dc current through a superconducting two-barrier system

PHYSICAL REVIEW B, VOLUME 65, 174505

Direct current through a superconducting two-barrier system

Elena Bascones1,2,3 and Francisco Guinea2

1Departamento de Fı´sica de la Materia Condensada, Universidad Auto´noma de Madrid, E-28049 Madrid, Spain2Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientı´ficas, Cantoblanco, E-28049 Madrid, Spain

3Departament of Physics. University of Texas at Austin, Austin, Texas 78712~Received 2 August 2001; revised manuscript received 7 December 2001; published 15 April 2002!

We analyze the influence of the structure within a superconductor–normal-metal–superconductor junctionon the multiple Andreev resonances in the subgapI -V characteristics. Coherent interference processes andincoherent propagation in the normal region are considered. The detailed geometry of the normal region wherethe voltage drops in superconducting contacts can lead to observable effects in the conductance at low voltages.

DOI: 10.1103/PhysRevB.65.174505 PACS number~s!: 68.37.Ef, 73.40.Cg

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I. INTRODUCTION

The physics of superconductor–normal-metasuperconductor~SNS! junctions of microscopic size has atracted a greal deal of attention recently.1–3 A variety of ex-periments can be understood by modelling the constricby a number of one-dimensional superconducting chanin parallel.2 Each channel is described in terms of its tranmission coefficient. This model assumes that, at finite vages, the potential drop occurs in a normal region closethe barrier. The size of this region is implicitly fixed whespecifying the boundary conditions for the quasipartiwave functions in the two superconducting leads. The maing conditions used so far are equivalent to the assumpthat the width of the normal region is much smaller thother relevant length scales in the system, such as the coence lengthj0. If this is the case, Anderson’s theorem canused to justify the lack of variation of the superconductiproperties of the leads in the proximity of the constrictio4

This model has, in fact, proved to be very valuable in intpreting the I -V curves of superconducting quantum pocontacts.1–3

Corrections to the preceding picture are expected wthe potential drop and the effective barrier region are of sicomparable to the scales which determine the superconding properties. We can expect that, near the constrictionsizeL, the mean free path of the superconductor,l, will notexceedL. The effective coherence length is given byj5Aj0l;Aj0L. According to Anderson’s theorem, the reltive corrections to the superconducting properties due toperfections of sizeL are of orderO(L/j). Thus, the smallparameter which justifies the existence of an abrupt barriethe junction is}AL/j0. For contact regionsL'10 nm andlarger, these effects need not be negligible for materials sas Al and should be more relevant for Pb or Nb junction

In this work, we analyze the leading corrections to tabrupt barrier limit by assuming that the barrier region han internal structure. We allow the transmission of the cenregion to depend on energy. The simplest such situationconsider that the junction is made up of two barriers, whdefine the central region of the contact and which the traport is ballistic in between the barriers. Recent experimesuggest that such a geometry can be manufactured withisting technologies.5–9 If there are no dephasing processes

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between the barriers, the quantum interference amongmal scattering events at the barriers will inducemomentum-~and consequently energy-! dependent normatransmission. We study how the interplay between Andrereflection processes and scattering at both barriers showin the I -V curves. The dependence on energy of the condtance of quantum point contacts in the normal state has brecently studied.10,11 Multiple-scattering processes in thessystems arise from reflection at impurities close to the ctact. These effects should also be present in the superducting state.

The study of microscopic models of Andreev reflectiowas started in 1982 by Blonderet al.,12 who analyzed thecurrent which flows through a normal-metal interface. Thconsidered the case in which scattering takes place onlthe interface. Landauer’s approach is generalized to accfor Andreev reflection. An electron incident from the normpart reaches the interface and its probability to cross itbeen reflected is calculated. Electrodes are in equilibriuThus the probability of each departing process and the ocpation of the states is controlled by the Fermi function. Ading all possible processes current is calculated.

In that reference, Ref. 12, it was shown that at large voages theI -V curves corresponding to a NS interface are lear with a slope equal to the normal-state conductance. Hever, they are displaced with respect to the normal-stateI -Vcurve by an amount called excess current. At low voltagV<D the shape of theI -V curve is strongly dependent on thtransmission of the interface. At large values of the transmsion, the current at low voltages is finite.13

Later Octavioet al.,14 proposed a model to analyze tranport in superconducting constrictions. They modeled the stem as a SNS system and analyzed the transport throughSN systems connected in series. Normal scattering was meled withd barriers at the NS interfaces.

