Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 17, No. 5, October 2004 ( C© 2004)

Proximity Effect From an Andreev Perspective

T. M. Klapwijk1

Received August 15, 2004; accepted September 4, 2004

The current understanding of the superconducting proximity effect is reviewed taking intoaccount recent experimental and theoretical results obtained for mesoscopic normal metal-superconductor junctions as well as superconducting weak links. Although known for 40 yearsthe phenomenon remained poorly understood. Current insights are the result of theoreticaldevelopments leading to the nonequilibrium quasiclassical theory, getting experimental ac-cess to proximity structures on a submicron scale as well as by combining it with the knowl-edge developed in the 80s on quantum transport in disordered and ballistic systems.

KEY WORDS: proximity-effect; Andreev reflection; mesoscopic superconductivity; Michael Tinkham.

1. INTRODUCTION

The proximity-effect is a well-known phe-nomenon in superconductivity and part of the stan-dard vocabulary. A summary of early work [l] startswith the following defining line: If a normal metalN is deposited on top of a superconductor S, and ifthe electrical contact between the two is good, Cooperpairs can leak from S to N. In some cases a morespecific analysis has emerged in particular for tunnel-junctions using a NS double-layer as one electrode[2], which has also entered the standard niobiumjunction technology [3]. Nevertheless, despite its fa-miliarity, the subject hardly appears in textbooks onsuperconductivity. In an often-used textbook writ-ten by Tinkham, Introduction to Superconductivity[4] one finds, searching the index for proximity effect,a single sentence: in which Cooper pairs from a su-perconducting metal in close proximity diffuse into thenormal metal. According to the author [private com-munication] a more extensive treatment of the sub-ject is absent because in his view the phenomenonremained poorly understood. Another textbook byWaldram [5] puts the subject in a Chapter titled Fur-ther theory and properties in a paragraph The prox-imity effect. Interestingly, this paragraph is followedby one with the title Andreev reflection although no

1Kavli Institute of Nanoscience, Faculty of Applied Sciences, DelftUniversity Technology, Delft, The Netherlands.

connection is made between the two topics. Due tothe more recent studies of transport in mesoscopicstructures it has now become common wisdom thatAndreev reflection and the proximity effect are inti-mately connected and certainly not two distinct phe-nomena [6]. The most-cited research paper of Tin-kham [7], coauthored with G. E. Blonder and thepresent author, is on the subject of Andreev reflec-tions. In a recent citation-study [8] this article ap-pears as “hot,” meaning that it has been cited at aremarkable rate in the past few years and belongsto a group of which some soon might join the top-100 citation-impact articles. Within this context itseems appropriate to use this celebratory occasion tosummarize some recent work on the proximity-effectfrom an Andreev perspective.

In retrospect a number of developments havecontributed to the present level of understanding.First, in the 80s a substantial body of knowledgehas been developed on quantum coherent transportin disordered normal systems. It has elucidated therole played by the Thouless length and the inelas-tic scattering length. Conductors have been studiedwhich are small compared to those length scales evendown to the Fermi wavelength. The crucial role isplayed by the single particle phase and the corre-lation between electrons in the disordered systems.Secondly, various experiments have been performedwhich probe the penetration of the superconductingproperties on submicron length scales and objects are

593

0896-1107/04/1000-0593/0 C© 2004 Springer Science+Business Media, Inc.

594 Klapwijk

constructed which are smaller than the characteris-tic lengths. Thirdly, various successful attempts havebeen made to bridge the gap between the heavy the-oretical framework of the quasi-classical equationsand the more accessible conceptual framework basedon transmission matrices which assists experimental-ists in designing and interpreting their experiments.

The aim of this contribution is to sketch adevelopment since the original Blonder-Tinkham-Klapwijk article from ballistic Andreev toward dif-fusive normal metal-superconductor systems. On ourway we highlight a few conceptually important ex-perimental results. We apologize that the selectionhas some personal bias. To correct for that extensivereference is made to a number of important recentreview articles. We hope that this contribution canserve as an introduction to the subject to newcomersin the field. We close with a historical note.

2. EARLY KNOWLEDGE

Most of the early knowledge about the prox-imity effect has been summarized by Deutscher andDe Gennes [l] in 1969. It is based on the theoreti-cal treatment developed by De Gennes [9,10] validclose to the critical temperature. In Fig. 1 the resultsare shown for two different NS interfaces and for aso-called SNS contact. On the vertical axis the pair

Fig. 1. Figure taken from DeGennes [9]. Spatial dependence of thepair potential �(x) at temperatures close to the critical tempera-ture in a NS sandwich (a) and in an SNS junction (b). For the NSsandwich two cases, one for a repulsive and one for an attractiveelectron-electron interaction in N are shown (we ignore the f (x)curve).

potential �(�r) is shown defined as:

�(�r) = VNF(�r) = VN〈ψ(�r ↑)ψ(�r ↓)〉 (1)

with F(�r) the condensation amplitude, the probabil-ity amplitude of finding two electrons in the con-densed state at a point �r and VN the electron-electroninteraction constant which may be either negative(repulsive) or positive (attractive). Obviously, Fig. 1shows that �(�r) is expected to have a finite value in Nand is depressed in S. The full normal state value of� = 0 and the full superconducting state of � = �0

are recovered far away from the interface over somecharacteristic length.

The basic experimental facts on a NS interfaceare:

• The observation that the critical tempera-ture of a bilayer of a low Tcl superconduc-tor such as Al and a higher Tch supercon-ductor, such as Pb, gradually increases fromTcl to Tch with increasing thickness of thePb on the Al film. The same experimentcould be done with a fully normal metalsuch as Cu covered by Pb. The effectivenessof the proximity effect could be influencedby adding magnetic impurities to the normalmetal.

• The second observation regards tunneling ex-periments through which one can measurethe excitation spectrum in a normal metal Nbacked by a superconducting film S.

• The third observation is the Meissner effectwhich accounts for the diamagnetic screeningrealized by a “proximitized” normal metal.

An additional body of knowledge results fromthe Josephson-effect. It has already been shown byDe Gennes [9] in 1964 that as a consequence of theproximity-effect a supercurrent should flow througha normal metal of thickness dN. The maximum su-percurrent carried by such a normal metal woulddepend exponentially on thickness with the normalmetal coherence length ξN as the characteristic lengthscale. This prediction has been confirmed in early ex-periments by Clarke [11] including its relationshipto the Josephson-effect [12]. However, unlike theJosephson-effect in tunnel junctions, a detailed com-parison with the microscopic theory has only beenmade recently [13]. This is partially due to lack ofpractical interest. Josephson junctions based on SNSjunctions have a relatively low impedance, far below1 �, and are not very useful in any practical appli-cation where usually 50-� impedances are required.

Proximity Effect From an Andreev Perspective 595

Due to the recent advances in microfabrication-techniques these experimental systems are muchmore accessible.

A more fundamental reason is that the origi-nal proximity-effect theory was much too simple. AnSNS junction forms a problem of inhomogeneous su-perconductors. A current is driven from S to N andvice versa and high-frequency time-dependent pro-cesses may take place. In the early 60s one reliedon the full Gorkov equations or used the simpli-fied version of the Ginzburg-Landau equations onlyvalid close to Tc. Subsequent theoretical develop-ments have provided a much more advanced theoret-ical framework [14]. The equilibrium theory, knownas the quasi-classical equations, was developed byEilenberger [15] and for dirty superconductors byUsadel [16]. In this approach it is assumed that all rel-evant variations occur on a length scale larger thanthe Fermi wavelength. Hence, dependences on thescale of the Fermi wavelength can be averaged out.Usadel simplified the equations further by assum-ing that the system is dirty and impurity-averagedGreen’s functions can be defined.

This approach came to its full maturity in theearly 70s to describe transport problems and time-dependent processes by Larkin and Ovchinnikov[17–19], Eliashberg [20], and Schmid and Schon [21].We will refer to this theoretical approach as the“quasi-classical theory,” also when it includes thenonequilibrium version. In its application to inho-mogeneous problems, such as the proximity-effect,these equations have to be supplemented by suitableboundary conditions. An example are those intro-duced by Zaitsev [22].

