The probe technique far from equilibrium: Magnetic field symmetries of nonlinear transport

Regular Article

Abstract

The probe technique is a simple mean to incorporate elastic and inelastic processes into quantum transport problems. Using numerical simulations, we demonstrate that this tool can be employed beyond the analytically tractable linear response regime, providing a stable solution for the probe parameters: temperature and chemical potential. Adopting four probes: dephasing, voltage, temperature, and voltage-temperature, mimicking different elastic and inelastic effects, we provide a systematic analysis of magnetic field and gate voltage symmetries of charge current and heat current in Aharonov-Bohm interferometers, potentially far from equilibrium. Considering electron current, we prove that in the linear response regime inelastic scattering processes do not break the Onsager symmetry. Beyond linear response, even (odd) conductance terms obey an odd (even) symmetry with the threading magnetic flux, as long as the system acquires a spatial inversion symmetry. When spatial asymmetry is introduced, particle-hole symmetry assures that nonlinear conductance terms maintain certain symmetries with respect to magnetic field and gate voltage. These analytic results are supported by numerical simulations. Analogous results are obtained for the electron heat current. Finally, we demonstrate that a double-dot Aharonov-Bohm interferometer can act as a charge rectifier when two conditions are met simultaneously: (i) many-body effects are included, here in the form of inelastic scattering; and (ii) time reversal symmetry is broken.

