Why does a metal–superconductor junction have a resistance?

C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden

P.O. Box 9506, 2300 RA Leiden, The Netherlands

Instituut-Lorentz, Universiteit Leiden

P.O. Box 9506, 2300 RA Leiden, The Netherlands

September 1999

Abstract: This is a tutorial article based on a lecture delivered in June 1999 at the NATO Advanced Study Institute in Ankara. The phenomenon of Andreev reflection is introduced as the electronic analogue of optical phase-conjugation. In the optical problem, a disordered medium backed by a phase-conjugating mirror can become completely transparent. Yet, a disordered metal connected to a superconductor has the same resistance as in the normal state. The resolution of this paradox teaches us a fundamental difference between phase conjugation of light and electrons.

To be published in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, edited by I. O. Kulik and R. Ellialtioglu (Kluwer, Dordrecht).

To be published in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, edited by I. O. Kulik and R. Ellialtioglu (Kluwer, Dordrecht).

The recent revival of interest in the proximity effect has produced a deeper understanding into how the proximity-induced superconductivity of non-interacting electrons differs from true superconductivity of electrons having a pairing interaction. Clearly, the proximity effect does not require two superconductors. One should be enough. Consider a junction between a normal metal and a superconductor (an NS junction). Let the temperature be zero. What is the resistance of this junction? One might guess that it should be smaller than in the normal state, perhaps even zero. Isn't that what the proximity effect is all about?

The answer to this question has been in the literature since 1979 [3], but it has been appreciated only in the last few years. A recent review [4] gives a comprehensive discussion within the framework of the semiclassical theory of superconductivity. A different approach, using random-matrix theory, was reviewed by the author [5]. In this lecture we take a more pedestrian route, using the analogy between Andreev reflection and optical phase-conjugation [6, 7] to answer the question: Why does an NS junction have a resistance?

It was first noted by Andreev in 1963 [1] that an electron is reflected from a superconductor in an unusual way. The differences between normal reflection and Andreev reflection are illustrated in Fig. 1. Let us discuss them separately.

- Charge is conserved in normal reflection but not in Andreev reflection. The reflected particle (the hole) has the opposite charge as the incident particle (the electron). This is not a violation of a fundamental conservation law. The missing charge of is absorbed into the superconducting ground state as a Cooper pair. It is missing only with respect to the excitations.
- Momentum is conserved in Andreev reflection but not in normal reflection. The conservation of momentum is an approximation, valid if the superconducting excitation gap is much smaller than the Fermi energy of the normal metal. The explanation for the momentum conservation is that the superconductor can not exert a significant force on the incident electron, because is too small compared to the kinetic energy of the electron [8]. Still, the superconductor has to reflect the electron somehow, because there are no excited states within a range from the Fermi level. It is the unmovable rock meeting the irresistible object. Faced with the challenge of having to reflect a particle without changing its momentum, the superconductor finds a way out by transforming the electron into a particle whose velocity is opposite to its momentum: a hole.
- Energy is conserved in both normal and Andreev reflection. The electron is at an energy above the Fermi level and the hole is at an energy below it. Both particles have the same excitation energy . Andreev reflection is therefore an elastic scattering process.
- Spin is conserved in both normal and Andreev reflection. To conserve spin, the hole should have the opposite spin as the electron. This spin-flip can be ignored if the scattering properties of the normal metal are spin-independent.

The NS junction has an optical analogue known as a phase-conjugating mirror [9]. Phase conjugation is the effect that an incoming wave is reflected as a wave , with opposite sign of the phase . Since , this is equivalent to reversing the sign of the time , so that phase conjugation is sometimes called a time-reversal operation. The reflected wave has a wavevector precisely opposite to that of the incoming wave, and therefore propagates back along the incoming path. This is called retro-reflection. Phase conjugation of light was discovered in 1970 by Woerdman and by Stepanov, Ivakin, and Rubanov [10, 11].