Many of the experimental problems studiedrecently can be understood with the help of thequasi-classical equations. However, some of theseproblems, in particular those where one could as-sume, ballistic electron transport could be under-stood more readily by using the Bogoliubov-DeGennes equations and using the concept of An-dreev reflection [23]. The current understanding ofthe proximity effect focuses on the following aspects.At the boundary between a normal metal and a su-perconductor Andreev reflection occurs. In this pro-cess electrons above the Fermi level are convertedinto holes below the Fermi level and a Cooper pairis formed inside the superconductor. This is one cru-cial ingredient of the proximity effect and describeshow the two electronic reservoirs are communicatingat the boundary itself. The second ingredient is howin the normal metal this electron-hole pair looses

its correlated properties. The electron and the holeform phase-conjugated pairs, a property which willbe lost at a certain distance from the interface. Theproximity-effect is now understood as a process atthe interface itself, in essence Andreev reflection,and how this coherence is propagated and lost dueto dephasing processes in the normal metal. The lat-ter are in principle identical to processes known fromphase-coherent transport in normal metals. How-ever, the presence of disorder in the normal metalinfluences also the processes at the boundary itselfbecause incident electron waves may make repeatedattempts which must be added phase-coherently tofind the probability for Andreev reflection. Hence,the proximity effect is on a microscopic level the in-timate connection between Andreev reflection at theinterface and the phase coherence maintained over acertain length scale in the normal metal.

3. BALLISTIC ANDREEV INTERFACE

As a starting point it is apprpriate to recall theresults of Blonder et al. [7] for a ballistic Andreev in-terface. The microscopic description of the supercon-ducting state starts with the Bogoliubov-De Gennesequations [11]. The wave-function ψ is defined as a2-component wavefunction:

ψ=(

u

v

)e

i2Eth

+φs (2)

with u and v representing the electron-like (k > kF)and the hole-like (k < kF) component, E the energywith respect to the Fermi-level, and φs the macro-scopic phase of the superconducting state.

The Bogoliubov-De Gennes equations, writtenin one dimension, are two coupled Schrodinger equa-tions:

Eψ(x) =(

− h2

2md

dx2− µ

)u(x) + �(x)v(x)

Eψ(x) =(

h2

2md

dx2+ µ

)v(x) + �(x)u(x) (3)

with �(x) the pair potential of Eq. (1) and µ thechemical potential, which serves as a reference forthe energy E.

In general Eq. (3) allows for four types of quasi-particle waves for a given energy:

ψ±k+ =(

u

v

)e±ik+x (4)

596 Klapwijk

and

ψ±k− =(

u

v

)e±ik−x (5)

where ψ is the position space representation of theBCS quasiparticles.

We assume that we will analyze a NS interfaceat which plane waves arrive from infinity and returnto infinity. In addition it will be understood that anyreal interface of two dissimilar materials will havedifferent Fermi-momenta, leading to an unavoidableelectronic mismatch. In addition it is very likely thata practical interface has some level of elastic scat-tering due to lattice mismatch or excess impuritiesdue to the fabrication process. The consequence isthat any interface will have some elastic scattering.In addition we will assume that there is no elasticscattering in the superconductor nor in the normalmetal itself and only at the interface. This is mod-elled by a delta-function potential at the interface:V(x) = Hδ(x). The direction of the particle flow isdetermined by the group velocity. At the interfacethe appropriate boundary conditions are continuityof ψ at x = 0 i.e ψS(0) = ψN(0) and in view of theδ function potential h/2m(ψ ′

S(0) − ψ ′N(0)) = Hψ(0),

with ψ ′S,N the spatial derivatives.

With these assumptions we may distinguish anincident plane wave ψinc, a reflected wave ψrefl con-sisting of an electron part due to elastic scattering anda hole part due to interaction with the superconduc-tor, and a transmitted wave ψtrans into the supercon-ductor with two possible solution:

ψinc =(

1

0

)eiq+x

ψrefl = a

(0

1

)eiq−x + b

(1

0

)e−iq+x

ψtransm = c

(u

v

)eik+x + d

(v

u

)e−ik−x (6)

with

hk± =√

2m[µ ± (E2 − �2)1/2]1/2

(7)

and

hq± =√

2m[µ ± E]1/2. (8)

These three waves have to be matched withthe given boundary conditions to determine theamplitudes. Working out the algebra the follow-ing expressions are found for the amplitudes (with

γ = u2 + (u2 − v2)Z2):

a = uvγ

b = − (u2 − v2)(Z2 + iZ)γ

c = u(1 − iZ)γ

d = ivZγ

(9)

In determining Eqs. (9) it is assumed that the nor-mal state Fermi velocities are identical for N and S.A possible difference is absorbed in the adjustableparameter Z, defined as Z ≡ H/hvF. In the absenceof elastic scattering at the interface, Z = 0, we ob-serve that both b and d are zero. Hence, the onlyexisting reflection is Andreev reflection, conversionfrom electron to hole, a condition which can only berealized with two dissimilar materials with identicalnormal state densities of states. The amplitude de-pends on the energy since u and v are given by:

u2 = 1 − v2 = 12

{1 + (E2 − �2)1/2

E

}. (10)

This energy-dependence will be one of the impor-tant ingredients in understanding the relationship be-tween the proximity-effect and Andreev reflection.The interaction at an interface cannot be representedby single parameter �, as in Fig. 1, because the effectsat the interface are different for different energies.

A very important discovery of Blonder et al. [7]was that the properties of a ballistic Andreev inter-face can be measured directly in pointcontact experi-ments. In the current concepts of quantum transporta mesoscopic conductor, a scatterer, is distinguishedfrom the equilibrium reservoirs where thermaliza-tion and phase-randomization occurs. An early ex-ample is the so-called Sharvin pointcontact, in whicha short narrow conductor is connected to two largeelectronic reservoirs. Electrical transport is the dif-ference between electrons originating from a reser-voir at voltage V to a reservoir at voltage V = 0 andvice versa:

I = G0

e

∫ ∞

0dE[f (E) − f (E + eV)] (11)

with f (E) the Fermi-function and G0 a quantitywhich counts the number of modes depending onthe dimensionality of the sample. For 3-dimensionalreservoirs connected by a small hole of area S it is

Proximity Effect From an Andreev Perspective 597

given by:

G0 = 2eh2

kF2 S4π

(12)

with kF the Fermi momentum. Returning to the im-portant case of a short channel of zero length Eq. (12)is now better known as the multichannel Landauerformula developed for mesoscopic conductors:

G0 = 2e2

h

m∑n=0

Tn (13)

with Tn the transmission coefficient per mode n andm the number of modes. Each mode carries a currente/h, the factor 2 takes care of the two different spindirections. Equation (11) follows from the Landauerformula assuming that each mode has a transmissioncoefficient equal to unity, the number of modes fol-lows from the area S compared to kF.

The same reasoning used to derive Eq. (11)leads to the current-voltage characteristics for apointcontact between a normal metal and a super-conductor. The probabilities A, B, C, and D are de-termined from the amplitudes, Eq. (9), and used tocalculate their contribution to the currents for a givenapplied voltage. Hence the current-voltage charac-teristic is given by:

I = G0

e

∫ ∞

0dE[f (E) − f (E + eV)][1 + A(E) − B(E)]

(14)with A the extra current per incident electron wavedue to the Andreev process and B the reflected elec-tron current due to elastic scattering. For low tem-peratures the 1st term in square brackets reduces toa δ-function around eV. Hence, scanning the voltageis also a direct probe of the energy-dependent reflec-tion coefficients:

dIdV

= GNS = G0[1 + A(eV) − B(eV)] (15)

and the conductivity of the pointcontact is a directmeasure of the reflection coeflicients. Note the simi-larity with the conventional expression for a NIS tun-neljunction in which the 2nd term within brackets isreplaced by the BCS density of states, E/

√E2 − �2,

in the superconductor. Indeed this term reduces tothe BCS density of states for large values of Z i.e. forthe tunnelbarrier regime. It is sometimes convenientto re-express the quantity Z in terms of a transmis-sion coefficient for the normal state Tn. Evaluating

Eq. (15) for � = 0 one finds for the conductance:

GN = G0(1 − B(E)) = G01

1 + Z2(16)

which means that Tn is equal to 1/(1 + Z2) and theconductance in the normal state is given by:

GN = G0Tn (17)

for finite Z.In the past 20 years Eq. (14) has been used

extensively to study new superconducting materialswith pointcontacts. From Eq. (14) one finds the su-perconducting energy gap allowing for various in-terface imperfections with the adjustable parameterZ, using the relatively simple technique of a pointcontact. Previously, the superconducting gap had tobe determined by developing a suitable tunnel bar-rier [2]. The discovery of many new superconductingmaterials and the pointcontact technique accountsfor much of the success of Eq. (14). A recent ex-ample is the application to the new superconductorMgB2 with a critical temperature of 38 K [24,25]. InFig. 2 temperature-dependent pointcontact conduc-tance measurements are shown for a single-crystalMgB2 and comparison with Eq. (14) assuming the

Fig. 2. (a) Temperature dependence of the conductance of a c-axisjunction at a magnetic field of 1 T. (b) Comparison of conductanceat B = 0 T and at 1 T. The full lines are fits to Eq. (14). The bot-tom panel shows the two energy gaps as a function of temperature.Taken from Ref. [25].

