Entangled electron current through finite size normal-superconductor tunneling structures

  • E. Prada
  • F. Sols


We investigate theoretically the simultaneous tunneling of two electrons from a superconductor into a normal metal at low temperatures and voltages. Such an emission process is shown to be equivalent to the Andreev reflection of an incident hole. We obtain a local tunneling Hamiltonian that permits to investigate transport through interfaces of arbitrary geometry and potential barrier shapes. We prove that the bilinear momentum dependence of the low-energy tunneling matrix element translates into a real space Hamiltonian involving the normal derivatives of the electron fields in each electrode. The angular distribution of the electron current as it is emitted into the normal metal is analyzed for various experimental setups. We show that, in a full three-dimensional problem, the neglect of the momentum dependence of tunneling causes a violation of unitarity and leads to the wrong thermodynamic (broad interface) limit. More importantly for current research on quantum information devices, in the case of an interface made of two narrow tunneling contacts separated by a distance r, the assumption of momentum-independent hopping yields a nonlocally entangled electron current that decays with a prefactor proportional to r -2 instead of the correct r -4.


Normal Derivative Normal Metal Emission Process Electron Field Momentum Dependence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.F. Andreev, Zh. Ekps. Teor. Fiz. 46, 1823 (1964) [Sov. Phys. JETP 19, 1228 (1964)]; A.F. Andreev, Zh. Ekps. Teor. Fiz. 49, 655 (1966) [22, 455 (1966)]Google Scholar
  2. 2.
    J. Demers, A. Griffin, Can. J. Phys. 49, 285 (1970)Google Scholar
  3. 3.
    G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25, 4515 (1982)CrossRefGoogle Scholar
  4. 4.
    M. Tinkham, Introduction to Superconductivity, 2nd edn. (McGraw-Hill, New York, 1996)Google Scholar
  5. 5.
    J.M. Byers, M.E Flatté, Phys. Rev. Lett. 74, 306 (1995)CrossRefGoogle Scholar
  6. 6.
    J. Torres, T. Martin, Eur. Phys. J. B 12, 319 (1999)Google Scholar
  7. 7.
    G. Deutscher, D. Feinberg, Appl. Phys. Lett. 76, 487 (2000)CrossRefGoogle Scholar
  8. 8.
    P. Recher, E.V. Sukhorukov, D. Loss, Phys. Rev. B 63, 165314 (2001)CrossRefGoogle Scholar
  9. 9.
    G. Falci, D. Feinberg, F.W.J. Hekking, Europhys. Lett. 54, 255 (2001)CrossRefGoogle Scholar
  10. 10.
    R. Mélin, J. Phys.: Condens. Matter 13, 6445 (2001)CrossRefGoogle Scholar
  11. 11.
    G.B. Lesovik, T. Martin, G. Blatter, Eur. Phys. J. B 24, 287 (2001)CrossRefGoogle Scholar
  12. 12.
    V. Apinyan, R. Mélin, Eur. Phys. J. B 25, 373 (2002)CrossRefGoogle Scholar
  13. 13.
    R. Mélin, D. Feinberg, Eur. Phys. J. B 26, 101 (2002)CrossRefGoogle Scholar
  14. 14.
    D. Feinberg, G. Deutscher, Physica E 15, 88 (2002)CrossRefGoogle Scholar
  15. 15.
    P. Recher, D. Loss, Phys. Rev. B 65, 165327 (2002)CrossRefGoogle Scholar
  16. 16.
    N.M. Chtchelkatchev, G. Blatter, G.B. Lesovik, T. Martin, Phys. Rev. B 66, 161320 (2002)CrossRefGoogle Scholar
  17. 17.
    D. Feinberg, Eur. Phys. J. B 36, 419 (2003)CrossRefGoogle Scholar
  18. 18.
    P. Recher, D. Loss, Phys. Rev. Lett. 91, 267003 (2003)CrossRefGoogle Scholar
  19. 19.
    P. Samuelsson, E.V. Sukhorukov, M. Büttiker, Phys. Rev. Lett. 91, 157002 (2003)CrossRefGoogle Scholar
  20. 20.
    M. Kupka, Phys. C 221, 346 (1994)CrossRefGoogle Scholar
  21. 21.
    M. Kupka, Phys. C 281, 91 (1997)CrossRefGoogle Scholar
  22. 22.
    P.G. de Gennes, Superconductivity of Metals and Alloys, (Addison-Wesley, Reading, 1989)Google Scholar
  23. 23.
    C.W.J. Beenakker, in Mesoscopic Quantum Physics, edited by E. Akkermans, G. Montamboux, J.L. Pichard (North-Holland, Amsterdam, 1995)Google Scholar
  24. 24.
    J. Sánchez-Cañizares, F. Sols, J. Phys.: Condens. Matter 7, L317 (1995)Google Scholar
  25. 25.
    J. Sánchez-Cañizares, F. Sols, Phys. Rev. B 55, 531 (1997)CrossRefGoogle Scholar
  26. 26.
    R. Kümmel, Z. Physik 218, 472 (1969)Google Scholar
  27. 27.
    F. Sols, J. Sánchez-Cañizares, Superlatt. and Microstruct. 25, 627 (1999)CrossRefGoogle Scholar
  28. 28.
    C.J. Lambert, J. Phys.: Condens. Matter 3, 6579 (1991)CrossRefGoogle Scholar
  29. 29.
    C.W.J. Beenakker, Phys. Rev. B 46, 12841 (1992)CrossRefGoogle Scholar
  30. 30.