Keywords

Mesoscopic and Nanoscale Systems 

References

  1. 1.
    S. Andergassen, V. Meden, H. Schoeller, J. Splettstoesser, M.R. Wegewijs, Nanotechnology 21, 272001 (2010), and references thereinADSCrossRefGoogle Scholar
  2. 2.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland Pub., Amsterdam, 1981)Google Scholar
  3. 3.
    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)Google Scholar
  4. 4.
    M. Büttiker, Phys. Rev. B 32, 1846 (1985) ADSCrossRefGoogle Scholar
  5. 5.
    M. Büttiker, Phys. Rev. B 33, 3020 (1986) ADSCrossRefGoogle Scholar
  6. 6.
    M. Büttiker, IBM J. Res. Dev. 32, 63 (1988)CrossRefGoogle Scholar
  7. 7.
    J.L. D’Amato, H.M. Pastawski, Phys. Rev. B 41, 7411 (1990) ADSCrossRefGoogle Scholar
  8. 8.
    T. Ando, Surf. Sci. 361-362, 270 (1996) ADSCrossRefGoogle Scholar
  9. 9.
    P.A. Jacquet, J. Stat. Phys. 134, 709 (2009) ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    K. Saito, G. Benenti, G. Casati, T. Prosen, Phys. Rev. B 84, 201306 (2011) ADSCrossRefGoogle Scholar
  11. 11.
    V. Balachandran, G. Benenti, G. Casati, Phys. Rev. B 87, 165419 (2013) ADSCrossRefGoogle Scholar
  12. 12.
    K. Brandner, K. Saito, U. Seifert, Phys. Rev. Lett. 110, 070603 (2013) ADSCrossRefGoogle Scholar
  13. 13.
    J.P. Bergfield, S.M. Story, R.C. Stafford, C.A. Stafford, ACS Nano 7, 4429 (2013)CrossRefGoogle Scholar
  14. 14.
    Y. Ming, Z. Wang, Z. Ding, H. Li, New J. Phys. 12, 103041 (2010) ADSCrossRefGoogle Scholar
  15. 15.
    M. Bandyopadhyay, D. Segal, Phys. Rev. E 84, 011151 (2011) ADSCrossRefGoogle Scholar
  16. 16.
    E. Pereira, H.C.F. Lemos, R.R. Avila, Phys. Rev. E 84, 061135 (2011) ADSCrossRefGoogle Scholar
  17. 17.
    Ph.A. Jacquet, C.-A. Pillet, Phys. Rev. B 85, 125120 (2012) ADSCrossRefGoogle Scholar
  18. 18.
    K. Saaskilahti, J. Oksanen, J. Tulkki, Phys. Rev. E 88, 012128 (2013) ADSCrossRefGoogle Scholar
  19. 19.
    P. Roulleau, F. Portier, P. Roche, A. Cavanna, G. Faini, U. Gennser, D. Mailly, Phys. Rev. Lett. 102, 236802 (2009) ADSCrossRefGoogle Scholar
  20. 20.
    M.J.M. de Jong, C.W.J. Beenakker, Physica A 230, 219 (1996) ADSCrossRefGoogle Scholar
  21. 21.
    S.A. van Langen, M. Büttiker, Phys. Rev. B 56, R1680 (1997) ADSCrossRefGoogle Scholar
  22. 22.
    H.-L. Engquist, P.W. Anderson, Phys. Rev. B 24, 1151 (1981) ADSCrossRefGoogle Scholar
  23. 23.
    D. Roy, A. Dhar, Phys. Rev. B 75, 195110 (2007) ADSCrossRefGoogle Scholar
  24. 24.
    F. Bonetto, J. Lebowitz, L. Rey-Bellet, Mathematical Physics 2000 (World Scientific, Singapore, 2000), pp. 128150Google Scholar
  25. 25.
    F. Bonetto, J.L. Lebowitz, J. Lukkarinen, J. Stat. Phys. 116, 783 (2004) ADSCrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    A. Dhar, D. Roy, J. Stat. Phys. 125, 801 (2006) ADSCrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    D. Roy, Phys. Rev. E 77, 062102 (2008) ADSCrossRefGoogle Scholar
  28. 28.
    D. Segal, Phys. Rev. E 79, 012103 (2009) ADSCrossRefGoogle Scholar
  29. 29.
    S. Pilgram, P. Samuelsson, H. Förster, M. Büttiker, Phys. Rev. Lett. 97, 066801 (2006) ADSCrossRefGoogle Scholar
  30. 30.
    L. Onsager, Phys. Rev. 37, 405 (1931)ADSCrossRefGoogle Scholar
  31. 31.
    L. Onsager, Phys. Rev. 38, 2265 (1931) ADSCrossRefMATHGoogle Scholar
  32. 32.
    H.B.G. Casimir, Rev. Mod. Phys. 17, 343 (1945)ADSCrossRefGoogle Scholar