A phase-conjugating mirror for light (see Fig. 2) consists of a cell containing a liquid or crystal with a large nonlinear susceptibility. The cell is pumped by two counter-propagating beams at frequency . A third beam is incident with a much smaller amplitude and a slightly different frequency . The non-linear susceptibility leads to an amplification of the incident beam, which is transmitted through the cell, and to the generation of a fourth beam, which is reflected. This non-linear optical process is called ``four-wave mixing''. Two photons of the pump beams are converted into one photon for the transmitted beam and one for the reflected beam. Energy conservation dictates that the reflected beam has frequency . Momentum conservation dictates that its wavevector is opposite to that of the incident beam. Comparing retro-reflection of light with Andreev reflection of electrons, we see that the Fermi energy plays the role of the pump frequency , while the excitation energy corresponds to the frequency shift .

A phase-conjugating mirror can be used for wavefront reconstruction. Imagine an incoming plane wave, that is distorted by some inhomogeneity. When this distorted wave falls on the mirror, it is phase conjugated and retro-reflected. Due to the time-reversal effect, the inhomogeneity that had distorted the wave now changes it back to the original plane wave. An example is shown in Fig. 3. Complete wavefront reconstruction is possible only if the distorted wavefront remains approximately planar, since perfect time reversal upon reflection holds only in a narrow range of angles of incidence for realistic systems. This is an important, but not essential complication, that we will ignore in what follows.

For a simple discussion it is convenient to replace the disordered medium by a tunnel barrier (or semi-transparent mirror) and consider the phase shift accumulated by an electron (or light wave) that bounces back and forth between the barrier and the superconductor (or phase-conjugating mirror). A periodic orbit (see Fig. 5) consists of two round-trips, one as an electron (or light at frequency ), the other as a hole (or light at frequency ). The miracle of phase conjugation is that phase shifts accumulated in the first round trip are cancelled in the second round trip. If this were the whole story, one would conclude that the net phase increment is zero, so all periodic orbits would interfere constructively and the tunnel barrier would become transparent because of resonant tunneling.

But it is not the whole story. There is an extra phase shift of acquired upon Andreev reflection that destroys the resonance. Since the periodic orbit consists of two Andreev reflections, one from electron to hole and one from hole to electron, and both reflections have the same phase shift , the net phase increment of the periodic orbit is and not zero. So subsequent periodic orbits interfere destructively, rather than constructively, and tunneling becomes suppressed rather than enhanced. In contrast, a phase-conjugating mirror adds a phase shift that alternates between and from one reflection to the next, so the net phase increment of a periodic orbit remains zero.

For a more quantitative description of the conductance we need to compute the probability that an incident electron is reflected as a hole. The matrix of probability amplitudes can be constructed as a geometric series of multiple reflections:

(1) |

(2) |

(3) |

(4) |

(5) |

It is remarkable that a small difference in phase shifts has such far reaching consequences. Note that one needs to consider multiple reflections in order to see the difference: The first term in the series is the same in Eqs. (1) and (4). That is probably why this essential difference between Andreev reflection and optical phase-conjugation was not noticed prior to Ref. [19].

(6) |

(7) |

Since for , we can immediately conclude that . If there is no disorder, then all 's are equal to unity, hence reaches its maximum value of . For a tunnel barrier all 's are , hence drops far below . A disordered metal will lie somewhere in between these two extremes, but where?

We have already alluded to the answer in the previous section, that for a disordered metal in the zero-temperature limit. To derive this remarkable equality, we parameterize the transmission eigenvalue in terms of the localization length ,

(8) |

(9) | (10) |

(11) |

The restriction to the range is the restriction to the regime of diffusive transport: For smaller we enter the ballistic regime and rises to ; For larger we enter the localized regime, where tunneling takes over from diffusion and becomes .

The research on which this lecture is based was done in collaboration with J. C. J. Paasschens. It was supported by the ``Stichting voor Fundamenteel Onderzoek der Materie'' (FOM) and by the ``Nederlandse organisatie voor Wetenschappelijk Onderzoek'' (NWO).

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