    H. Nakano, H. Takayanagi, Phys. Rev. B 50, 3139 (1994)CrossRefGoogle Scholar
  31. 31.
    A. Levy Yeyati, A. Martín-Rodero, F.J. García-Vidal, Phys. Rev. B 51, 3743 (1995)CrossRefGoogle Scholar
  32. 32.
    J. Sánchez-Cañizares, F. Sols, J. Low Temp. Phys. 122, 11 (2001)CrossRefGoogle Scholar
  33. 33.
    J.R. Kirtley, Phys. Rev. B 47, 11379 (1992)CrossRefGoogle Scholar
  34. 34.
    J. Bardeen, Phys. Rev. Lett. 6, 57 (1961)CrossRefGoogle Scholar
  35. 35.
    G.D. Mahan, Many-Particle Physics, 3rd edn. (Kluwer Academic/Plenum, New York, 2000), p. 561Google Scholar
  36. 36.
    R.E. Prange, Phys. Rev. 131, 1083 (1963)CrossRefGoogle Scholar
  37. 37.
    A. Galindo, P. Pascual, Quantum Mechanics (Springer, Berlin, 1990)Google Scholar
  38. 38.
    P.V. Gray, Phys. Rev. 140, 179 (1965)CrossRefGoogle Scholar
  39. 39.
    The low-energy linear dependence \(T(E_z)\sim E_z\) and the related bilinear dependence \(T_{\mathbf{k}\mathbf{q}}\sim k_z q_z\) is implicit in reference [40]. We note, however, that here we find perfect agreement between Bardeen’s perturbative method and the exact results in the tunneling limitGoogle Scholar
  40. 40.
    C.B. Duke, Tunneling in Solids (Academic Press, New York and London, 1969), p. 218Google Scholar
  41. 41.
    C.J. Chen, Phys. Rev. B 42, 8841 (1990)CrossRefGoogle Scholar
  42. 42.
    E. Merzbacher, Quantum Mechanics, 3rd edn. (John Wiley & Sons, New York, 1998), Chap. 20Google Scholar
  43. 43.
    Note that, as defined in equation (28), the final state \(({\bf k}_1,{\bf k}_2)\) is identical to the state \(({\bf k}_2,{\bf k}_1)\). Thus, when summing over final states, one must avoid double counting. Specifically, in equation (25), \(\sum_f\) is to be understood as \(\frac{1}{2}\sum_{{\bf k}_1,{\bf k}_2}\), where \(\sum_{{\bf k}_1,{\bf k}_2}\) is an unrestricted sum over indices \({\bf k}_1,{\bf k}_2\) Google Scholar
  44. 44.
    A study of the angular dependence of Andreev reflection in the broad interface limit has been presented in reference [45]. However, it is restricted to a delta barrier interface and consider the case where N is a doped semiconductorGoogle Scholar
  45. 45.
    N.A. Mortensen, K. Flensberg, A.P. Jauho, Phys. Rev. B 59, 10176 (1999)CrossRefGoogle Scholar
  46. 46.
    There, under the constraint of a given total current, entropy is maximized by the electron system adopting a displaced Fermi sphere configuration which exactly yields the \(\cos\theta\) lawGoogle Scholar
  47. 47.
    R. Landauer, IBM J. Res. Dev. 1, 223 (1957); M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986); A. Szafer, A.D. Stone, IBM J. Res. Dev. 32, 384 (1988)Google Scholar
  48. 48.
    For a broad interface, \({\bf k}_{\parallel}\) is conserved and A n can be identified with T n. However, invoking trace invariance confersgeneral validity to the argument and makes it more rigorous in the particular case of a broad interface embedded in an even broader wireGoogle Scholar
  49. 49.
    Together with the factor appearing in the definition (33), this factor \(1/2\) yields the \(1/4\) geometrical correction given by reference [50] and cited in reference [3]. We emphasize however that such \(1/4\) correction applies only to the normal conductance, as derived in reference [50], but not to the NS interface where, rather, the correct geometrical correction is \(1/12\), as implicitly noted in equations (33) and (41)Google Scholar
  50. 50.
    Yu.V. Sharvin, Zh. EKsp. Teor. Fiz. 48, 984 (1965) [Sov. Phys.-JETP 21, 655 (1965)]Google Scholar
  51. 51.
    H.A. Bethe, Phys. Rev. 66, 163 (1944)CrossRefzbMATHGoogle Scholar
  52. 52.
    We do not rule out, however, that the errors derived from the use of equation (61) may cancel in the calculation of some physical quantities such as e.g. the ratio between the critical current and the normal conductance of a superconducting tunnel junction [53,35]Google Scholar
  53. 53.
    V. Ambegaokar, A. Baratoff, Phys. Rev. Lett. 10, 486 (1963)CrossRefGoogle Scholar
  54. 54.
    F.W.J. Hekking, Yu. V. Nazarov, Phys. Rev. B 49, 6847 (1994)CrossRefGoogle Scholar
  55. 55.
    F. Sols, M. Macucci, U. Ravaioli, K. Hess, J. Appl. Phys. 66, 3892 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Departamento de Física Teórica de la Materia CondensadaC-VMadridSpain
  2. 2.Instituto Nicolás CabreraUniversidad Autónoma de MadridMadridSpain

Personalised recommendations