Their model is semiclassical. All possible scattering pcesses are added, weighted by its probability. Probabiliand not quantum mechanical amplitudes are considered.interference between scattering processes is neglected.this model, it was possible to explain qualitatively the apearance of subharmonic gap structure in theI -V character-istics of superconducting weak links.15 The computation ofI -V curves in superconducting constrictions was analyzedArnold16,17within the formalism of Green’s functions. In th

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ELENA BASCONES AND FRANCISCO GUINEA PHYSICAL REVIEW B65 174505

last years, several authors have improved18–20 these calcula-tions, have calculated the fully coherent quantum mechanI -V curves using different approaches,21–26 and have generalized the calculations to the case in which transport taplace by resonant tunneling.27,28

In this paper we generalize previous calculations tocase in which there are two barriers in a one-dimensiosuperconducting constriction, superconductor-insulasuperconductor-insulator-superconductor~SISIS! system.Another interesting and related system has been recestudied by Ingermanet al.30 In that work the current througha SNINS with finite-length normal regions is analyzed. Thfind that in these systems the subgap current is enhancecomparison to that of superconducting constrictions. Thefect is most pronounced in low-transparency junctions.

We will consider two cases: in Sec. II the case in whiquantum coherence is maintained between the barriers29 isanalyzed. The details of the calculation are given in the Apendix. In Sec. III we study the case in which both barriare added in series, without considering quantum interence between the scattering processes at both barriersend with the main conclusions.

II. QUANTUM-COHERENT PROPAGATION BETWEENTHE BARRIERS

In this section, we generalize the previous calculationsthe case in which there are two barriers, separated by atanceL, assuming that phase coherence in maintainedtween the barriers; see Fig. 1. Except for the existencethese two barriers, the superconductor is perfectly clean.is the simplest model which accounts for the quantum inference associated to multiple-scattering processes. A simmodel was used in Ref. 10 to explain the non-Ohmic behior of gold atomic contacts.

FIG. 1. Model system considered to analyze transport in aperconducting constriction when two barriers are present. The hzontal axis represents the position in the one-dimensional probconsidered, while the vertical axis corresponds to the potentialfile as is viewed by a conduction electron. Due to the lack of stain the superconductors at energies lower than the gap, the gapis drawn as a potential step. The reflection processes at the intewith the superconductor are of the Andreev type, in contrast wnormal reflection processes which take place at the potential bers. For convenience to satisfy matching conditions between wfunctions at both sides of each barrier, normal regions of negliglength have been introduced in between the superconductorsthe barriers; see text. In between the barriers~region II!, these nor-mal regions are labeleda andb. The distance between the barriedetermines the size of the constriction.

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Each superconductor is described by Bogoliubov–Gennes~BdG! equations and is assumed to be in equilibriuat its own chemical potential. Boundary conditions on twave function and its derivative must be given at the intface. If there is a finite potential dropV between two super-conductors, the chemical potential of both superconductonot equal,eV5m22m1. Here i 51,2 label the left and rightelectrodes. A common reference level must be chosen. Wboth superconductors are referred to the same chemicatential the order parameter acquires a time-dependphase.23 Thus the time-dependent BdG equations mustsolved. The time dependence of the relative phasedf5f22f1 is given by the Josephson relation

ddf

dt5

2eV

\. ~1!

To visualize the processes which contribute to the currand perform the calculations, it is easier to work in a scating formalism scheme, in which each barrier is inside a nmal region~see Fig. 1!, as developed in Ref. 21. This normregion, of lengthd!j, with j the coherence length of thsuperconductor, can be introduced without losing generaas the conditiond!j makes the superconducting propertiof the constriction irrelevant. The superconductor acts asource of quasiparticles. These quasiparticles arrive atbarrier and can be transmitted or reflected. Moreover, Adreev reflection processes can occur at the NS interfaAssuming thatD!m i , the Andreev approximation can bused. For simplicity, in the following, we assume that all tsuperconducting regions are ideal BCS ones and thatorder parameter in all regions is equal.

