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.Comment: 12 pages, 5 postscript figures [v2: all figures inline

AA Abrikosov

AF Andreev

BI Stepanov

CWJ Beenakker

DM Pepper

E Toyoda

GE Blonder

H Courtois

H Houten van

IO Kulik

J Feinberg

J. C

JCJ Paasschens

JP Woerdman

SN Artemenko

Y Takane

YV Nazarov

English

arXiv

C. W. J. Beenakker

Crossref

Why Does a Metal—Superconductor Junction Have a Resistance?

Beenakker, C.W.J.

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Why does a metal-superconductor junction have a resistance?

Leiden University Scholary Publications

Chapter 4WHY DOES A METAL-SUPERCONDUCTORJUNCTION HAVE A RESISTANCE?C. W. J. BeenakkerInstituut-Lorentz, Universiteit LeidenP. 0. Box 9506, 2300 RA Leiden, The NetherlandsAbstract The phenomenon of Andreev reflection is introduced äs the electronicanalogue of optical phase-conjugation. In the optical problem, a disor-dered medium backed by a phase-conjugating mirror can become com-pletely transparent. Yet, a disordered metal connected to a supercon-ductor has the same resistance äs in the normal state. The resolutionof this paradox teaches us a fundamental difference between phase con-jugation of light and electrons.1. INTRODUCTIONIn the late sixties, Kulik used the mechanism of Andreev reflection[1] to explain how a metal can carry a dissipationless current betweentwo superconductors over arbitrarily long length scales, provided thetemperature is low enough [2]. One can say that the normal metal hasbecome superconducting because of the proximity to a superconductor.This proximity effect exists even if the electrons in the normal metalhave no interaction. At zero temperature the maximum supercurrentthat the metal can carry decays only algebraically with the Separationbetween the superconductors — rather than exponentially, äs it does athigher temperatures.The recent revival of interest in the proximity effect has produced adeeper understanding into how the proximity-induced superconductivityof non-interacting electrons differs from true superconductivity of elec-trons having a pairing interaction. Clearly, the proximity effect does notrequire two superconductors. One should be enough. Consider a junc-5l7. O. Kulik and R. Ellialtioglu (eds.),Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, 51-60.© 2000 Kluwer Academic Publishers. Printed in the Netherlands.52 C. W. J. Beenakkertion 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 recentreview [4] gives a comprehensive discussion within the framework of thesemiclassical theory of superconductivity. A different approach, usingrandom-matrix theory, was reviewed by the author [5]. In this lecturewe take a more pedestrian route, using the analogy between Andreevreflection and optical phase-conjugation [6, 7] to answer the question:Why does an NS junction have a resistance?2. ANDREEV REFLECTION AND OPTICALPHASE-CONJUGATIONIt was first noted by Andreev in 1963 [1] that an electron is reflectedfrom a superconductor in an unusual way. The differences between nor-mal reflection and Andreev reflection are illustrated in Fig. 4.1. Let usdiscuss them separately.• Charge is conserved in normal reflection but not in Andreev re-flection. The reflected particle (the hole) has the opposite Chargeäs the incident particle (the electron). This is not a violation ofa fundamental conservation law. The missing charge of 2e is ab-sorbed into the superconducting ground state äs a Cooper pair. Itis missing only with respect to the excitations.Figure 4-1 Normal reüection by an insulator (I) versus Andreev reflection by a su-perconductor (S) of an electron excitation in a normal metal (N) near the Fermi level.Normal reflection (left) conserves Charge but does not conserve momentum. Andreevreflection (right) conserves momentum but does not conserve Charge: The electron(e) is reflected äs a hole (h) with the same momentum and opposite velocity. Themissing charge of 2e is absorbed äs a Cooper pair by the superconducting condensate.Why Does a Metal-Superconductor Junction Have a Resistance? 53• Momentum is conserved in Andreev reflection but not in normalreflection. The conservation of momentum is an approximation,valid if the superconducting excitation gap Δ is much smaller thanthe Fermi energy E-p of the normal metal. The explanation for themomentum conservation is that the superconductor can not exerta significant force on the incident electron, because Δ is too smallcompared to the kinetic energy Ep of the electron [8]. Still, thesuperconductor has to reflect the electron somehow, because thereare no excited states within a ränge Δ from the Fermi level. It isthe unmovable rock meeting the irresistible object. Faced with thechallenge of having to reflect a particle without changing its mo-mentum, the superconductor finds a way out by transforming theelectron into a particle whose velocity is opposite to its momentum:a hole.