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Shot Noise for Entangled and Spin-Polarized Electrons

  • J. C. Egues
  • P. Recher
  • D. S. Saraga
  • V. N. Golovach
  • G. Burkard
  • E. V. Sukhorukov
  • D. Loss
Chapter
Part of the NATO Science Series book series (NAII, volume 97)

Abstract

We review our recent contributions on shot noise for entangled electrons and spinpolarized currents in novel mesoscopic geometries. We first discuss some of our recent proposals for electron entanglers involving a superconductor coupled to a double dot in the Coulomb blockade regime, a superconductor tunnel-coupled to Luttinger-liquid leads, and a triple-dot setup coupled to Fermi leads. We briefly survey some of the available possibilities for spin-polarized sources. We use the scattering approach to calculate current and shot noise for spin-polarized currents and entangled/unentangled electron pairs in a novel beam-splitter geometry with a local Rashba spinorbit (s-o) interaction in the incoming leads. For single-moded incoming leads, we find continuous bunching and antibunching behaviors for the entangled pairs — triplet and singlet — as a function of the Rashba rotation angle. In addition, we find that unentangled triplets and the entangled one exhibit distinct shot noise; this should allow their identification via noise measurements. Shot noise for spin-polarized currents shows sizable oscillations as a function of the Rashba phase. This happens only for electrons injected perpendicular to the Rashba rotation axis; spin-polarized carriers along the Rashba axis are noiseless. The Rashba coupling constant α is directly related to the Fano factor and could be extracted via noise measurements. For incoming leads with s-o induced interbandcoupled channels, we find an additional spin rotation for electrons with energies near the crossing of the bands where interband coupling is relevant. This gives rise to an additional modulation of the noise for both electron pairs and spin-polarized currents. Finally, we briefly discuss shot noise for a double dot near the Kondo regime.

Keywords

Beam Splitter Shot Noise Cooper Pair Fano Factor Luttinger Liquid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W. Schottky, Ann. Phys. 57 (1918) 541.CrossRefGoogle Scholar
  2. 2.
    Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).ADSCrossRefGoogle Scholar
  3. 3.
    D. Loss and E.V. Sukhorukov, Phys. Rev. Lett. 84, 1035 (2000), cond-mat/9907129.ADSCrossRefGoogle Scholar
  4. 4.
    G. Burkard, D. Loss, and E.V. Sukhorukov, Phys. Rev. B 61,R16303 (2000), condmat/9906071. For an early account see D. P. DiVincenzo and D. Loss, J. Magn. Magn. Mat. 200, 202 (1999), cond-mat/9901137.ADSCrossRefGoogle Scholar
  5. 5.
    W. D. Oliver et al., in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, vol. 559 of NATO ASI Series C: Mathematical and Physical Sciences, eds. I. O. Kulik and R. Ellialtioglu (Kluwer, Dordrecht, 2000), pp. 457-466.Google Scholar
  6. 6.
    F. Taddei and R. Fazio, Phys. Rev. B 65, 075317 (2002).ADSCrossRefGoogle Scholar
  7. 7.
    J. C. Egues, G. Burkard, and D. Loss, to appear in the Journal of Superconductivity; condmat/0207392.Google Scholar
  8. 8.
    J. C. Egues, G. Burkard, and D. Loss, Phys. Rev. Lett. 89, 176401 (2002); cond-mat/0204639.ADSCrossRefGoogle Scholar
  9. 9.
    B. R. Bulka et al. Phys. Rev. B 60,12246 (1999).ADSCrossRefGoogle Scholar
  10. 10.
    F. G. Brito, J. F. Estanislau, and J. C. Egues, J. Magn. Magn. Mat. 226-230,457 (2001).ADSCrossRefGoogle Scholar
  11. 11.
    K.M. Souza, J. C. Egues, and A. P. Jauho, cond-mat/0209263.Google Scholar
  12. 12.
    J. J. Sakurai, Modern Quantum Mechanics, San Fu Tuan, Ed., (Addison-Wesley, New York, 1994); (Ch. 3, p. 223). See also J. I. Cirac, Nature 413, 375 (2001).Google Scholar
  13. 13.
    Semiconductor Spintronics and Quantum Computation, Eds. D. D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002).Google Scholar
  14. 14.
    P. Recher, E.V. Sukhorukov, and D. Loss, Phys. Rev. B 63,165314 (2001); cond-mat/0009452.ADSCrossRefGoogle Scholar
  15. 15.
    D. S. Saraga and D. Loss, cond-mat/0205553.Google Scholar
  16. 16.
    R. Fiederling et al., Nature 402, 787 (1999); Y. Ohno et al., Nature 402, 790 (1999).ADSCrossRefGoogle Scholar
  17. 17.