Andreev reflection at the NS interface is given by tamplitudea(e) of suffering an Andreev reflection process.the superconductor is of the BCS type21

a~e!51

D H e2sgn~e!~e22D2!1/2, ueu.D,

e2 i ~D22e2!1/2, ueu,D.~2!

In the absence of a magnetic field, which breaks timreversal invariance, each barrier is characterized by a scaing matrix

S r i t i

t i 2r i* t i

t i*D . ~3!

r i andt i are the reflection and transmission amplitudes ofbarrier when a plane wave incides from the left. They aassumed to be independent of the momentum. The cosponding reflection and transmission probabilities areRi5ur i u2 andTi5ut i u2. In the neck, between the barriers, eletrons propagate without suffering scattering events, butquire a phaseeikL, wherek is the electronic momentum.

As a result of all multiple Andreev reflection~MAR! pro-cesses and their interference with the normal scattering othe wave function in the normal region, in zones I, II~a andb!, and III can be written as21

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DIRECT CURRENT THROUGH A SUPERCONDUCTING . . . PHYSICAL REVIEW B65 174505

C Ie5(

m,n@~a2n

2mAnm1J0dm0dn0!eikx

1Bnme2 ikx#ei (e12neV112meV2)t,

C Ih5(

m,n@An

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me2 ikx#e2 i [ e12neV112meV2] t,

C II 2ae 5(

m,n@Jn

meikx1Gnme2 ikx#e2 i [ e1(2n11)eV112meV2] t,

C II 2ah 5(

m,n@Kn21

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3e2 i [ e1(2n21)eV112meV2] t,

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m,n@ Jn

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m,n@Kn21

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3e2 i [ e1(2n21)eV112meV2] t,

C IIIe 5(

mn@En

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me2 ikx#

3e2 i [ e1(2n11)eV11(2m11)eV2] t,

C IIIh 5(

m,n@a2n21

2m21En21m21eikx1Fn21

m21e2 ikx#

3e2 i [ e1(2n21)eV112meV2] t, ~4!

whereV1 and V2 are the voltage drops at the two barrieandan

m(e)5a(e1meV11neV2) with a(e) the Andreev re-flection amplitude. Region IIa refers to the superconductpart situated just after the left barrier, while region IIb refeto that one just before the right barrier.

The current is a time-dependent quantity, which oscillawith all the harmonics of the Josephson frequencyvJ52eV/\:

I ~ t !5(n

I neinv j . ~5!

The time dependence of the current arises from the tdependence of the superconducting phase induced byvoltage drop. Here we will only be interested in the dc crent (I 0 component! which is the one experimentally measured.

The scattering matrices relate electron and hole coecients at regions I, IIa,b, and III. Scattering matrices of eltrons and holes are related bySh(e)5Sel* (2e). As explainedin the Appendix the matching conditions lead to a setmatrix equations between the coefficients in the wave futions, which can be recursively solved.

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There it is shown that if the barriers are a distanceL farapart, the problem can be mapped onto another one wimomentum-dependent single barrierTe f f given by

Te f f~k!5T1T2

11R1R222AR1R2 cos~f12kL!. ~6!

Heref is defined as

r 18r 25AR1R2eif, ~7!

where r 1852r 1* t1 /t1* is the amplitude of probability of re-flection for those electrons which incide from the right onthe first barrier.f is a relative phase which controls thquantum interference between the two barriers and is preeven whenL50 and the transmission does not dependmomentum; see below. It changes from configuration to cfiguration. This effective transmission would be the samethe case of a normal central region and/or normal left aright electrodes. The superconducting nature of the etrodes enters in the calculation of the wave function coecients.

Due to the dependence on the momentum, the transsion depends on energy and the equations of previous semust be generalized to the case in which transmissionreflection are energy dependent. A similar situation wfound in Ref. 30. We assume that Andreev approximatstill holds, which implies that the chemical potential is mularger than any other energy scale in the problem. Assuma linear dispersion, relation~6! can be written as

T~e!5T1T2

11R1R222AR1R2 cos~f12 f e/D!, ~8!

with f 5DL/\vF .The procedure to calculate the coefficientsAn andBn and

the current is detailed in the Appendix. Only those coecientsAn5Am

n dnm andBn5Bmn dmn are nonzero.