• Energy is conserved in both normal and Andreev reflection. Theelectron is at an energy ε above the Fermi level and the hole is at anenergy ε below it. Both particles have the same excitation energyε. Andreev reflection is therefore an elastic scattering process.• Spin is conserved in both normal and Andreev reflection. To con-serve spin, the hole should have the opposite spin äs the electron.This spin-flip can be ignored if the scattering properties of thenormal metal are spin-independent.The NS Junction has an optical analogue known äs a phase-conjugatingmirror [9]. Phase conjugation is the effect that an incoming wave occos(/ca; — ωί) is reflected äs a wave oc cos(—kx — ωΐ), with opposite signof the phase kx. Since cos(—kx—ωί) = cos(fco;+Ci;i), this is equivalent toreversing the sign of the time i, so that phase conjugation is sometimescalled a time-reversal Operation. The reflected wave has a wavevectorprecisely opposite to that of the incoming wave, and therefore propagatesback along the incoming path. This is called retro-reflection. Phase con-jugation of light was discovered in 1970 by Woerdman and by Stepanov,Ivakin, and Rubanov [10, 11].A phase-conjugating mirror for light (see Fig. 4.2) consists of a cellcontaining a liquid or crystal with a large nonlinear susceptibility. Thecell is pumped by two counter-propagating beams at frequency WQ. Athird beam is incident with a much smaller amplitude and a slightlydifferent frequency ωό + δω. The non-linear susceptibility leads to anamplification of the incident beam, which is transmitted through thecell, and to the generation of a fourth beam, which is reflected. Thisnon-linear optical process is called "four-wave mixing". Two photons ofPCM54 C. W. J. BeenakkerFigure 4.Z Schematic drawing ofoptical phase-conjugation by meansof four-wave mixing. The phase-conjugating mirror (PCM) consistsof a cell filled by a medium witha third-order non-linear susceptibil-ity X3· (Examples are BaTiOa andCS2.) The medium is pumped bytwo counter-propagating beams atfrequency ωο· Α probe beam inci-dent at frequency ωρ = ωό + δω isthen retro-reflected äs a conjugatebeam at frequency wc = ωό — δω.Pump Prom Ref. [12].the pump beams are converted into one photon for the transmitted beamand one for the reflected beam. Energy conservation dictates that thereflected beam has frequency ωό — δω. Momentum conservation dictatesthat its wavevector is opposite to that of the incident beam. Comparingretro-reflection of light with Andreev reflection of electrons, we see thatthe Fermi energy Ep plays the role of the pump frequency WQ, while theexcitation energy ε corresponds to the frequency shift δω.A phase-conjugating mirror can be used for wavefront reconstruction.Imagine an incoming plane wave, that is distorted by some inhomogene-ity. When this distorted wave falls on the mirror, it is phase conjugatedand retro-reflected. Due to the time-reversal effect, the inhomogeneitythat had distorted the wave now changes it back to the original planewave. An example is shown in Fig. 4.3. Complete wavefront reconstruc-tion is possible only if the distorted wavefront remains approximatelyplanar, since perfect time reversal upon reflection holds only in a narrowränge of angles of incidence for realistic Systems. This is an important,but not essential complication, that we will ignore in what follows.3. THE RESISTANCE PARADOXWe have learned that a disordered medium (such äs the frosted glass inFig. 4.3) becomes transparent when it is backed by a phase-conjugatingmirror. By analogy, one would expect that a disordered metal backedby a superconductor would become "transparent" too, meaning that itsresistance should vanish (up to a small contact resistance that is presenteven without any disorder). This does not happen. Upon decreasing thetemperature below the superconducting transition temperature, the re-sistance drops slightly but then rises again back to its high-temperaturevalue. (A recent experiment is shown in Fig. 4.4, where the conduc-Why Does a Metal-Superconductor Junction Have a Resistance? 55Figure 4-3 Example of wavefront reconstruction by optical phase-conjugation. Inboth photographs the image of a cat was distorted by transmitting it through a pieceof frosted glass, and reflecting it back through the same piece of glass. This givesan unrecognizable image when reflected by an ordinary mirror (leffc panel) and theoriginal image when reflected by a phase-conjugating mirror (right panel). FromRef. [13].tance is plotted instead of the resistance.) This so-called "re-entranceeffect" has been reviewed recently by Courtois et al. [4], and we refer to4..