{"id":68,"date":"2010-03-30T12:37:08","date_gmt":"2010-03-30T12:37:08","guid":{"rendered":"http:\/\/lptms.u-psud.fr\/christophe_texier\/?page_id=68"},"modified":"2023-09-04T14:33:04","modified_gmt":"2023-09-04T14:33:04","slug":"effect-of-electron-electron-interaction","status":"publish","type":"page","link":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/quantum-transport-in-networks-of-metallic-wires\/effect-of-electron-electron-interaction\/","title":{"rendered":"Electronic interactions"},"content":{"rendered":"<h2><span style=\"color: #0000ff;\">A. Decoherence by electronic interactions<br \/>\n<\/span><\/h2>\n<p>From the experimental point of view, the great interest in weak localization is to provide a powerful tool to probe decoherence mechanisms in weakly disordered metals or semiconductors. In the works listed above, the dephasing is described by introducing a phenomenological parameter in the cooperon. This procedure is correct for some dephasing mechanisms spin-flip, spin-orbit scattering, effect of penetration of a magnetic field in the wires) however it does not describe correctly the effect of electron-electron interaction, which is the dominant mechanism at low temperature (below 1K).<\/p>\n<p>A description of electron-electron interaction was provided in a pioneering paper by Al&rsquo;tshuler, Aronov&amp;Khmel&rsquo;nitskii in 1982: these authors modeled the dephasing in a one-particle picture through the interaction of the electron with a classically fluctuating electric field whose fluctuations are given by the fluctuation-dissipation theorem. They showed that in a wire, this leads to a phase coherence length behaving like L<sub>\u03c6<\/sub>\u221d T <sup>-1\/3<\/sup>. It is only very recently that Ludwig&amp;Mirlin (PRB, 2004) noticed that the effect of the nontrivial geometry of a ring and the effect of electron-electron interaction combine in a nontrivial way and lead to unexpected behaviour of the magnetoconductance harmonics. We re-examine this question in the following paper, correct Ludwig&amp;Mirlin&rsquo;s magnetoconductance&rsquo;s prefactor, and predict a new behaviour in a regime not studied by Ludwig&amp;Mirlin.<\/p>\n<ul>\n<li>Christophe Texier and Gilles Montambaux,<br \/>\n<strong>Dephasing due to electron-electron interaction in a diffusive ring<\/strong><br \/>\n<a href=\"http:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.72.115327\">Phys. Rev. B <strong>72<\/strong>, 115327 (2005).<\/a><br \/>\n<a href=\"http:\/\/xxx.lanl.gov\/abs\/cond-mat\/0505199\">cond-mat\/0505199<\/a>.<\/li>\n<li>Erratum <a href=\"http:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.74.209902\">Phys. Rev. B <strong>74<\/strong>, 209902(E) (2006).<\/a><\/li>\n<\/ul>\n<ul>\n<li>Christophe Texier and Gilles Montambaux,<br \/>\n<strong>Comment on \u00ab\u00a0Interaction-induced dephasing of Aharonov-Bohm oscillations\u00a0\u00bb [Ludwig and Mirlin, Phys. Rev.~B <strong>69<\/strong>, 193306 (2004).]<\/strong><br \/>\n<a href=\"\/christophe_texier\/files\/2010\/03\/comment.pdf\">pdf file<\/a>.<br \/>\nIn this <em>unpublished comment<\/em>, we show that Ludwig &amp; Mirlin did a mistake when evaluating the preexponential factor of their path integral within semiclassical method. This affects the temperature dependence of the preexponential factor of AB\/AAS harmonics of conductance.<\/li>\n<li>Christophe Texier,<br \/>\n<strong>Effect of connecting wires on the decoherence due to electron-electron interaction in a metallic ring<\/strong><br \/>\n<a href=\"http:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.76.153312\">Phys. Rev. B <strong>76<\/strong>, 153312 (2007).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0707.2916\">cond-mat arXiv:0707.2916<\/a>.<br \/>\nIn this short paper I re-examine in detail the effect of the connecting arms in the problem of decoherence due to electron-electron interaction in a metallic ring. The two results of Ludwig &amp; Mirlin (2004), corrected according to our comment, and [us, PRB72, 115327 (2005)] are rediscussed. The conclusion is that connecting wires do not affect decoherence in realistic situations.