  33. 33.
    Y. Imry, Introduction to Mesoscopic Physics, 2nd edn. (Oxford University Press, Oxford, 2002)Google Scholar
  34. 34.
    A. Yacoby, M. Heiblum, D. Mahalu, H. Shtrikman, Phys. Rev. Lett. 74, 4047 (1995) ADSCrossRefGoogle Scholar
  35. 35.
    H. Linke, W.D. Sheng, A. Svensson, A. Löfgren, L. Christensson, H.Q. Xu, P. Omling, P.E. Lindelof, Phys. Rev. B. 61, 15914 (2000) ADSCrossRefGoogle Scholar
  36. 36.
    A. Löfgren, C.A. Marlow, I. Shorubalko, R.P. Taylor, P. Omling, L. Samuelson, H. Linke, Phys. Rev. Lett. 92, 046803 (2004) ADSCrossRefGoogle Scholar
  37. 37.
    C.A. Marlow, R.P. Taylor, M. Fairbanks, I. Shorubalko, H. Linke, Phys. Rev. Lett. 96, 116801 (2006) ADSCrossRefGoogle Scholar
  38. 38.
    J. Wei, M. Shimogawa, Z. Wang, I. Radu, R. Dormaier, D.H. Cobden, Phys. Rev. Lett. 95, 256601 (2005) ADSCrossRefGoogle Scholar
  39. 39.
    R. Leturcq, D. Sanchez, G. Götz, T. Ihn, K. Ensslin, D.C. Driscoll, A.C. Gossard, Phys. Rev. Lett. 96, 126801 (2006) ADSCrossRefGoogle Scholar
  40. 40.
    R. Leturcq, R. Bianchetti, G. Götz, T. Ihn, K. Ensslin, D.C. Driscoll, A.C. Gossard, Physica E 35, 327 (2006)ADSCrossRefGoogle Scholar
  41. 41.
    D.M. Zumbhl, C.M. Marcus, M.P. Hanson, A.C. Gossard, Phys. Rev. Lett. 96, 206802 (2006) ADSCrossRefGoogle Scholar
  42. 42.
    L. Angers, E. Zakka-Bajjani, R. Deblock, S. Gueron, H. Bouchiat, A. Cavanna, U. Gennser, M. Polianski, Phys. Rev. B 75, 115309 (2007) ADSCrossRefGoogle Scholar
  43. 43.
    M. Sigrist, T. Ihn, K. Ensslin, M. Reinwald, W. Wegscheider, Phys. Rev. Lett. 98, 036805 (2007) ADSCrossRefGoogle Scholar
  44. 44.
    T. Ihn, M. Sigrist, K. Ensslin, W. Wegscheider, M. Reinwald, New J. Phys. 9, 111 (2007)ADSCrossRefGoogle Scholar
  45. 45.
    G.M. Gusev, Z.D. Kvon, E.B. Olshanetsky, A.Y. Plotnikov, Europhys. Lett. 88, 47007 (2009) ADSCrossRefGoogle Scholar
  46. 46.
    F.G.G. Hernandez, G.M. Gusev, Z.D. Kvon, J.C. Portal, Phys. Rev. B 84, 075332 (2011) ADSCrossRefGoogle Scholar
  47. 47.
    D. Sanchez, M. Büttiker, Phys. Rev. Lett. 93, 106802 (2004) ADSCrossRefGoogle Scholar
  48. 48.
    M. Büttiker, D. Sanchez, Int. J. Quantum Chem. 105, 906 (2005) ADSCrossRefGoogle Scholar
  49. 49.
    B. Spivak, A. Zyuzin, Phys. Rev. Lett. 93, 226801 (2004) ADSCrossRefGoogle Scholar
  50. 50.
    A.R. Hernandez, C.H. Lewenkopf, Phys. Rev. Lett. 103, 166801 (2009) ADSCrossRefGoogle Scholar
  51. 51.
    J.S. Lim, D. Sanchez, R. Lopez, Phys. Rev. B 81, 155323 (2010). Note that in this work the authors used a different convention for the current expansion with voltage, I = G0Δμ + G1Δ μ2 + ... ADSCrossRefGoogle Scholar
  52. 52.
    T. Kubo, Y. Ichigo, Y. Tokura, Phys. Rev. B 83, 235310 (2011) ADSCrossRefGoogle Scholar
  53. 53.
    V. Puller, Y. Meir, M. Sigrist, K. Ensslin, T. Ihn, Phys. Rev. B 80, 035416 (2009) ADSCrossRefGoogle Scholar
  54. 54.
    M. Terraneo, M. Peyrard, G. Casati, Phys. Rev. Lett. 88, 094302 (2002) ADSCrossRefGoogle Scholar
  55. 55.
    D. Segal, A. Nitzan, Phys. Rev. Lett. 94, 034301 (2005) ADSCrossRefGoogle Scholar
  56. 56.
    C.W. Chang et al., Science 314, 1121 (2006) ADSCrossRefGoogle Scholar
  57. 57.
    X.-F. Li, X. Ni, L. Feng, M.-H. Lu, C. He, Y.-F. Chen, Phys. Rev. Lett. 106, 084301 (2011) ADSCrossRefGoogle Scholar
  58. 58.