    See J. C. Egues Phys. Rev. Lett. 80, 4578 (1998) and J. C. Egues et al. Phys. Rev. B 64,195319 (2001) for ballistic spin filtering in semimagnetic heterostruc ures.ADSCrossRefGoogle Scholar
  18. 18.
    P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. Lett. 85, 1962 (2000), condmat/0003089.ADSCrossRefGoogle Scholar
  19. 19.
    P. Recher and D. Loss, Phys. Rev. B 65, 165327 (2002), cond-mat/0112298.ADSCrossRefGoogle Scholar
  20. 20.
    V.N. Golovach and D. Loss, cond-mat/0109155.Google Scholar
  21. 21.
    R. C. Liu et al., Nature (London), 391,263 (1998).ADSCrossRefGoogle Scholar
  22. 22.
    M. Henny et al., Science 284,296 (1999); W. D. Oliver et al., Science 284, 299 (1999). See also M. Büttiker, Science 284, 275 (1999).ADSCrossRefGoogle Scholar
  23. 23.
    G. Fève et al. (cond-mat/0108021) also investigate transport in a beam splitter configuration. These authors assume a “global” s-o interaction and formulate the scattering approach using Rashba states in single-moded leads.Google Scholar
  24. 24.
    S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).ADSCrossRefGoogle Scholar
  25. 25.
    L.P. Kouwenhoven, G. Schön, L.L. Sohn, Mesoscopic Electron Transport, NATO ASI Series E: Applied Sciences-Vol.345, 1997, Kluwer Academic Publishers, Amsterdam.Google Scholar
  26. 26.
    D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998), cond-mat/9701055.ADSCrossRefGoogle Scholar
  27. 27.
    M.-S. Choi, C. Bruder, and D. Loss, Phys. Rev. B 62, 13569 (2000); cond-mat/0001011.ADSCrossRefGoogle Scholar
  28. 28.
    C. Bena, S. Vishveshwara, L. Balents, and M.P.A. Fisher, Phys. Rev. Lett. 89, 037901 (2002).ADSCrossRefGoogle Scholar
  29. 29.
    G.B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B 24, 287 (2001).ADSCrossRefGoogle Scholar
  30. 30.
    R. Mélin, cond-mat/0105073.Google Scholar
  31. 31.
    V. Bouchiat et al., cond-mat/0206005.Google Scholar
  32. 32.
    W.D. Oliver, F. Yamaguchi, and Y. Yamamoto, Phys. Rev. Lett. 88, 037901 (2002).ADSCrossRefGoogle Scholar
  33. 33.
    S. Bose and D. Home, Phys. Rev. Lett. 88, 050401 (2002).MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    In principle, an entangler producing entangled triplets or orbital entanglement would also be desirable.Google Scholar
  35. 35.
    This condition reflects energy conservation in the Andreev tunnelling event from the SC to the two QDs.Google Scholar
  36. 36.
    This reduction factor of the current I 2 compared to the resonant current I 1 reflects the energy cost in the virtual states when two electrons tunnel via the same QD into the same Fermi lead and are given by U and/or Δ. Since the lifetime broadenings γ1 and γ2 of the two QDs 1 and 2 are small compared to U and Δ such processes are suppressed.Google Scholar
  37. 37.
    P. Recher and D. Loss, Journal of Superconductivity: Incorporating Novel Magnetism 15(1): 49–65, February 2002; cond-mat/0205484.Google Scholar
  38. 38.
    A.F. Volkov, P.H.C. Magne, B.J. van Wees, and T.M. Klapwijk, Physica C 242, 261 (1995).ADSCrossRefGoogle Scholar
  39. 39.
    M. Kociak, A.Yu. Kasumov, S. Guron, B. Reulet, I.I. Khodos, Yu.B. Gorbatov, V.T. Volkov, L. Vaccarini, and H. Bouchiat, Phys. Rev. Lett. 86, 2416 (2001).ADSCrossRefGoogle Scholar
  40. 40.
    M. Bockrath et al., Nature 397, 598 (1999).ADSCrossRefGoogle Scholar
  41. 41.
    R. Egger and A. Gogolin, Phys. Rev. Lett. 79, 5082 (1997); R. Egger, Phys. Rev. Lett. 83, 5547 (1999).ADSCrossRefGoogle Scholar
  42. 42.
    C. Kane, L. Balents, and M.P.A. Fisher, Phys. Rev. Lett. 79, 5086 (1997).ADSCrossRefGoogle Scholar
  43. 43.
    L. Balents and R. Egger, Phys. Rev. B, 64 035310 (2001).ADSCrossRefGoogle Scholar
  44. 44.
    For a review see e.g. HJ. Schulz, G. Cuniberti, and P. Pieri, cond-mat/9807366; or J. von Delft and H. Schoeller, Annalen der Physik, Vol. 4, 225-305 (1998).Google Scholar
  45. 45.