<\/li>\n<li>Christophe Texier and Gilles Montambaux,<br \/>\n<strong>Magnetoconductance oscillations in metallic rings and decoherence due to electron-electron interaction<\/strong><br \/>\nproceedings of the 6th rencontres du Vietnam \u00ab\u00a0Nanophysics: from fundamentals to applications\u00a0\u00bb, Hanoi, 6-12 August 2006.<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0704.0742\">cond-mat arXiv:0704.0742<\/a>.<br \/>\nIn this article, decoherence due to electronic interactions is studied in chain of rings. When rings are separated by long (&gt;&gt;L<sub>\u03c6<\/sub>) arms of short arms (&lt;&lt; L<sub>\u03c6<\/sub>).<\/li>\n<\/ul>\n<ul>\n<li>Christophe Texier, Pierre Delplace and Gilles Montambaux<br \/>\n<strong>Quantum oscillations and decoherence due to electron-electron interaction in metallic networks and hollow cylinders<\/strong><br \/>\n<a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v80\/i20\/e205413\">Phys. Rev. B <strong>80<\/strong>, 205413 (2009).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0907.3133\">cond-mat arXiv:0907.3133<\/a><\/li>\n<li>Maximilian Treiber, Christophe Texier, Oleg M. Yevtushenko, Jan von Delft and Igor V. Lerner<br \/>\n<strong>Thermal noise and dephasing due to electron interactions in non-trivial geometries <\/strong><br \/>\n<a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v84\/i5\/e054204\">Phys. Rev. B <strong>84<\/strong>, 054204 (2011).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1105.0554\">cond-mat arXiv:1105.0554<\/a><br \/>\nWe propose a real-space derivation of the fluctuation-dissipation theorem in metals. The result is used in order to obtain a <strong>decoherence rate formula<\/strong> describing <strong>electronic interactions<\/strong> in complex geometries (decoherence by e-e interaction must be accounted for through a <em>functional<\/em> of the electronic trajectories). This result generalizes the result of\u00a0 (Marquardt, von Delft <em>et al<\/em>, <a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v76\/i19\/e195331\">Phys. Rev. B 76, 195331 (2007)<\/a> ; von Delft, Marquardt <em>et al<\/em>, <a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v76\/i19\/e195332\">Phys. Rev. B 76, 195332 (2007)<\/a>\u00a0; von Delft, <a href=\"http:\/\/www.worldscinet.com\/ijmpb\/22\/2207\/S0217979208039083.html\">Int. J. Mod. Phys. B 22, 727 (2008)<\/a>\u00a0; Treiber <em>et al<\/em>, <a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v80\/i20\/e201305\">Phys. Rev. B 80, 201305(R) (2009)<\/a>) providing a description of decoherence accounting for classical and quantum noise regimes in simple geometries, and our results (<a href=\"http:\/\/scitation.aip.org\/getabs\/servlet\/GetabsServlet?prog=normal&amp;id=PRBMDO000072000011115327000001&amp;idtype=cvips&amp;gifs=yes\">Phys. Rev. B <strong>72<\/strong>, 115327 (2005)<\/a> ; <a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v80\/i20\/e205413\">Phys. Rev. B <strong>80<\/strong>, 205413 (2009)<\/a>) accounting for classical Johnson-Nyquist noise in arbitrary geometries.<\/li>\n<\/ul>\n<h2><span style=\"color: #008000;\">Experiments<\/span><\/h2>\n<p>The temperature dependences of the various length scales have been <strong><span style=\"color: #008000;\">observed in experiments<\/span><\/strong> in the team of <a href=\"http:\/\/chercheurs.lps.u-psud.fr\/Spm\/spip.php?rubrique1\">H\u00e9l\u00e8ne Bouchiat, Sophie Gu\u00e9ron and Meydi Ferrier<\/a> (<span style=\"color: #008000;\"><strong>weak localization of large square networks<\/strong><\/span>):<\/p>\n<ul>\n<li>Meydi Ferrier, Alistair C.H. Rowe, Sophie Gu\u00e9ron, H\u00e9l\u00e8ne Bouchiat, Christophe Texier and Gilles Montambaux,<br \/>\n<strong>Geometrical dependence of decoherence by electronic interactions in a GaAs\/GaAlAs square network<\/strong><br \/>\n<a href=\"http:\/\/journals.aps.org\/prl\/abstract\/10.1103\/PhysRevLett.100.146802\">Phys. Rev. Lett. <strong>100<\/strong>, 146802 (2008).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0707.3890\">cond-mat arXiv:0707.3890<br \/>\n<\/a>Experimental evidence of result of Ludwig&amp;Mirlin (2004) and us (PRB72, 2005).