598 Klapwijk

existence of two gaps in different crystal directions.Note the excellent agreement between data and the-ory over a large temperature range.

A 2nd important extension has been in the fieldof spin polarization. De Jong and Beenakker [26]pointed out that Andreev reflection should be sup-pressed at an interface between a ferromagnet anda superconductor. Since one has in general in a fer-romagnet both spin-up and spin-down channels andAndreev reflection should still be partially possi-ble and can become a measure of the degree ofspin-polarization of a ferromagnet. This was recog-nized and successfully used by Soulen et al. [27] andUpadhyay et al. [28]. A further theoretical basis ofthe technique has been developed by Mazin et al.[29].

Clearly, rather than being an artificial model sys-tem the ballistic Andreev interface can be accesseddirectly by point contact spectroscopy. This applica-tion has turned out be very flexible and useful fora large range of new superconducting materials andalso useful for the determination of the degree ofspin-polarization in ferromagnets or half-metals.

4. SPATIAL EXTENT OF THEANDREEV PROCESSES

In a point contact geometry it is assumed thatthe reservoirs are three dimensional. The Andreevinterface represents a weak link between the reser-voirs. The one-dimensional solution of Eq. (3) canbe used provided translational invariance is assumedin the plane parallel to the interface. In addition thechoice for this geometry made it acceptable to ignorean important equation which in principle should beadded to Eq. (3):

�(�r) = VN(�r)∑E>0

v∗(�r)u(�r)[1 − 2f (E)] (18)

with �(�r) the pair potential which couples thetwo Schrodinger equations. It is called the self-consistency-equation while it is dependent on thesolutions found from Eq. (3). In three-dimensionalreservoirs the induced values for u and v are di-luted by solutions for the unconnected reservoirsand therefore one can ignore the spatial decay of� at both the N and the S-side. In comparison withFig. 1 � is a step-function rising from 0 to a finitebulk-value �0 over a very short distance. To simplifythis problem further one can choose the electron-electron interaction constant VN = 0.

However, let us now return to the hypothet-ical situation of a ballistic system in a true one-dimensional model. The first thing to notice is thatEq. (18) with Eqs. (6) leads to a finite value for �

in N assuming a finite value for the electron-electroninteraction VN �= 0. For each energy E the coherencegets lost over a characteristic decay length hvF/E. Fora given energy E above the Fermi energy EF the wavevector for the electron qe = qF + δk/2 differs slightlyfrom the Fermi wave vector. The reflected hole alsohas a slightly smaller value qh = qF − δk/2. The dif-ference in momentum of δk is given by qF(E/EF)which leads to a dephasing length depending on en-ergy. Equation 18 clearly shows that to determine thespatial extent of the quantity � one needs to knowu and v, which both are energy-dependent and thedistribution-function. Each energy weighs differentlyinto the sum and therefore one cannot define a char-acteristic decay length, except for the limit close tothe critical temperature [9]. The same applies to thesuperconducting side, where a gradual rise of � is ex-pected but again the energy dependence and the oc-cupation numbers will play a role.

Another quantity of interest is the current car-ried by quasiparticles. At a normal metal-supercon-ductor interface current carried by quasiparticles isconverted into current carried by Cooper-pairs. Thecharge current carried by Bogoliubov quasiparticlesis the sum of the part carried by the electron part uand the part carried by the hole part v:

JQ = ehm

[Im(u∗∇u) + Im(v∗∇v)] (19)

For E < � both k+ and k− have small imaginary partswhich lead to an exponential decay of the quasiparti-cle amplitude in the superconductor on a length scaleλ given by:

λ = hvF

2�

[1 −

(E�

)2]1/2

(20)

of which the leading term is hvF/2� ≈ ξ(T). Theseevanescent waves carry quasiparticle current, whichis taken over by an increasing supercurrent of this en-ergy. To determine the full value of the quasiparticlecurrent the fraction carried by each energy should betaken together with the probability of these states tobe occupied.

In closing this section we return to the self-consistency equation. If we assume that in N VN = 0we will have also �(�r) = 0. However, since we findat any point close to the interface a finite value for

Proximity Effect From an Andreev Perspective 599

the amplitudes of electrons and holes, we will obtaina finite value for the condensation amplitude F:

F = 〈ψ(�r ↑)ψ(�r ↓)〉 =∑E>0

v∗(�r)u(�r)[1 − 2f (E)] (21)

which is known as the order parameter or alternati-vely the “Cooper pair density.” Hence, we findthat in the absence of any attractive interactionthere is still a finite probability of finding Cooperpairs in N, which is equivalent to stating thatthe Andreev-reflected electrons and holes maintainphase-coherence over a certain length leading toa finite contribution to F. This is the “leakage ofCooper pairs” alluded to in various textbooks onsuperconductivity.

5. IMPURITY-SCATTERING ANDTRANSMISSION MATRIX APPROACH

In Section 3 the equivalent of a transmissionmatrix for a quantum-coherent scattering problemwas used. A great deal of quantum-coherent trans-port theory has been developed to understand quan-tized conductance, Aharonov-Bohm oscillations anduniversal conductance fluctuations in, for example,metallic rings [30]. In recent years a similar approachhas evolved for NS interfaces in which elastic scatter-ing in N occurs. This approach has been developedand summarized by Beenakker [31,32] and Lambert[33]. To appreciate the nature of the problem we re-call the early work of Van Wees et al. [34].

In Fig. 3 three relevant elements of the geome-try are shown. We will assume that the inelastic scat-tering length Linel is always smaller than the devicesize. Hence, the single particle phase is conserved

Fig. 3. Interface between a superconductor and a normal metalreservoir separated by a diffusive scattering region and an inter-face scatterer (barrier). is the applied magnetic flux.

in crossing the object while being scattered by im-purities. An equilibrium superconductor is separatedfrom a normal metal by an interface barrier as usedin Section 3. In the normal metal a contact is indi-cated, called a reservoir from which waves will emitand which will absorb waves. The middle region is thescattering region. It is assumed that � = 0(VN = 0).An incident electron wave will scatter from the impu-rities and eventually reach the interface with the su-perconductor undergoing Andreev reflection. Sincethe hole wave is essentially phase-conjugate withthe incident electron wave it will retrace the origi-nal path. However, since the incident electron waveis also partially reflected as an electron wave impu-rity scattering might scatter the partial electron waveagain to the superconductor and so forth. The scat-tered wave will be the coherent superposition of thevarious contributions and since for small energy dif-ferences phase coherence is maintained the additionof impurities to the scattering problem of Section 3 isthe enhancement of the Andreev reflection probabil-ity and hence, enhanced conductance (“reflectionlesstunneling” [31]). Depending on the path length andthe magnetic field a total phase shift of

�φ = 2eLhvF

+ 4πBSφ0

(22)

is accumulated with L the path length, B the ap-plied magnetic field, S the enclosed area, and 0

the flux quantum. For increased voltages (increasedenergies) or applied magnetic field phase coherenceis destroyed and the enhanced conductance is sup-pressed. Such a behavior had been first reported byKastalsky et al. [35]. In calculating the actual conduc-tance an energy-average over the available occupiedstates is taken. The correlations of these electronsis determined by the Thouless energy ETh = hD/L2.Or one can define for a particular energy value E adecay length �T = √

hD/E , which is analogous tothe phase-correlation length identified in the previ-ous Section, but now for a dirty system. The corre-lation between an electron and an Andreev-reflectedhole in a diffusive system is given by the normal metalcoherence length ξN = √

hD/kT if kT would be thedominant energy.

For small voltages very simple expressions havebeen derived by Beenakker [31]. For small voltagesthe probabilities of Section 3 A(E) reduces to 1/(1 +2Z2)2 and B(E) to 2(1 + Z2)/(1 + 2Z2)2 and hence

GNS = G02

(1 + 2Z2)2(23)

600 Klapwijk

which by using our previous observation that Tn =1/(1 + Z2) leads to

GNS = G02T2

n

(2 − Tn)2(24)

Subsequently Beenakker has generalized this ex-pression for any arbitrary scatterer such as a meso-scopic phase-coherent conductor containing elasticimpurities:

GNS = 2e2

h

m∑n=1

2T2n

(2 − Tn)2(25)

which is a generalization of the Landauer formula tothe NS case with Tn the transmission coefficients ap-plicable to the particular distribution of transmissionchannels.