    L. Feng, Y.-L. Xu, W.S. Fegadolli, M.-H. Lu, J.E.B. Oliveira, V.R. Almeida, Y.-F. Chen, A. Scherer, Nat. Mater. 12, 108 (2013)ADSCrossRefGoogle Scholar
  59. 59.
    S. Bedkihal, M. Bandyopadhyay, D. Segal, Phys. Rev. B 88, 155407 (2013) ADSCrossRefGoogle Scholar
  60. 60.
    R. Landauer, IBM J. Res. Dev. 1, 223 (1957)CrossRefMathSciNetGoogle Scholar
  61. 61.
    M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986) ADSCrossRefGoogle Scholar
  62. 62.
    S. Datta, Electric transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995)Google Scholar
  63. 63.
    W.H. Press, B.P. Flannery, S.A. Teukosky, W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, 1992)Google Scholar
  64. 64.
    Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992) ADSCrossRefGoogle Scholar
  65. 65.
    J. Fransson, Non-equilibrium nano-physics, a many body approach, Lecture Notes in Physics (Springer, 2010), Vol. 809 Google Scholar
  66. 66.
    S. Bedkihal, D. Segal, Phys. Rev. B 85, 155324 (2012) ADSCrossRefGoogle Scholar
  67. 67.
    M.W.-Y. Tu, W.-M. Zhang, J. Jin, O. Entin-Wohlman, A. Aharony, Phys. Rev. B 86, 115453 (2012) ADSCrossRefGoogle Scholar
  68. 68.
    S. Bedkihal, M. Bandyopadhyay, D. Segal, Phys. Rev. B 87, 045418 (2013) ADSCrossRefGoogle Scholar
  69. 69.
    B. Kubala, J. König, Phys. Rev. B 65, 245301 (2002) ADSCrossRefGoogle Scholar
  70. 70.
    D. Sanchez, R. Lopez, Phys. Rev. Lett. 110, 026804 (2013) ADSCrossRefGoogle Scholar
  71. 71.
    S.-Y. Hwang, D. Sanchez, M. Lee, R. Lopez, arXiv: 1306.6558v1 (2013)Google Scholar
  72. 72.
    S. Bedkihal, D. Segal, unpublished Google Scholar
  73. 73.
    E. Deyo, B. Spivak, A. Zyuzin, Phys. Rev. B 74, 104205 (2006) ADSCrossRefGoogle Scholar
  74. 74.
    A. Ueda, M. Eto, Phys. Rev. B 73, 235353 (2006) ADSCrossRefGoogle Scholar
  75. 75.
    A. Ueda, M. Eto, New J. Phys. 9, 119 (2007)ADSCrossRefGoogle Scholar
  76. 76.
    T. Kubo, Y. Tokura, S. Tarucha, J. Phys. A 43, 354020 (2010) CrossRefGoogle Scholar
  77. 77.
    D. Sanchez, K. Kang, Phys. Rev. Lett. 100, 036806 (2008) ADSCrossRefGoogle Scholar
  78. 78.
    V.I. Puller, Y. Meir, Phys. Rev. Lett. 104, 256801 (2010) ADSCrossRefGoogle Scholar
  79. 79.
    O. Hod, R. Baer, E. Rabani, Phys. Rev. Lett. 97, 266803 (2006) ADSCrossRefGoogle Scholar
  80. 80.
    O. Hod, R. Baer, E. Rabani, J. Phys.: Condens. Matter 20, 383201 (2008) ADSGoogle Scholar
  81. 81.
    O. Entin-Wohlman, A. Aharony, Phys. Rev. B 85, 085401 (2012) ADSCrossRefGoogle Scholar
  82. 82.
    K. Saito, Y. Utsumi, Phys. Rev. B 78, 115429 (2008) ADSCrossRefGoogle Scholar
  83. 83.
    D. Sanchez, L. Serra, Phys. Rev. B 84, 201307(R) (2011) ADSCrossRefGoogle Scholar
  84. 84.
    R.S. Whitney, Phys. Rev. B 87, 115404 (2013) ADSCrossRefGoogle Scholar
  85. 85.
    Z. Zimboras, M. Faccin, Z. Kadar, J.D. Whitefield, B.P. Lanyon, J. Biamonte, Sci. Rep. 3, 2361 (2013)ADSCrossRefGoogle Scholar
  86. 86.
    V. Kashcheyevs, A. Aharony, O. Entin-Wohlman, Phys. Rev. B 73, 125338 (2006) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Salil Bedkihal
    • 1
  • Malay Bandyopadhyay
    • 2
  • Dvira Segal
    • 1
  1. 1.Chemical Physics Theory Group, Department of ChemistryUniversity of TorontoTorontoCanada
  2. 2.School of Basic SciencesIndian Institute of TechnologyBhubaneswarIndia

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