O UTLINE Transport through a mesoscopic Luttinger liquid with Rashba spin-orbit coupling Nicolas Pauget 1 Mikhail Pletyukhov 1 Vladimir Gritsev 2 1 Institut f¨ ur Theoretische Festk¨ orperphysik Universit¨ at Karlsruhe 2 Condensed Matter Theory Group Harvard University 2nd February 2006 P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
O UTLINE Outline 1 Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling 2 Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique 3 Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
O UTLINE Outline 1 Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling 2 Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique 3 Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
O UTLINE Outline 1 Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling 2 Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique 3 Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Outline 1 Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling 2 Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique 3 Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Conductance of a quasi 1-dimensional ring External parameters Gate voltage ( V G ), V B, � � chemical potential µ , E Electric field � E (Rashba Φ SO-coupling), Charging energy E C . V G , C G Magnetic field ( � B ), Magnetic flux Φ , Zeeman effect (negligible). P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Conductance of a quasi 1-dimensional ring External parameters Gate voltage ( V G ), V B, � � chemical potential µ , E Electric field � E (Rashba Φ SO-coupling), Charging energy E C . V G , C G Magnetic field ( � B ), Magnetic flux Φ , Zeeman effect (negligible). P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Experiments J. Nitta et al., Physica E 12 Contour plot: maxima (dark), minima (white). (2002) 753-757 Measure of the magnetoresistance, InGaAs/InAlAs heterostructure. P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Experiments M. K¨ onig et al., arXiv:cond-mat/0508396 Contour plot: Measure of the maxima (yellow), minima (blue). magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Experiments Contour plot: M. K¨ onig et al., arXiv:cond-mat/0508396 maxima (yellow), minima (blue). Measure of the magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Experiments M. K¨ onig et al., Contour plot: arXiv:cond-mat/0508396 maxima (yellow), minima (blue). Measure of the magnetoconductance, HgTe/HgCdTe heterostructure (strong Rashba coupling). P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Outline 1 Tunneling conductance through a mesoscopic ring Experimental motivations Flux and gate voltage dependencies of conductance peaks Effect of Spin-Orbit coupling 2 Linear response theory for an interacting ring Kubo formula Keldysh expansion Bosonisation technique 3 Results: Position of the conductance peaks vs. external parameters Effect of interactions Introduction of Rashba Spin-Orbit coupling Introduction of charging energy P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks E.A. Jagla et al., PRL 70, 5, 639 (1993) Y. Aharonov et al., PR 115, 3, 485 (1959) G/G 0 Effect of the magnetic field 1 Interferences: Aharonov-Bohm effect. 0 . 5 Phase accumulated ( � = 1): 0 − e Z − 2 − 1 1 2 Φ / Φ 0 A · dx c ± Flux quantum Φ 0 = 2 π c e , + B ( � � A ) Kinetic Hamiltonian: H c = ( k − k Φ ) 2 , k Φ = Φ 2 π 2 m ∗ L Φ 0 L : perimeter of the ring. P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks E.A. Jagla et al., PRL 70, 5, 639 (1993) Y. Aharonov et al., PR 115, 3, 485 (1959) E Effect of the magnetic field Interferences: Aharonov-Bohm k F E F effect. k Phase accumulated ( � = 1): − e Z A · dx c ± Flux quantum Φ 0 = 2 π c e , + B ( � � Kinetic Hamiltonian: A ) H c = ( k − k Φ ) 2 , k Φ = Φ 2 π 2 m ∗ L Φ 0 L : perimeter of the ring. P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks E.A. Jagla et al., PRL 70, 5, 639 (1993) E Y. Aharonov et al., PR 115, 3, 485 (1959) Effect of the magnetic field Interferences: Aharonov-Bohm k F E F effect. k Phase accumulated ( � = 1): − e Z A · dx c ± k Φ Flux quantum Φ 0 = 2 π c e , Kinetic Hamiltonian: + B ( � � A ) H c = ( k − k Φ ) 2 , k Φ = Φ 2 π 2 m ∗ L Φ 0 L : perimeter of the ring. P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks (2) G/G 0 1 0 . 5 Effect of the magnetic field Spectral flow, 0 − 2 − 1 1 2 z µ 2 π k µ = ( µ − v F k F ) L L z µ = + 1 2 , 2 π Dispersion relation shifted vertically. 2 πv F L P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks (2) E k F E F Effect of the magnetic field k Spectral flow, 2 π k µ = ( µ − v F k F ) L L z µ = + 1 2 , 2 π Dispersion relation shifted vertically. 2 πv F L P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
T UNNELING CONDUCTANCE E XPERIMENTAL MOTIVATIONS L INEAR RESPONSE C ONDUCTANCE PEAKS A NALYTICAL RESULTS E FFECT OF S PIN -O RBIT COUPLING S UMMARY Modulation of the conductance peaks (2) E k F E F k Effect of the magnetic field Spectral flow, ∆ µ 2 π k µ = ( µ − v F k F ) L L z µ = + 1 2 , k µ 2 π Dispersion relation shifted vertically. 2 πv F L P AUGET , P LETYUKHOV , G RITSEV T RANSPORT THROUGH A MESOSCOPIC L UTTINGER LIQUID
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