Evidently, impurity scattering has a stronginfluence on the Andreev scattering probability andwill also lead to a spatial dependence of the Cooper-pair density. It is the result of phase-coherentscattering processes, which interact with a nearbyphase-coherent superconductor. An interestingextension has been the introduction of so-called An-dreev interferometry. In Fig. 4 an example is shown.A T-shaped phase coherent conductor is connectedat the two ends of the T-bar with a superconductor.The superconductor is part of a loop, which allowsthe application of a well-defined macroscopic phase-difference at both ends of the T-bar. In the Andreevreflection process the macroscopic superconductingphase φs appears. The transmitted wave in Eq. (2)

Fig. 4. At the left hand side a schematic picture of a T-shaped2-dimensional electron gas with an interrupted superconductingniobium loop. The contacts 1 and 2 are connected to the T-shapedconductor and the contacts 3 and 4 to the superconducting nio-bium loop. The indicated dimensions L and W are 0.7 µm and0.3 µm respectively. The right hand side picture shows a scanningelectron microscope image of the actual device. Taken from DenHartog et al. [36]

should read:

ψtransm = c

(ueiφs1

v

)eik+x + d

(veiφs1

u

)e−ik−x (26)

and analogously for the 2nd superconductor withφs1 replaced by φs2. Consequently, the reflectedwave amplitude a and b carry the information ofthe superconducting phase. The diffusive scatteringleads to coherent superposition of wavelets whichinteract with both superconductors and the conduc-tance will depend on their phase-difference φs1 − φs2.Den Hartog et al. [36] observed a conductancewhich depends on the macroscopic phases of thesuperconducting loop, changed by the enclosedmagnetic field. The amplitude of these Aharonov-Bohm oscillations is shown in Fig. 5. With increasingmagnetic field the main contribution decays untila field of about 120 G is reached. Subsequently arandom fluctuating amplitude continues to be visibleup to the highest field measured. These two regimesmark the difference between ensemble-averagedand sample-specific resistance oscillations, whichhave been discussed in detail by Beenakker [31].

Andreev-interferometry has also been used toidentify a phase coherent contribution to the conduc-tance in normal metal-superconductor tunnel junc-tions. For low values of the transmission coeffi-cient the conductance at low voltages or the subgap

Fig. 5. (a) Magnetoresistance R13,24 minus the background re-sistance at T = 50 mK. (b) Autocorrelation function C(�B) ≡<

δR(B)δR(B + �B) > / < δR(B)2 > between 200 and 3000 G forthe Andreev-mediated conductance oscillations of the trace shownin (a) (solid line) and for the fluctuations in the background resis-tance (dashed line). Taken from Den Hartog et al. [36].

Proximity Effect From an Andreev Perspective 601

current in a tunnel junction is small. An Andreevcomponent will be present, although small since itis 2nd order in the transmission coefficient. Hekkingand Nazarov [37] recognized that the probability forthese 2nd order processes would depend on the im-purity scattering in the metals used in the experi-ments. Pothier et al. [38] studied two closely spacedNIS tunneljunctions with the S electrodes connectedby a superconducting loop. They demonstrated thatthe subgap current has a contribution which de-pends on the applied magnetic flux, which thereforecould be identified as the Andreev-contribution, inperfect agreement with the theory of Hekking andNazarov.

6. DIFFUSIVE USADEL-EQUATIONS

The previous Sections have demonstrated that afull understanding of the interactions between a nor-mal metal and a superconductor must face the spa-tial dependence, the energy dependence, the impurityscattering, the dependence on the macroscopic super-conducting phase, and on the occupation-numbers.This theoretical complexity is dealt with in the theoryof nonequilibrium superconductivity based on impu-rity averaged Green’s functions. A relatively accessi-ble introduction has been developed by Esteve et al.[39,40]. Further details of this approach can also befound in Gueron [41] and Anthore [42].

Rather than by using two-component planewaves with u and v, the diffusive superconductingstate is described by the impurity-averaged Greenfunctions introduced by Usadel [16]: gR

S , gAS and gK

S .The first two, gR

S , and gAS describe the equilibrium

states of the system, whereas the third one gKS de-

scribes the occupation of the states. A convenientway to proceed with these quantities, first introducedby Nazarov [43], is the parameterization by two com-plex parameters θ(x, E) and ϕ(x, E), both being po-sition dependent and energy dependent. Each of thefunctions gR

S and gAS is now given by:

gRS =

(cos θ sin θeiϕ

sin θe−iϕ − cos θ

)(27)

and

gAS =

(− cos θ∗ sin θ∗eiϕ∗

sin θ∗e−iϕ∗cos θ∗

)(28)

where θ(x, E) represents the complex superconduct-ing order and ϕ(x, E) is an energy-dependent super-

conducting phase. The third (Keldysh) function gKS is

given by:

gKS = gR

S h − hgAS (29)

with the distribution matrix h defined as:

h =(

1 − 2f e 0

0 2f h − 1

)(30)

with f e and f h the distribution functions for electronsand holes.

Once θ(x, E) and ϕ(x, E) are known a set of ex-pressions can be used to determine several quantitiescharacteristic for the superconducting state. The den-sity of states follows from:

n(x, E) = N(0)Re[cos θ(x, E)] (31)

with N(0) the density of states in the normal state.For a normal metal θ = 0 and for a bulk superconduc-tor sin θBCS(E) = �/

√�2 − E2. As argued by Esteve

et al. [39] Im[sin2 θ] can be interpreted as an effectiveenergy dependent pair density.

Finally, the self-consistency equation for �(x)depends on θ and ϕ according to:

�(x) = N(0)VN

∫ hωD

0dE tanh

E2kBT

Im(sin θ)eiϕ (32)

which can be compared to Eq. (18).For a normal metal-superconductor interface θ

follows from solving the Usadel-equation:

hD2

∇2θ +(

iE − hτsf

cosθ)

sin θ + �(x)cos θ = 0

(33)

allowing for spin-flip scattering. In N it is assumedthat � = 0 with no electron-electron interaction,whereas in S �(x) is given by Eq. (32).

6.1. Induced Local Density of States

This analysis has been performed by Gueronet al. [44] and applied to experimental results.Figure 6 shows a superconducting wire (aluminium)connected to a normal wire (copper). The contactslabeled F1, F2, and F3 are tunnelcontacts, which en-able the measurement of the local density of states ofEq. (31).

Experimental results are shown in the upperpanel of Fig. 7. The inset shows, for reference, astandard BCS density of states measurement. In thelower panel the results from the Usadel theory are

602 Klapwijk

Fig. 6. SEM photograph of a sample used by Gueron et al. [44]to measure the local density of states in a normal metal in con-tact with a superconductor. F1, F2, and F3 are tunnel junctions tomeasure the density of states.

shown. These results clearly show how the pairing-correlations manifest themselves over a length ofabout 1 µm and are visible in the density of states.In this experiment, in which equilibrium propertiesare measured the agreement between theory and ex-periment is strikingly good.

Fig. 7. Differential conductance of the tunnel junctions at the lo-cations F1, F2, and F3. The inset shows the differential conduc-tance of a reference tunnel junction on a BCS superconductor.The lower panel shows the calculated density of states, assuminga small degree of spin flip scattering in the normal metal. Takenfrom Gueron et al. [44].

Fig. 8. Conductance of a normal metal wire in direct contact witha superconductor. Upon approaching T = 0 the full normal resis-tance is reestablished. The inset shows the actual sample. Takenfrom Ref. [45].

6.2. Reentrant Resistance

The position-dependent density of states impliesthat the normal metal wire has properties reminis-cent of a superconductor. It is natural to expect thata normal wire in contact with a superconductor willwith decreasing temperature become gradually lessresistive. An example of such an experiment is shownin Fig. 8, which clearly shows counterintuitive non-monotonous behavior. A short normal metal wireof copper is attached to the superconductor alu-minium. At the Tc of aluminium the conductance in-creases sharply reaching a maximum around 400 mKand then decreasing again. This reentrant resistiv-ity was first pointed out by Artemenko et al. [46]and more recently treated by Stoof and Nazarov[47,48].

Transport is a nonequilibrium problem, whichmeans that the occupation-numbers f e and f h needto be determined. In terms of θ(θ1 and θ2 are the realand imaginary part, respectively) the following equa-tions need to be solved:

∇[cos2θ1∇f odd] = 0 (34)

and

D∇[cosh2(θ2)∇f even] − 2δ f even Re[sin θ] = 0 (35)

in which f even and f odd represent the symmetric andasymmetric parts of the distribution-function. The2nd term in Eq. (35) is only present in a superconduc-tor and is needed when there is conversion from nor-mal current to supercurrent (Section 6.3). In N the

Proximity Effect From an Andreev Perspective 603

normal current is given by:

JN(x) = σ

e

∫ ∞

0dE∇f even cosh2(θ2) (36)

which determines the conductivity for a given volt-age. The result of such a calculation is shown in Fig. 8.