    The interaction dependent constants A b are of order one for not too strong interaction between electrons in the LL but are decreasing when interaction in the LL-leads is increased [19]. Therefore in the case of substantially strong interaction as it is present in metallic carbon nanotubes, the pre-factors A b can help in addition to suppress I 2 Google Scholar
  46. 46.
    Since γp-> γP+, it is more probable that two electrons coming from the same Cooper pair travel in the same direction than into different directions when injected into the same LL-lead.Google Scholar
  47. 47.
    In order to have exclusively singlet states as an input for the beamsplitter setup, it is important that the LL-leads return to their spin ground-state after the injected electrons have tunnelled out again into the Fermi leads. For an infinite LL, spin excitations are gapless and therefore an arbitrary small bias voltage μ between the SC and the Fermi liquids gives rise to spin excitations in the LL. However, for a realistic finite size LL (e.g. a nanotube), spin excitations are gapped on an energy scale ∼ ħ VF/L, where L is the length of the LL. Therefore, if κBT,μ < ħVF/L only singlets can leave the LL again to the Fermi leads, since the total spin of the system has to be conserved. For metallic carbon nanotubes, the Fermi velocity is ∼ 106m/s, which gives an excitation gap of the order of a few meV for L ∼ μm; this is large enough for our regime of interest.Google Scholar
  48. 48.
    A singlet-triplet transition for the ground state of a quantum dot can be driven by a magnetic field; see S. Tarucha et al., Phys. Rev. Lett. 84,2485 (2000).ADSCrossRefGoogle Scholar
  49. 49.
    This symmetric setup of the charging energy U is obtained when the gate voltages are tuned such that the total Coulomb charging energies in D c are equal with zero or two electrons.Google Scholar
  50. 50.
    K. Blum, Density Matrix Theory and Applications (Plenum, New York, 1996).MATHGoogle Scholar
  51. 51.
    T.H. Oosterkamp et al., Nature (London) 395,873 (1998); T. Fujisawa et al., Science 282,932 (1998).ADSCrossRefGoogle Scholar
  52. 52.
    J.M. Kikkawa and D.D. Awschalom, Phys. Rev. Lett. 80,4313 (1998).ADSCrossRefGoogle Scholar
  53. 53.
    I. Malajovich, J. M. Kikkawa, D. D. Awschalom, J. J. Berry, and D. D. Awschalom, Phys. Rev. Lett. 84,1015 (2000); I. Malajovich, J. J. Berry, N. Samarth, and D. D. Awschalom, Nature 411,770 (2001).ADSCrossRefGoogle Scholar
  54. 54.
    M. Johnsson and R. H. Silsbee, Phys. Rev. Lett. 55,1790 (1985); M. Johnsson and R. H. Silsbee, Phys. Rev. B 37, 5326 (1988); M. Johnsson and R. H. Silsbee, Phys. Rev. B 37, 5712 (1988).ADSCrossRefGoogle Scholar
  55. 55.
    F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature 410, 345 (2001); F. J. Jedema, H. B. Heersche, J. J. A. Baselmans, and B. J. van Wees, Nature 416, 713 (2002).ADSCrossRefGoogle Scholar
  56. 56.
    In addition, for fully spin-polarized leads the device can act as a single spin memory with read-in and read-out capabilities if the dot is subjected to a ESR source.Google Scholar
  57. 57.
    This is true as long as the Zeeman splitting in the leads is much smaller than their Fermi energies.Google Scholar
  58. 58.
    H.-A. Engel and D. Loss, Phys. Rev. B 65, 195321 (2002), cond-mat/0109470.ADSCrossRefGoogle Scholar
  59. 59.
    S. Kawabata, J. Phy. Soc. Jpn. 70, 1210 (2001).ADSCrossRefGoogle Scholar
  60. 60.
    N.M. Chtchelkatchev, G. Blatter, G.B. Lesovik, and T. Martin, cond-mat/0112094.Google Scholar
  61. 61.
    M. Büttiker, Phys. Rev. B 46, 12485 (1992); Th. Martin and R. Landauer, Phys. Rev. B 45,1742 (1992). For a recent comprehensive review on shot noise, see Ref. [2].ADSCrossRefGoogle Scholar
  62. 62.
    Our noise definition here differs by a factor of two from that in the review article by Blanter and Büttiker (Ref. [2]); these authors define their power spectral density of the noise with a coefficient two in front (see definition following Eq. (49) and footnote 4 in Ref. [2]). We use a standard Fourier transform (no factor of two in front) to define the noise spectral density.Google Scholar
  63. 63.
    For a discrete energy spectrum we need to insert a density-of-states factor v in the current and noise definitions; see Ref. [4].Google Scholar
  64. 64.
