{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T03:00:22Z","timestamp":1778036422824,"version":"3.51.4"},"reference-count":16,"publisher":"IOP Publishing","issue":"22","license":[{"start":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T00:00:00Z","timestamp":1747008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T00:00:00Z","timestamp":1747008000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["J. Phys.: Condens. Matter"],"published-print":{"date-parts":[[2025,6,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We calculate the non-linear charge currents in a two-dimensional electron system with Rashba (<jats:italic>\u03b1<\/jats:italic>) and Dresselhaus (<jats:italic>\u03b2<\/jats:italic>) linear spin\u2013orbit interactions (SOIs) in the presence of in-plane electric <jats:bold>E<\/jats:bold> and magnetic <jats:bold>B<\/jats:bold> fields. Working in a rotated system of coordinates that introduces <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>\u03b1<\/mml:mi>\n                           <mml:mo>\u00b1<\/mml:mo>\n                           <mml:mi>\u03b2<\/mml:mi>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula> as effective couplings on perpendicular directions, we show that the currents, quadratic in the electric field, exist for all values of the chemical potential <jats:italic>\u00b5<\/jats:italic>, above or below the band crossing energy <jats:italic>E<\/jats:italic>\n                  <jats:sub>0<\/jats:sub>. Our formalism uses a second order correction to the particle distribution function <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>\u03b4<\/mml:mi>\n                           <mml:msup>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mrow>\n                                 <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                 <mml:mn>2<\/mml:mn>\n                                 <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                           <\/mml:msup>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula> derived in a semi-classical approximation that takes into account the local change in the single-particle energy under the action of the electric field. In a quantum well, where <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>\u03bc<\/mml:mi>\n                           <mml:mo>&gt;<\/mml:mo>\n                           <mml:msub>\n                              <mml:mi>E<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                           <\/mml:msub>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula>, for a spin\u2013orbit energy larger than the Zeeman splitting, the non-linear currents that flow perpendicular on the direction of the magnetic field are found to be proportional with <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>\u03b1<\/mml:mi>\n                           <mml:mo>\u00b1<\/mml:mo>\n                           <mml:mi>\u03b2<\/mml:mi>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula> when <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">E<\/mml:mi>\n                           <\/mml:mrow>\n                           <mml:mo>\u22a5<\/mml:mo>\n                           <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">B<\/mml:mi>\n                           <\/mml:mrow>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula> and with <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mo stretchy=\"false\">(<\/mml:mo>\n                           <mml:mi>\u03b1<\/mml:mi>\n                           <mml:mo>\u00b1<\/mml:mo>\n                           <mml:mi>\u03b2<\/mml:mi>\n                           <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                           <\/mml:msup>\n                           <mml:mrow>\n                              <mml:mo>\/<\/mml:mo>\n                           <\/mml:mrow>\n                           <mml:mo stretchy=\"false\">(<\/mml:mo>\n                           <mml:mi>\u03b1<\/mml:mi>\n                           <mml:mo>\u2213<\/mml:mo>\n                           <mml:mi>\u03b2<\/mml:mi>\n                           <mml:mo stretchy=\"false\">)<\/mml:mo>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula> when <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">E<\/mml:mi>\n                           <\/mml:mrow>\n                           <mml:mo>\u2225<\/mml:mo>\n                           <mml:mrow>\n                              <mml:mi mathvariant=\"bold\">B<\/mml:mi>\n                           <\/mml:mrow>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula>, a result independent of the presence of any additional SOIs, such as the cubic Dresselhaus or warping.<\/jats:p>","DOI":"10.1088\/1361-648x\/add3e9","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T23:02:07Z","timestamp":1746226927000},"page":"225301","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Non-linear charge currents and the planar Hall effect in quantum wells with Rashba and Dresselhaus spin\u2013orbit interactions"],"prefix":"10.1088","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-2880-7899","authenticated-orcid":true,"given":"Abhishek","family":"Khanal","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8358-8553","authenticated-orcid":true,"given":"D 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Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2024-12-09","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2025-05-01","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2025-05-12","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}