6.3. Resistance of the SuperconductorNear the Interface

In Section 4 the decay of quasiparticle waves ina superconductor appeared. These evanescent wavescan scatter from elastic impurities and contribute toa finite resistance of the superconducting wire. Wehave recently studied this resistive contribution [49].We chose to study samples (Fig. 9) made of super-conducting (S) aluminium (Al) because of its longcoherence length. Thick and wide normal (N) con-tacts are used with a negligible contribution to thenormal state resistance. To minimize interface resis-tances due to electronic mismatch of both materi-als bilayers of aluminium covered with thick normalmetal (Cu) are used. In such a geometry the super-conducting aluminium wire is directly connected tonormal aluminium.

Figure 10 shows typical R – T measurements.Samples with different RRR values show qualita-tively identical behavior. All wires show a finite re-maining resistance down to low temperatures as themost striking result. Similar results have been ob-tained by Siddiqi et al. [50] in studying hot-electronbolometers. Obviously, despite the differences in

Fig. 9. SEM picture of a device (slightly misaligned), showing thecoverage of the thin aluminium film with the thick Cu layer (exceptfor one the measured devices are carefully lined up). The insetshows a schematic picture of an ideal device.

Fig. 10. Measured R – T curves for four different bridge lengths.The intrinsic Tc0 is indicated by the vertical dashed line. The in-set shows the measured critical temperature of the wire vs. 1/L2,which is used to determine Tc0 by letting L → ∞.

length, the resistances at low temperatures haveidentical values and follow the same trace. It indi-cates that the origin of this remaining resistance isdue to the region in the S-wire attached to the in-terface with the normal reservoir. The resistance at600 mK is equal to a normal segment of the supercon-ductor with a length of about 200 nm. Given the resis-tivity of the superconductor the coherence length ξ =√

(hD)/(2πkBTc) = 124 nm. Figure 10 (Inset) alsoshows that the critical temperature of the wire de-creases linearly with increasing the inverse square ofthe wire length, as expected from a straightforwardGinzburg-Landau analysis.

Since the studied nanowires show diffusivetransport the Usadel equations should apply to thesystem. It is convenient to calculate the normal cur-rent for a given applied voltage difference (assuminglinear response). The strength of the pairing interac-tion, the proximity angle θ is determined by solvingEq. (33) together with the self-consistency equationEq. (32). The density of states as a function of posi-tion is determined with Eq. (31).

Our main interest is the question how the cur-rent conversion process contributes to the resistance.First of all, the presence of decaying normal electronstates suppresses the gap in the density of states.

In Fig. 11, the calculated density-of-statesN1(E) = n(E) is shown given for several positionsalong the wire of L = 2 µm, D = 160 cm2/s, �0 =192 µeV, and T/Tc = 0.4. Clearly, moving away fromthe normal contacts the density of states resem-bles more and more the well-known BCS densityof states. Note however, that a finite subgap value

604 Klapwijk

Fig. 11. The calculated density-of-states N1 at various distancesfrom the reservoirs (x = 100, 200, 300, 400, 500, 1000 nm) for a 2-µm long wire (t = 04, D = 160 cm2/s and �0 = 192 µeV). Notethe exponentially small but finite subgap density-of-states in themiddle (at x = 1 µm; see inset).

remains in the middle (x = 1 µm) even for very longwires. This is an intrinsic result for any NSN system.

The normal current in S is found using Eq. (36)with the distribution function determined from theBoltzmann-like Usadel equation Eq. (35). The ap-plied voltage V, is taken into account via the bound-ary conditions for Eq. (35):

δ f (0, L; E) = ±eV/2

4kBT cosh2(E/2kBT). (37)

Hence, the normal metal leads are taken as equilib-rium reservoirs.

The local voltage is calculated using:

eV(x) =∫ L

0dEf even(E, x)Re[cos θ(x, E)] (38)

In Fig. 12, we show the results of such a cal-culation as a function of position along the wirefor two different temperatures: t = 0.4. and t = 0.9with D = 160 cm2/s, and �0 = 192 µeV. At the tem-perature close to the transition temperature, theelectric field penetrates the sample completely andthe resistance is close to the normal state value.At low temperatures, the electric field still pene-trates the superconductor over a finite length, leav-ing a middle piece with hardly any voltage drop.The penetration length is of the order of the co-herence length. The inset shows the position de-pendent normal currents (full line) and supercur-rents (dashed line) illustrating the current conversionprocesses.

Fig. 12. The voltage in the superconducting wire as a functionof position for two different temperatures (t = 0.4 and 0.9). Att = 0.9 the wire behaves as a normal metal and for t = 0.4 thevoltage is clearly present to a depth ξ (wire length 2 µm withD = 160 cm2/s and �0 = 192 µeV. The inset shows the positiondependent normal currents and supercurrents.

In Fig. 13, a comparison is made between the cal-culated resistance as a function of temperature andthe measurement for a L = 2 µm wire. The calcula-tion is done with parameters D = 160 cm2/s, as de-termined from the impurity resistivity, and �(0) =1.764 kBTc = 192 µeV with Tc = 1.26 K determinedfrom the length dependence. Without any fitting pa-rameter, the agreement between the model (dots)and the experiment (data points: open symbols) is

Fig. 13. The measured R – T curve for the 2-µm bridge togetherwith the model calculation using the boundary condition θ(x =0, L) = 0 (dots), and using the boundary condition ∂xθ(x = 0, L) =θ(x = 0, L)/a (triangles) with a = 75 nm. The value for T = 0 isfound from Eq. (8) to be 0.323 and numerically 0.3255 for the hardboundary conditions(dots). For the soft boundary conditions wehave numerically 0.2609 (triangles).

Proximity Effect From an Andreev Perspective 605

encouragingly good. Only at the lower tempera-tures the observed resistance is slightly less thanthe theoretically predicted values. The most likelycause is that the rigid boundary conditions imposedat the interfaces should be relaxed. There is a fi-nite possibility for superconducting correlations toextend into the normal metal reservoirs, which wouldmean that the boundary condition θ(x = 0, L) = 0 istoo rigid. Since the correlations will extend into a3-dimensional volume we assume that using theboundary conditions ∂xθ(x = 0, L) = θ(x = 0, L)/ai.e. a decay over a fixed characteristic length a, is arealistic assumption. It assumes a geometric dilutionof the correlations. The result is shown in Fig. 13by filled triangles. The best agreement between mea-surement and calculation model is obtained for a =75 nm. This value appears reasonable for a decaylength since it is comparable to the dimensions (100nm × 250 nm) of the wire which emits into thereservoirs.

7. BALLISTIC SUPERCURRENTS

In Section 2 it was pointed out that DeGennes[9] in 1964 recognized that a Josephson supercur-rent could flow through a normal metal sandwichedbetween two superconductors. The strength of theJosephson supercurrent was found to be exponen-tially dependent on the length reflecting the fi-nite correlation length for penetrating Cooper pairs.In some of the experiments described above twosuperconductors were used to perform Andreev-interferometry and by doing that it was demonstratedthat the normal conductance has a component whichis phase-dependent. However, a phase-coherentcomponent should be the Josephson-current. Appar-ently the normal conductance discussed so far canbe studied without having to consider the supercur-rent. The reason is that the characteristic length LT

plays its role differently. The supercurrent decays ex-ponentially with LT, whereas the conductance is pro-portional to LT. If the length of the normal domainis short enough the supercurrent will appear. Forlarger lengths only the phase-coherent normal con-ductance is observable. Experimentally, this depen-dence has been clearly demonstrated in experimentsby Dimoulas et al. [51] and in subsequent work.

In the spirit of Section 3 the supercurrentthrough SNS systems has been studied by start-ing with the concept of Andreev reflections. Theearly work was based on a normal domain free ofany elastic scattering, a condition reachable in cur-

rent 2-dimensional semiconductors. The early workwas done by Kulik [52], Ishii [53], and Bardeenand Johnson [54]. It is assumed that we have per-fect interfaces between N and S leading to perfectAndreev-reflection at the interfaces i.e. determinedby Eq. (9) and Eq. (26) with Z = 0. For simplicity aone-dimensional system is assumed. Because of thefinite length of N discrete energy levels are formed.