Ϊ62D electron gasInAIAs/InGaAs0.5 1.5temperature (K)Figure 4-4 Temperature dependence of the conductance of an NS Junction, show-ing the re-entrance effect. The superconductor is Nb, the normal metal is a two-dimensional electron gas. A gate creates a strongly disordered region in the 2D gasthat dominates the conductance of the Junction. Upon lowering the temperature theconductance fast rises and then drops again. Under ideal circumstances the low- andhigh-temperature limits would be the same. From Ref. [16].56 C. W. J. Beenakkerthat review for an extensive list of references. The theoretical prediction[3, 14, 15] is that at zero temperature the resistance of the normal-metal-superconductor junction is the same äs in the normal state. How can wereconcile this with the notion of Andreev reflection äs a "time-reversing"process, analogous to optical phase-conjugation? To resolve this para-dox, let us study the analogy more carefully, to see where it breaksdown.For a simple discussion it is convenient to replace the disorderedmedium by a tunnel barrier (or semi-transparent mirror) and considerthe phase shift accumulated by an electron (or light wave) that bouncesback and forth between the barrier and the superconductor (or phase-conjugating mirror). A periodic orbit (see Fig. 4.5) consists of two round-trips, one äs an electron (or light at frequency WQ + δω), the other äs ahole (or light at frequency ωό — δω). The miracle of phase conjugationis that phase shifts accumulated in the first round trip are cancelled inthe second round trip. If this were the whole story, one would concludethat the net phase increment is zero, so all periodic orbits would interfereconstructively and the tunnel barrier would become transparent becauseof resonant tunneling.But it is not the whole story. There is an extra phase shift of —π/2acquired upon Andreev reflection that destroys the resonance. Since theperiodic orbit consists of two Andreev reflections, one from electron tohole and one from hole to electron, and both reflections have the samephase shift — π/2, the net phase increment of the periodic orbit is —π andnot zero. So subsequent periodic orbits interfere destructively, ratherthan constructively, and tunneling becomes suppressed rather than en-hanced. In contrast, a phase-conjugating mirror adds a phase shift thatalternates between +π/2 and — π/2 from one reflection to the next, sothe net phase increment of a periodic orbit remains zero.barrierFigure 4· 5 Periodic orbit consistingof two normal reflections and tworetro-reflections. The net phase in-crement is zero in the optical caseand —π in the electronic case. Hencethe periodic orbits interfere con-structively for light and destruc-tively for electrons. This explainswhy the barrier becomes transpar-ent for light but not for electrons.Why Does a Metal-Superconductor Junction Have a Resistance? 57For a more quantitative description of the conductance we need tocompute the probability Ähe that an incident electron is reflected äs ahole. The matrix of probability amplitudes r^ can be constructed äs ageometric series of multiple reflections:tl tl l + 1 11 Γ l t l l2rhe = V-t + V-r-ri-t + V- ΓΤΓΤ- Hl 1 1 1 l L l i j(4.1)Each factor l/i = exp(—ϊττ/2) corresponds to an Andreev reflection. Thematrices t, tf and r, r^ are the NxN transmission and reflection matricesof the tunnel barrier, or more generally, of the disordered region in thenormal metal. (The number N is related to the cross-sectional areaA of the junction and the Fermi wavelength Xp by N ~ A/Xp.) Thematrices i, r pertain to the electron and the matrices i^, r t to the hole.The resulting reflection probability R^e = N~l Tr rher^e *s given by [14](4.2)We have used the relationship ίί^ + rr^ = l, dictated by current conser-vation. The conductance GNS of the NS junction is related to R^e by[17, 18](4.3)In the optical analogue one has the probability R± for an incident lightwave with frequency ωό + δω to be reflected into a wave with frequencyWQ — δω. The matrix of probability amplitudes is given by the geometricseriestl tl tl l Γ tl]2Jt " · * ' ! * ' « ' · * · ' . " I 'S m l " 'i 1 1 i L i jl Γ II"1= V- l-rir f T t. (4.4)i L i jThe only difference with Eq. (4.1) is the alternation of factors l/i andi, corresponding to the different phase shifts exp(±ivr/2) acquired at thephase-conjugating mirror. The reflection probability R± = N~l Trr±rj.now becomes independent of the disorder [19],N V l - rrt58 C. W. J. BeenakkerThe disordered medium has become completely transparent.It is remarkable that a small difference in phase shifts has such faxreaching consequences. Note that one needs to consider multiple reflec-tions in order to see the difference: The first term in the series is