou_Gerstner" title="Lou Gerstner">Louis V. Gerstner Jr.</a> (1993–2002)</li> <li><a href="/wiki/Samuel_J._Palmisano" title="Samuel J. Palmisano">Samuel J. Palmisano</a> (2002–2011)</li> <li><a href="/wiki/Ginni_Rometty" title="Ginni Rometty">Ginni Rometty</a> (2012–2020)</li> <li><a href="/wiki/Arvind_Krishna" title="Arvind Krishna">Arvind Krishna</a> (since 2020)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Board of<br />directors</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/Thomas_Buberl" title="Thomas Buberl">Thomas Buberl</a></li> <li><a href="/wiki/David_Farr_(businessman)" title="David Farr (businessman)">David Farr</a></li> <li><a href="/wiki/Alex_Gorsky" title="Alex Gorsky">Alex Gorsky</a></li> <li><a href="/wiki/Michelle_Howard" title="Michelle Howard">Michelle J. Howard</a></li> <li><a href="/wiki/Arvind_Krishna" title="Arvind Krishna">Arvind Krishna</a></li> <li><a href="/wiki/Andrew_Liveris" title="Andrew Liveris">Andrew Liveris</a></li> <li><a href="/wiki/Martha_E._Pollack" class="mw-redirect" title="Martha E. Pollack">Martha E. Pollack</a></li> <li><a href="/wiki/Joseph_R._Swedish" title="Joseph R. Swedish">Joseph R. Swedish</a></li> <li><a href="/wiki/Peter_Voser" title="Peter Voser">Peter R. Voser</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/A_Boy_and_His_Atom" title="A Boy and His Atom">A Boy and His Atom</a></i></li> <li>Big Blue sports teams <ul><li><a href="/wiki/IBM_Big_Blue_(X-League)" title="IBM Big Blue (X-League)">American football</a></li> <li><a href="/wiki/IBM_Big_Blue_(rugby_union)" title="IBM Big Blue (rugby union)">Rugby union</a></li></ul></li> <li><a href="/wiki/Common_Public_License" title="Common Public License">Common Public License</a>/<a href="/wiki/IBM_Public_License" title="IBM Public License">IBM Public License</a></li> <li><a href="/wiki/Deep_Blue_(chess_computer)" title="Deep Blue (chess computer)">Deep Blue</a></li> <li><a href="/wiki/Deep_Thought_(chess_computer)" title="Deep Thought (chess computer)">Deep Thought</a></li> <li><a href="/wiki/Dynamic_infrastructure" title="Dynamic infrastructure">Dynamic infrastructure</a></li> <li><a href="/wiki/GlobalFoundries" title="GlobalFoundries">GlobalFoundries</a></li> <li><a href="/wiki/GUIDE_International" title="GUIDE International">GUIDE International</a></li> <li><i><a href="/wiki/IBM_and_the_Holocaust" title="IBM and the Holocaust">IBM and the Holocaust</a></i></li> <li><a href="/wiki/IBM_international_chess_tournament" title="IBM international chess tournament">International chess tournament</a></li> <li><a href="/wiki/Lucifer_(cipher)" title="Lucifer (cipher)">Lucifer cipher</a></li> <li><i><a href="/wiki/Mathematica:_A_World_of_Numbers..._and_Beyond" title="Mathematica: A World of Numbers... and Beyond">Mathematica</a></i></li> <li><a href="/wiki/IBM_Plex" title="IBM Plex">IBM Plex</a></li> <li><a href="/wiki/SHARE_(computing)" title="SHARE (computing)">SHARE computing</a></li> <li><a href="/wiki/ScicomP" title="ScicomP">ScicomP</a></li> <li><a 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