For E < � the continuity conditions for thewavefunctions and their derivatives at x = ±L/2 weobtain the dispersion relation:

exp(2iα(E)) exp[i(k+ − k−)L] exp(±iφ) = 1 (39)

where k+ and k− are the wave vectors of the elec-tron and the hole respectively. The phase-factorφ is equal to the differences of the macroscopicphases of the two superconductors (φs1 − φs2). Theenergy-dependent phase factor α(E) is given byarccos(E/�). Since the considered energy E is usu-ally much less than the Fermi-energy, the energyeigenvalues can be calculated by using the relationhδk = h(k+ − k−) = 2E/vF. Under this condition thedispersion relation becomes:

En± = hvF

2L[2(πn + α) ± φ], n = 0, 1, 2, . . . (40)

If we specify for the case of low energies E � � wefind:

En± = hvF

2L

[2π

(n + 1

2

)± φ

](41)

Hence we have a set of equally spaced energylevels until the levels approach �. Note that the po-sition of the energy level is set by the phase differ-ence φ.

In recent years this approach has been exten-sively studied for the limit of very short contacts. Theavailability of mechanical break junctions has led tovery detailed studies of both the supercurrent as wellas the voltage carrying state in single-atom point con-tacts [55]. These junctions consist of adjustable pointcontacts in which the distance between the outermostatoms can be adjusted with a precision in the orderof 0.001 A. It has therefore become feasible to studycontinuously the transition from tunneling to metal-lic contact in superconducting junctions. These ballis-tic contacts are short compared to ξ0 = hvF/�0. Sincewe are weakly connecting two bulk superconductorsthe contact-region itself can be viewed as a normalmetal part. Hence, the dispersion relation (Eq. (39))has only one solution within the bound E < �:

α = arccos(E/�) = ±(φ/2) (42)

606 Klapwijk

which leads to the very simple relation that

E = ±� cos(φ/2) (43)

All these expressions have been derived for the casethat elastic scattering is negligible. Beenakker [56]has extended this approach to the case with finitetransmission eigenvalues. He finds that the energylevels are given by:

En = �[1 − Tn sin2(φ/2)]1/2 (44)

with n = 1, 2, . . . m the number of allowable modes.The values for Tn might range between 1 and 0.

The most important aspect of these expressionsin the various limits is that the allowed energy lev-els depend on the macroscopic phase difference be-tween the two superconductors. The position of thelevels mediate the information about the differentphases of the two superconductors and form the coreof the Josephson-effect.

Despite their importance a direct experimen-tal demonstration of these energy-levels is difficultto provide. Several unpublished attempts have beenmade using far-infrared spectroscopy. Alternativelyone would like to use a tunneljunction as a spec-troscopic tool. The need for ballistic transport ispractically incompatible with needed geometrical ex-tension to fabricate a tunneljunction on top of the de-vice. Morpurgo et al. [57] have been able to demon-strate the precursor of the formation of an Andreevbound state.

Each energy level carries a contribution to thecurrent of evF/m either in positive or negative direc-tion. For φ = 0 there is an equal number of left andright-movers and the net current is zero. However, ifwe now allow a phase-difference between the two su-perconductors the energy of the levels changes. Somemove upwards and others move downwards. For agiven temperature these energy levels are occupieddetermined by the Fermi-Dirac distribution function.Hence, some levels move out of the occupied rangeof energies others move deeper inwards. As a conse-quence a net supercurrent flows in either the positiveor the negative direction dependent on the appliedphase-difference. The supercurrent carried by thesebound states is determined by the contribution of theelectron- and the hole-component:

I =m∑

n=0

[I+(En)f +(En) + I−(En)f −(En)] (45)

with I+ and I− the contribution of the hole and theelectron respectively, which are identical with op-

posite sign. In addition by definition f +(En) = 1 −f −(En), which leads to:

I =m∑

n=0

I(En)[1 − 2f (En)] (46)

The charge current carried by the allowable energystates follow from the probability-current-density:

Jq = hm

Im(u∗(r)∇u(r) − v∗(r)∇v(r)) (47)

Using this expression one finds for the supercur-rent, taking the equilibrium Fermi-functions for theoccupations:

I(φ) = 2eh

m∑n=0

∂En

∂φtanh

En

2kT(48)

which for a short junction with a transmission coeffi-cient 1 using Eq. (43) we find:

I(φ) = eh

N�(T) sin (φ/2) tanh�(T)cos(φ/2)

2kT(49)

Knowing the number of modes one also knows thenormal state conductance, the Sharvin-conductanceG0 = N(e2/πh), which leads to the expression firstderived by Kulik and Omelyanchuk [58]:

I(φ) = πG0�(T)

esin (φ/2) tanh

�(T)cos(φ/2)2kT

(50)

In the same spirit one finds [58] for systems witha finite transmission coefficient using Eq. (44) thatthe supercurrent is governed by:

I(φ) = e�2

2hsin(φ)

m∑n=0

Tn

Entanh

En

2kT(51)

In many cases the transmission coefficients areenergy dependent. For the well-known case of a tun-neljunction we have a single transmissioncoefficientfor all energies and one obtains En = � and hence:

I(φ) = πG0

4esin φ tanh

�

2kT(52)

the well-known expression first derived byAmbegaokar and Baratoff [59] with G0 = e2NTn.

This set of equations clearly shows that start-ing with the Andreev bound states one obtains well-known expressions for the supercurrent carried bythese systems. It also shows that the total supercur-rent is determined by the relative contribution of theelectron and the hole-like component.

Proximity Effect From an Andreev Perspective 607

Fig. 14. Two SNS junctions in parallel. Each SNS junction canbe individually controlled with a normal control wire in which anonequilibrium electron distribution can be realized. Taken fromBaselmans et al. [61].

8. DIFFUSIVE AND CONTROLLABLESUPERCURRENTS

An important ingredient of the supercurrent-carrying capacity appearing in Eq. (46) is thedependence on the distribution-function f(E). It con-trols the temperature dependence and it might en-able the control through a nonequilibrium distribu-tion function [60]. Figure 14 shows a recently studiedsample lay-out consisting of two SNS junctions inparallel in which each of them can be individuallycontrolled with a nonequilibrium distribution func-tion [61,62]. In earlier work a thermal control was re-alized with hot electrons by Morpurgo et al. [60]. Thisled Wilhelm et al. [63] to a nonequilibrium analysis,using a nonequilibrium distribution as measured byPothier et al. [64].

In a diffusive system the proximity-effect isagain expressed in the two parameters θ and φ intro-duced in Section 6. The supercurrent is given by:

�Js(x) = σ

e

∫ ∞

0dE[1 − 2f (E)]Im(sin2 θ(∇ϕ)) (53)

By solving the Usadel equations for θ and φ withthe appropriate boundary conditions at the interfacesone finds �Js. This program has been carried out byWilhelm et al. [63] in which a supercurrent-carrying

Fig. 15. Supercurrent-carrying density of states and distributionfunction. Left panel for thermal distribution and right panel fora steplike nonequilibrium function. The shaded area indicates thecontribution to the integral Eq. (54).

density of states is introduced Im(j E).

Is = d2Rd

∫ +∞

−∞dE[1 − 2f (E)]Im( jE) (54)

It means that for a dirty superconductor the energydependent spectral current Im(j E) replaces the dis-crete Andreev bound states of Eq. (46). It is also de-pendent on the phase difference φ. For φ = 0 thereis a minigap Eg given by Eg �= 3.2ETh with ETh theThouless energy associated with the length of N. Thisenergy-dependent current rises sharply at this mini-gap and oscillates for increasing energy around zerowith an exponentially decaying amplitude (Fig. 15).For increasing phase-difference the mini-gap closesfor φ = π. Evidently one way to vary the supercur-rent is to increase the temperature, which is modeledby replacing 1 − 2f (E) by tanh(E/2kT).

Wilhelm et al. [63] predicted that a suitably cho-sen distribution function would lead to a reversal ofthe direction of the supercurrent. In essence the left-moving parts of the contribution to the supercurrentweigh more heavily in the integral than the right-moving parts as illustrated in Fig. 15. This rever-sal of the direction of the supercurrent was demon-strated by Baselmans et al. [65]. The reversal of thesupercurrent implies a shift of the phase-differenceby π and the device is therefore called a control-lable π-junction. A detailed comparison of exper-iment and theory [66] is shown in Fig. 16. As afunction of control voltage (dots) the supercurrent-amplitude decreases, passes through zero, and con-tinues with a small negative amplitude. The Insetshows the distribution-functions assumed in the com-parison with theory. This experiment clearly illus-trates the importance of the distribution-function incontrolling the superconducting state in N.

608 Klapwijk

Fig. 16. Reversal of the direction of the supercurrent induced by anonequilibrium distribution function and comparison with theory.The inset shows the two-step distribution-functions used in thefits.

9. CONCLUDING HISTORICAL NOTE

The examples shown above illustrate the intri-cate interplay between energies, correlation-lengths,phase-dependence, and distribution-functions. Col-lectively they constitute the relationship betweenthe proximity-effect and Andreev-reflections. For methis interplay started in 1980 when I was a postdoc inMike Tinkham’s group.