thesame in Eqs. (4.1) and (4.4). That is probably why this essential differ-ence between Andreev reflection and optical phase-conjugation was notnoticed prior to Ref. [19].4. HOW BIG IS THE RESISTANCE?Now that we understand why a disordered piece of metal connected toa superconductor does not become transparent, we would like to go onestep further and ask whether the resistance (or conductance) is biggeror smaller than without the superconductor. To that end we compare,following Ref. [14], the expression for the conductance of the NS junction[obtained from Eqs. (4.2) and (4.3)],with the Landauer formula for the normal-state conductance,η. (4.7)n=lThe numbers TI, T2, . . . T ff are the eigenvalues of the matrix product t t f .These transmission eigenvalues are real numbers between 0 and l thatdepend only on the properties of the metal (regardless of the supercon-ductor). Both formulas (4.6) and (4.7) hold at zero temperature, so wewill be comparing the zero-temperature limits of GNS and GN·Since x2/(2 — x)2 < χ for χ e [0, 1], we can immediately conclude thatGNS < 2GN· If there is no disorder, then all Tn's are equal to unity,hence GNS reaches its maximum value of 2GN- For a tunnel barrier allTn's are <C l, hence GNS drops far below GN- A disordered metal willlie somewhere in between these two extremes, but where?We have already alluded to the answer in the previous section, thatGNS — GN for a disordered metal in the zero-temperature limit. Toderive this remarkable equality, we parameterize the transmission eigen-value Tn in terms of the localization length £n,Tn = 0 * . , (4-8)^Why Does a Metal-Superconductor Junction Have a Resistance? 59where L is the length of the disordered region. Substitution into Eqs.(4.6) and (4.7) gives the average conductances4e2 Γ00(GNS)L = -r-N l dC^L(C)cosh-2(2L/C), (4.9)h JQOp roo(GN)L = -j-N dC^(C)cosh-2(L/C). (4.10)h JQ(For Eq. (4.9) we have used that 2 cosh2 x—1 = cosh 2α;.) The probabilitydistribution PL(() of ζ is independent of L in a ränge of lengths between/ and N l [5]. It then follows immediately that)2L. (4.11)Since GN oc l/£, according to Ohm's law, we arrive at the equality ofGNS and GN·The restriction to the ränge / -C L -C Nl is the restriction to theregime of diffusive transport: For smaller L we enter the ballistic regimeand GNS rises to 2GNJ For larger L we enter the localized regime, wheretunneling takes over from diffusion and GNS becomes <C GN-5. CONCLUSIONWe have learned a fundamental difference between Andreev reflectionof electrons and phase-conjugation of light. While it is appealing to thinkof the Andreev reflected hole äs the time reverse of the incident electron,this picture breaks down upon closer inspection. The phase shift of —π/2acquired upon Andreev reflection spoils the time-reversing propertiesand explains why a disordered metal does not become transparent whenconnected to a superconductor.AcknowledgementsThe 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 derMaterie" (FOM) and by the "Nederlandse organisatie voor Wetenschappelijk Onder-zoek" (NWO).References[1] A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov. Phys.JETP 19, 1228 (1964)].[2] I. O. Kulik, Zh. Eksp. Teor. Fiz. 57, 1745 (1969) [Sov. Phys. JETP30, 944 (1970)].[3] S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, Solid StateComm. 30, 771 (1979).60 G. W. J. Beenakker[4] H. Courtois, P. Charlat, Ph. Gandit, D. Mailly, and B. Pannetier,J. Low Temp. Phys. 116, 187 (1999).[5] C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).[6] D. Lenstra, in: Huygens Principle 1690-1990; Theory and Applica-tions, edited by H. Blök, H. A. Ferweda, and H. K. Kuiken (North-Holland, Amsterdam, 1990).[7] H. van Houten and C. W. J. Beenakker, Physica B 175, 187 (1991).[8] A. A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988).[9] D. M. Pepper, Sei. Am. 254 (1), 56 (1986).[10] J. P. Woerdman, Optics Comm. 2, 212 (1970).[11] B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, Dokl. Akad. NaukUSSR 196, 567 (1971) [Sov. Phys. Dokl. 16, 46 (1971)].[12] J. C. J. Paasschens, Ph.D. Thesis (Universiteit Leiden, 1997).[13] J. Feinberg, Opt. Lett. 7, 486 (1982).[14] C. W. J. Beenakker, Phys. Rev. B 46, 12841 (1992).[15] Yu. V. Nazarov and T. H. Stoof, Phys. Rev. Lett. 76, 823 (1996).[16] E. Toyoda, H. Takayanagi, and H. Nakano, Phys. Rev. B 59, 11653(1999).[17] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25,4515 (1982).[18] Y. Takane and H. Ebisawa, J. Phys. Soc. Japan 61, 3466 (1992).[19] J. C. J. Paasschens, M. J. M. de Jong, P. W. Brouwer, and C. W.J. Beenakker, Phys. Rev. A 56, 4216 (1997).

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Why does a metal-superconductor junction have a resistance?

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