<\/li>\n<\/ul>\n<p>And also in the <strong><span style=\"color: #008000;\">experimental analysis of Aharonov-Bohm oscillations of a single mesoscopic ring<\/span><\/strong> :<\/p>\n<ul>\n<li>Thibaut Capron, Christophe Texier, Gilles Montambaux, Dominique Mailly, Andreas Wieck and Laurent Saminadayar,<br \/>\n<strong>Ergodic <em>vs<\/em> diffusive decoherence in mesoscopic devices<\/strong><br \/>\n<a href=\"http:\/\/prb.aps.org\/abstract\/PRB\/v87\/i4\/e041307\">Phys. Rev. B <strong>87<\/strong>, 041307(R) (2013).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1206.0757\">cond-mat arXiv:1206.0757<br \/>\n<\/a>Experimental evidence of result of Ludwig&amp;Mirlin (2004) and us (PRB72, 2005).<\/li>\n<\/ul>\n<h2><span style=\"color: #0000ff;\">B. Other interaction effects<\/span><\/h2>\n<h3>Al&rsquo;tshuler-Aronov correction in a large square network<\/h3>\n<p>All previous works on quantum transport in networks have discussed the weak localization correction (WK) or the universal conductance fluctuations (UCF). Both are phase coherent quantities. Another quantum correction is the Al&rsquo;tshuler-Aronov (AA) correction, which is due to the electron-electron interaction. Contrary WL and UCF, this quantum correction is not sensitive to phase coherence and does not feel the magnetic field, what allows to distinguish the two quantities experimentally. AA correction is computed in a large square network. Interpolation between 1d and 2d results is obtained. Comparison to some experimental results is provided.<\/p>\n<ul>\n<li>Christophe Texier and Gilles Montambaux,<br \/>\n<strong>Al&rsquo;tshuler-Aronov correction to the conductivity of a large metallic square network<\/strong><br \/>\n<a href=\"http:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.76.094202\">Phys. Rev. B <strong>76<\/strong>, 094202 (2007).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/0704.0741\">cond-mat arXiv:0704.0741<\/a>.<\/li>\n<\/ul>\n<p>The Altshuler-Aronov correction of large planar networks has been <strong><span style=\"color: #008000;\">studied experimentally<\/span><\/strong> in the team of <a href=\"http:\/\/neel.cnrs.fr\/spip.php?rubrique50\">Christopher B\u00e4uerle &amp; Laurent Saminadayar<\/a> (Institut N\u00e9el, Grenoble), during Fran\u00e7ois Mallet&rsquo;s PhD.<\/p>\n<h3>Non-linear transport in diffusive metal and magnetic field asymmetry<\/h3>\n<p>B\u00fcttiker &amp; Sanch\u00e9z and Spivak &amp; Zyuzin discovered in 2004 an interesting effect of electronic interaction, which induces an asymmetry of the nonlinear conductance under magnetic field reversal. We have studied the effect in diffusive sample<\/p>\n<ul>\n<li>Christophe Texier and Johannes Mitscherling<br \/>\n<strong>Non-linear conductance in mesoscopic weakly disordered wires &#8212; Interaction and magnetic field asymmetry<\/strong>,<br \/>\n<a href=\"https:\/\/journals.aps.org\/prb\/abstract\/10.1103\/PhysRevB.97.075306\">Phys. Rev. B <strong>97<\/strong>, 075306 (2018)<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1510.02214\">cond-mat arXiv:1510.02214<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>A. Decoherence by electronic interactions From the experimental point of view, the great interest in weak localization is to provide a powerful tool to probe decoherence mechanisms in weakly disordered metals or semiconductors. In the works listed above, the dephasing &hellip; <a href=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/quantum-transport-in-networks-of-metallic-wires\/effect-of-electron-electron-interaction\/\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":378,"menu_order":30,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-68","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/68","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/comments?post=68"}],"version-history":[{"count":40,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/68\/revisions"}],"predecessor-version":[{"id":2721,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/68\/revisions\/2721"}],"up":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/378"}],"wp:attachment":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/media?parent=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}