When I arrived in Tinkham’s group in 1979 hehad just returned from a sabbatical in Karlsruhe.During this sabbatical he had also travelled toMoscow to visit the Institute of Radio Engineer-ing and Electronics (IREE), in particular AnatolyVolkov, Sergey Artemenko, and Sacha Zaitsev.From this visit he had picked up the understand-ing that in their difficult to digest theoretical workAndreev reflection was an important concept tounderstand constrictions between superconductorsand between superconductors and normal metals.In Josephson contacts the voltage-carrying statewas mostly dominated by time-averaged quantitiessuch as in the celebrated resistively-shunted-junction(RSJ) model. However, the fact that quantities likean excess current appeared in both S-c-S as well asS-c-N contacts was attributed to a static process likeAndreev reflection. Tinkham had written a note fordiscussion in which he summarized his thinking onthis subject by starting with charge imbalance andhow that would be changed at lower temperatures.

My experimental work at Harvard was ini-tially directed to make and study one of the firstJosephson-arrays, in collaboration with Chris Lobb,

to mimic the Kosterlitz-Thouless phase transition. Itwas to be based on SNS junctions using a commer-cially available mesh as a shadow mask [67]. As afamily we kept a diary during our stay in the US toremember the responses of our kids to the new en-vironment. In this diary I find that Chris reports onour work in a Tinkham-group seminar on January29, 1980. But more importantly I write for the firsttime about the “bouncing ball picture.” That day Ihad been chatting with Greg Blonder who was try-ing to measure niobium point contacts as a func-tion of temperature. And again, as in so many pre-vious experiments including my own thesis work onmicrobridges, a phenomenon called “subharmonicgap structure” popped up abundantly. It had beenreported extensively since at least 1964 and wasnever conclusively explained. In retrospect multipar-ticle tunnelling was the correct track but there wereno experimental systems which allowed us to studythis phenomenon in sufficient isolation, neither wasthe theory sufficiently developed. This experimentalproblem was eventually resolved when the mechani-cal break junctions [55] became available, which alsostimulated a thorough theoretical treatment whichstill continues to the present day [68].

During my ride home on the Harvard ExpressBus to Lexington on the 29th I realized suddenlythat with two superconductors connected by a lit-tle Sharvin-like hole Andreev-reflection should takeplace at both superconductors and electron/hole par-ticles should bounce back and forth picking up thevoltage difference at each passage. With our kids Ishared my enthusiasm over dinner referring to ballschanging color upon reflection of the walls and ris-ing higher and higher until they would go over theroof. The next day on the 30th the discussion withGreg started followed by a discussion over lunch withMike, both of them swallowing the elementary idearather quickly.

We worked out the idea in more details by mak-ing lots of pictures illustrating various bias condi-tions and illustrating what would happen for 2�/n.It was obvious that the transition from higher orderto lower order multiple transitions was at the heartof the observed subharmonic gap structure. Mostsatisfying was the application to asymmetric junc-tions, contacts between superconductors with differ-ent gaps. The experiments had shown a peculiar se-ries of structure at the large gap, the sum gap andonly the even series of the smaller gap. The picturesclearly showed that this was naturally explained with

Proximity Effect From an Andreev Perspective 609

the concept of multiple Andreev reflections. This wascompletely convincing even without any mathemati-cal formulation.

It was our hope to get a paper written before myreturn to The Netherlands, early July 1980. Severalattempts were made, mostly descriptive and unsatis-fying. Computer simulations with an ad hoc reflec-tion coefficient at the interfaces looked encouragingbut we were not sure how much of the Josephson-effect one could throw out and the concept of dif-ferent chemical potentials associated with the chargeimbalance was still confusing us. We had been gradu-ally looking more and more into the Bogoliubov-DeGennes equations, which had been the vehicle to in-troduce the concept of Andreev reflections back in1964. We realized that we were using a reflection-coefficient for number current and not for chargecurrent.

A note was written by Greg Blonder, datedAug. 20, 1980, which summarized the relevant as-pects under the title A Bogoliubov Equation Primer.To this note he added an Appendix where he showedhow the matching of the wave functions at an in-terface with a δ-function worked (Section 3). Thiswas used to perform a computercalculation using thebouncing-ball picture, which we subsequently calledthe “trajectory method.” Everything worked fine andbeautiful, except for the fact that the subharmonicgap structure disappeared at low temperatures. Inthe computer calculations the reflection-coefficientswithout the δ-function were used and therefore therewas no cut-off for higher order processes. All ordersweighed equally heavy. But the δ-function was intro-duced to find some mechanism to cut off these higherorder processes. Since we modelled the SS contactwith S-c-S with c a normal metal like constrictionwe had two δ-functions at each interface. With thetrajectory method this led to the impossible task tokeep track of multiple elastic scattering paths andmultiple Andreev scattering paths. So we were stuckexcept that we were able to correctly describe in itsfull glory the NS contact with the trajectory methodand also the SS contacts provided we were willing toaccept that there was no cut-off for higher order pro-cesses. Having been impressed by the depth of theRussian theoretical work we were extremely pleasedto obtain for NS contacts exactly the same result asZaitsev had published in the course of 1980, even be-ing able to point out that his expressions must containa mistake. Moreover we were able to describe the NScontacts with an arbitrary transmission coefficient, an

accomplishment which Zaitsev [22] stimulated to de-velop his boundary conditions.

These results were included in a submitted con-tribution to LT16, in early 1981. The contribution gotupgraded to an invited talk held during this confer-ence in Los Angeles (Aug. 19–25, 1981) and pub-lished in the Proceedings [69]. On Oct. 19, 1981 theBTK paper was submitted. In parallel Mike devel-oped the Boltzmann equation approach to attack theSNS problem with elastic scattering at the interfacesfor which the trajectory method had turned out tobe too clumsy. The original note on this issue wasdated Feb. 18, 1981. It was further worked out withthe help of Miguel Octavio and submitted on Feb. 18,1983.

We believed that we had solved the last remain-ing puzzle in understanding superconducting con-tacts and that the subject was now closed. In retro-spect we did not anticipate that the BTK paper wouldbecome so popular because of pointcontact spec-troscopy of superconducting and magnetic materials.We neither did realize that it was an early example ofdealing with quantum transport problems in the spiritof the Landauer-Buttiker formalism [31]. We alsodid not realize that the mechanical break junctions[55] would provide such rich and detailed experimen-tal data, which stimulated a lot of theoretical workand which also clearly connected the multiple An-dreev concept with multiparticle tunneling. The ideaof multiple charge quanta have found strong sup-port through recent shot noise measruements [70,71].In view of the historical account in which exper-imental results on subharmonic gap structure be-tween dissimilar superconductors played such an en-couraging role it would be nice to see experimentswith controllable break junctions of two dissimilarsuperconductors.

ACKNOWLEDGMENTS

The author likes to express his gratitude toMichael Tinkham for his very personal style of do-ing experimental research in superconductivity witha sharp eye for the essential physics. In additionhe acknowledges the stimulating interactions withstudents and collaborators: J. J. A. Baselmans, W.Belzig, G. E. Blonder, G. R. Boogaard, A. Dimoulas,S. G. den Hartog, J. P. Heida T. T. Heikkilla,D. R. Heslinga, W. M. van Huffelen, A. T. Filip, R.S. Keizer, P. H. C. Magnee, A. F. Morpurgo, Yu. V.

610 Klapwijk

Nazarov, M. Octavio, S. Russo, A. H. Verbruggen,A. F. Volkov, B. J. van Wees, F. K. Wilhelm, andA. V. Zaitsev. Financial support has been providedby the Nederlandse Organisatie voor Wetenschap-pelijk Onderzoek (NWO) through the Stichting voorFundamenteel Onderzoek der Materie (FOM).

REFERENCES

1. G. Deutscher and P. G. de Gennes, in Superconductivity,R. D. Parks, ed. (Marcel Dekker, New York, 1969), Vol. 2,p. 1005.

2. E. L. Wolf and G. B. Arnold, Phys. Rep. 91, 31 (1982).3. M. Gurvitch, M. A. Washington, and H. A. Huggins, Appl.

Phys. Lett. 42, 472 (1983).4. M. Tinkham, Introduction to Superconductivity (McGraw-Hill,

New York, 1996).5. J. R. Waldram, Superconductivity of Metals and Cuprates

(Institute of Physics Publishing, London, 1996).6. B. Pannetier and H. Courtois, J. Low Temp. Phys. 118, 599

(2000).7. G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B

25, 4515 (1982).8. S. Redner, Citation Statistics From More Than a Century of

Physical Review, physics/040713/v1, July 27, 2004.9. P. G. de Gennes, Rev. Mod. Phys. 36, 225 (1964).