    Note that the uncorrelated-beam case here refers to a beam splitter configuration with only one of the incoming leads “open”. This is an important point since a beam splitter is noiseless for (unpolarized) uncorrelated beams in both incoming leads.Google Scholar
  65. 65.
    G. Engels et al. Phys. Rev. B 55, R1958 (1997); J. Nitta et al., Phys. Rev. Lett. 78, 1335 (1997); D. Grundler Phys. Rev. Lett. 84,6074 (2000); Y. sato et al. J. Appl. Phys. 89,8017 (2001).ADSCrossRefGoogle Scholar
  66. 66.
    A. V. Moroz and C. H. W. Barnes, Phys. Rev. B 60, 14272 (1999); F. Mireles and G. Kirczenow, ibid. 64,024426 (2001); M. Governale and U. Zülicke, Phys. Rev. B 66 073311 (2002).ADSCrossRefGoogle Scholar
  67. 67.
    G. Lommer et al., Phys. Rev. Lett. 60,728 (1988), G. L. Chen et al., Phys. Rev. B 47, 4084 (R) (1993), E. A. de Andrada e Silva et al., Phys. Rev. B 50,8523 (1994), and F. G. Pikus and G. E. Pikus Phys. Rev. B 51,16928 (1995).ADSCrossRefGoogle Scholar
  68. 68.
    Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39,78 (1984).ADSGoogle Scholar
  69. 69.
    L. W. Molenkamp et al., Phys. Rev. B 64, R121202 (2001); M. H. Larsen et al., ibid. 66,033304 (2002).ADSCrossRefGoogle Scholar
  70. 70.
    The Rashba-active region in lead 1 is (supposed to be) electrostatically induced. This implies that there is no band-gap mismatch between the Rashba region and the adjacent regions in lead 1 due to materials differences. There is, however, a small mismatch arising from the Rashba energy ∈r; this is the amount the Rashba bands are shifted down with respect to the bands in the absence of s-o orbit in the channel. Since typically ∈R ≪ εF, we find that the transmission is indeed very close to unity (see estimate in Ref. [8]).Google Scholar
  71. 71.
    Note that the velocity operator is not diagonal in the presence of the Rashba interaction.Google Scholar
  72. 72.
    J. C. Egues, G. Burkard, and D. Loss, cond-mat/0209692.Google Scholar
  73. 73.
    In the absence of the s-o interaction, we assume the wire has two sets of spin-degenerate parabolic bands for each κ vector. In the presence of s-o interaction but neglecting s-o induced interband coupling, there is a one-to-one correspondence between the parabolic bands with no spin orbit and the Rashba bands; hence they can both be labelled by the same indices.Google Scholar
  74. 74.
    N. W. Ashcroft and N. D. Mermin, Solid State Physics, Ch. 9. (Holt, Rinehart, and Winston, New York, 1976).Google Scholar
  75. 75.
    G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B 62, R4790 (2000).ADSCrossRefGoogle Scholar
  76. 76.
    L. P. Kouwenhoven, private communication.Google Scholar
  77. 77.
    L. I. Glazman and M.E. Raikh, JETP Lett. 47, 452 (1988); T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988).ADSGoogle Scholar
  78. 78.
    Y. Meir and A. Golub, Phys. Rev. Lett. 88, 116802 (2002).ADSCrossRefGoogle Scholar
  79. 79.
    F. Yamaguchi and K. Kawamura, Physica B 227, 116 (1996).ADSCrossRefGoogle Scholar
  80. 80.
    A. Schiller and S. Hershfield, Phys. Rev. B 58, 14978 (1998).ADSCrossRefGoogle Scholar
  81. 81.
    G. Burkard, D. Loss, and D.P. DiVincenzo, Phys. Rev. B 59, 2070 (1999), cond-mat/9808026.ADSCrossRefGoogle Scholar
  82. 82.
    W. Izumida and O. Sakai, Phys. Rev. B 62, 10260 (2000).ADSCrossRefGoogle Scholar
  83. 83.
    A. Georges and Y. Meir, Phys. Rev. Lett. 82, 3508 (1999).ADSCrossRefGoogle Scholar
  84. 84.
    T. Aono and M. Eto, Phys. Rev. B 63, 125327 (2001).ADSCrossRefGoogle Scholar
  85. 85.
    I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B 52, 9528 (1995).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • J. C. Egues
    • 1
    • 2
  • P. Recher
    • 1
  • D. S. Saraga
    • 1
  • V. N. Golovach
    • 1
  • G. Burkard
    • 1
  • E. V. Sukhorukov
    • 1
  • D. Loss
    • 1
  1. 1.Department of Physics and AstronomyUniversity of BaselBaselSwitzerland
  2. 2.Department of Physics and InformaticsUniversity of São Paulo at São CarlosSão Carlos/SPBrazil

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