10. P. G. de Gennes, Superconductivity of Metals and Alloys(Benjamin, New York, 1966).

11. J. Clarke, Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 308, 447(1969).

12. J. Clarke, Phys. Rev. B 4, 2963 (1971).13. P. Dubos, H. Courtois, B. Pannetier, F. K. Wilhelm, A. D.

Zaikin, and G. Schon, Phys. Rev. B 63, 064502 (2001).14. J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).15. G. Eilenberger, Z. Phys. 214, 195 (1968).16. K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).17. A. ILarkin and Yu. N. Ovchinnikov, Zh. Eksp. Theor. Fiz. 55,

2262 (1968); [Sov. Phys. JETP 28, 1200 (1969)].18. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Theor. Fiz. 68,

1915 (1975); [Sov. Phys. JETP 41, 960 (1975)].19. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Eksp. Theor. Fiz. 73,

299 (1977); [Sov. Phys. JETP 46, 155 (1977)].20. G. M. Eliashberg, Zh. Eksp. Thoer. Fiz. 61, 1254 (1971); [Sov.

Phys. JETP 34, 668 (1972)].21. A. Schmid and G. Schon, J. Low Temp. Phys. 20, 207 (1975).22. A. V. Zaitsev, Zh. Eksp. Theor. Fiz. 86, 1742 (1984); [Sov.

Phys. JETP 59, 1015 (1984)].23. A. F. Andreev, Zh. Eksp. Theor. Fiz. 46, 1823 (1964); [Sov.

Phys. JETP 19, 1228 (1964)].24. R. S. Gonnelli, D. Daghero, G. A. Ummarino, V. A. Stepanov,

J. Jun, S. M. Kazakov, and J. Karpinski, Phys. Rev. Lett. 89,247004 (2002).

25. P. Szabo, P. Samuely, J. Kacmarcik, T. Klein, J. Marcu, D.Fruchart, S. Miraglia, C. Marcenat, A. G. M. Jansen, Phys.Rev. Lett. 87, 137005 (2001).

26. M. J. M. de Jong and C. W. J. Beenakker, Phys. Rev. Lett. 74,1657 (1995).

27. R. J. Soulen, Jr., J. M. Byers, M. S. Osofsky, B. Nadgorny,T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J.Nowak, J. S. Moodera, A. barry, and J. M. D. Coey, Science282, 85 (1998).

28. S. K. Updahyay, A. Palanisami, R. N. Louie, and R. A.Buhrman, Phys. Rev. Lett. 81, 3247 (1998).

29. I. I. Mazin, A. A. Golubov, and B. Nadgorny, J. Appl. Phys.89, 7576 (2001).

30. Y. Imry, Introduction to Mesoscopic Physics (OxfordUniversity Press, Oxford, 2002).

31. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).32. C. W. J. Beenakker, Andreev Billiards, cond-mat/0406018.33. C. J. Lambert and R. Raimondi, J. Phys.: Condens. Matter 10,

901 (1998).34. B. J. van Wees, P. de Vries, P. Magnee, and T. M. Klapwijk,

Phys. Rev. Lett. 69, 510 (1992).35. A. Kastalsky, A. W. Kleinsasser, L. H. Greene, R. Bhat, F. P.

Milliken, and J. P. Harbison, Phys. Rev. Lett. 67, 3026 (1991).36. S. G. Den Hartog, C. M. A. Kapteyn, B. J. van Wees, and T.

M. Klapwijk, Phys. Rev. Lett. 76, 4592 (1996).37. F. W. J. Hekking and Yu. V. Nazarov, Phys. Rev. Lett. 71, 1625

(1993).38. H. Pothier, S. Gueron, D.Esteve, and M. H. Devoret, Phys.

Rev. Lett. 73, 2488 (1994).39. D. Esteve, H. Pothier, S. Gueron, Norman O. Birge, and M. H.

Devoret, in L. L. Sohn et al. (eds.), Introduction to MesoscopicElectron Transport (Kluwer Academic, Amsterdam, 1997).

40. D. Esteve, H. Pothier, S. Gueron, Norman O. Birge, andM. H. Devoret, in Correlated Fermions and Transport in Meso-scopic Systems, T. Martin, G. Montambaux, and J. Tran Van,eds. (Editions Frontieres, Paris, 1996).

41. S. Gueron, Quasiparticles in a Diffusive Conductor: Interactionand Pairing, Thesis, CEA-Saclay, 1997.

42. A. Anthore, Decoherence Mechanisms in Mesoscopic Conduc-tors, Thesis, CEA-Saclay, 2003.

43. Yu. N. Nazarov, Phys. Rev. Lett. 73, 1420 (1994).44. S. Gueron, H. Pothier, N. O. Birge, D. Esteve, and

M. H. Devoret, Phys. Rev. Lett. 96, 3025 (1996).45. H. Courtois, Ph. Gandit, B. Pannetier, and D. Mailly, Super-

lattices Microstruct. 25, 721 (1999).46. S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, Solid State

Commun. 30, 771 (1979).47. Yu. V. Nazarov and T. H. Stoof, Phys. Rev. Lett. 76, 823 (1996).48. T. H. Stoof and Yu. V. Nazarov, Phys. Rev. B 53, 14496 (1996).49. G. R. Boogaard, A. H. Verbruggen, W. Belzig, and T. M.

Klapwijk, Phys. Rev. B 69, R220503 (2004).50. I. Siddiqi, A. Verevkin, D. E. Prober, A. Skalare,

W. R. McGrath, P. M. Echternach, and H. G. LeDuc, J.Appl. Phys. 91, 4646 (2002).

51. A. Dimoulas, J. P. Heida, B. J. van Wees, T. M. Klapwijk, W.van de Graaf, and G. Borghs, Phys. Rev. Lett. 74, 602 (1995).

52. I. O. Kulik, Zh. Eksp. Theor. Fiz. 57, 1745 (1969); [Sov. Phys.JETP 30, 944 (1970)].

53. C. Ishii, Prog. Theor. Phys. 44, 1525 (1970).54. J. Bardeen and J. L. Johnson, Phys. Rev. B 5, 72 (1972).55. N. Agrait, A. Levy Yeyati, and J. M. van Ruitenbeek, Phys.

Rep. 377, 81 (2003).56. C. W. J. Beenakker, in Transport Phenomena in Mesoscopic

Systems, H. Fukuyama and T. Ando, eds. (Springer, Berlin.1992).

57. A. F. Morpurgo, B. J. van Wees, T. M. Klapwijk, and G.Borghs, Phys. Rev. Lett. 79, 4010 (1997).

58. I. O. Kulik and A. N. Omel’yanchuk, JETP Lett. 21, 96 (1975);Sov. J. Low Temp. Phys. 3, 459 (1978).

59. V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486[1963; erratum 11, 104 (1963)].

60. A. F. Morpurgo, T. M. Klapwijk, and B. J. van Wees, Appl.Phys. Lett. 72, 966 (1998).

61. J. J. A. Baselmans, B. J. van Wees, and T. M. Klapwijk, Appl.Phys. Lett. 79, 2940 (2001).

62. J. J. A. Baselmans, T. T. Heikkila, B. J. van Wees, and T. M.Klapwijk, Phys. Rev. Lett. 89, 207002 (2002).

63. F. K. Wilhelm, G. Schon, and A. D. Zaikin. Phys. Rev. Lett.81, 1682 (1998).

64. H. Pothier, S. Gueron, N. O. Birge, D. Esteve, and M. H.Devoret, Phys. Rev. Lett. 79, 3490 (1997).

Proximity Effect From an Andreev Perspective 611

65. J. J. A. Baselmans, A. F. Morpurgo, B. J. van Wees, and T. M.Klapwijk, Nature 397, 43 (1999).

66. J. J. A. Baselmans, B. J. van Wees, and T. M. Klapwijk, Phys.Rev. B 63, 094504 (2001).

67. D. W. Abraham, C. J. Lobb, M. Tinkham, and T. M. Klapwijk,Phys. Rev. B 26, 5268 (1982).

68. S. Pilgram and P. Samuelsson, cond-mat/0407029.

69. T. M. Klapwijk, G. E. Blonder, and M. Tinkham, Physica B+C 109–110, 1657 (1982).

70. P. Dieleman, H. G. Bukkems, T. M. Klapwijk, M.Schicke, and K. H. Gundlach, Phys. Rev. Lett. 79, 3486(1997).

71. R. Cron, M. F. Goffman, D. Esteve, and C. Urbina, Phys. Rev.Lett. 86, 4104 (2001).