If f 50 ~barriers are at the same point!, theI -V curves arethe ones corresponding to a superconducting constricwith a non-energy-dependent transmission of value

Tf 505T1T2

11R1R222AR1R2 cosf~9!

and no new features appear. NewI -V curves appear whenf Þ0. Then, the transmission depends on energy and olates between the valuesTmin andTmax given by

Tmax5T1T2

~12AR1R2!2, ~10!

Tmin5T1T2

~11AR1R2!2. ~11!

Note also that if the transmission of one of the barriersequal to unity, the current is controlled by the other barrand we again recover the single-constriction case. The eof a finite value off will be more important for a larger

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difference betweenTmax andTmin . Note that, in particular, ifboth barriers are equal, equal toT, Tmax51.0 and Tmin5T2/(22T)2, which decreases with decreasingT. Some ex-amples are shown in Figs. 2–5. The values off used arecompatible with the energy dependence observed in Ref

In Fig. 2 curves corresponding toT150.9, T250.1, andf50.0 are shown. The maximum and minimum values cresponding to these transmissions areTmax50.184 andTmin50.05. These values are relatively similar, and a vstrong effect of a finitef is not expected. In fact, asf in-creases the conductance is slightly reduced. The fact thaconductance is reduced and not increased is because westarted from the maximum value atf 50.

Another example is shown in Fig. 3, in which case curvfor T15T250.9, f50.0, and several values off are plotted.Maximum and minimum values areTmax51 and Tmin50.67, which are again not too different. The main effecta finite value off is the suppression of the current at zevoltage. This effect is also observed in Fig. 4 where sevcurves corresponding to the case in which both barriers

FIG. 2. I -V curve for the case of coherent transport in a supconducting constriction with two barriers with parametersT1

50.9, T250.1, f50, and several values off.

FIG. 3. I -V curve for the case of coherent transport in a supconducting constriction with two barriers with parametersT15T2

50.9, f50, and several values off .

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equal, f 50.1 andf50, are shown. In all these cases ttransmission forf 50 would be equal to unity. The deviatioof this curve is more pronounced for small transmissionthe barrier. For all transmissions, current at zero voltagecompletely suppressed. The reason of this suppression iseven atV50, all the energies contribute to the current athe transmission is not equal to unity at all energies.

The effect of a finite value off is specially strong in Fig.5 asTmin50.002. The maximum value is 500 times the minmum one. The effect is very strong even for a very smvalues off, as in the casef 50.02.

Together with the subharmonic structure, the other chacteristic feature ofI -V curves of single-barrier superconducting constrictions is the excess currentI exc. At voltagesmuch larger than the gap, theI -V curves are linear but do noextrapolate to zero current at zero voltage, but to a finvalue, equal toI exc. In such a system the excess currentproportional to the transmission of the barrier.22 In the coher-ent double-barrier constriction discussed in this section

-

-

FIG. 4. I -V curve for the case of coherent transport in a supconducting constriction with two equal barriers for the case of50.1, f50 and several values of the transmission.

FIG. 5. I -V curve for the case of coherent transport in a supconducting constriction with two barriers with parametersT15T2

50.1, f50, and several values off.

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energy dependence of the transmission prevents theI -Vcurve of being linear even at voltages much larger thansuperconducting gap. Thus an excess current cannot befined in the usual way. In the single-barrier case the relaI S5I N1I exc ~with superindicesN andS referring to normaland superconducting states, respectively! could also be usedto define the excess current. In our system, however,computedI exc depends on energy. This dependence onergy can be easily understood by taking into accountdependence of excess current on transmission and thatransmission on energy. As a consecuence,the definition oexcess current in the present system is meaningless.

The I -V curves presented have been calculated at ztemperature. Single-barrierI -V curves calculated at zertemperature have been shown to remain valid at temptures small compared to the critical one (T<0.2Tc).

1–3 Weexpect for our double-barrier calculations the same rangvalidity.

III. INCOHERENT PROPAGATION BETWEEN THEBARRIERS

The calculation and features of theI -V curves in the casein which propagation between the barriers is not coherentcompletely different to those of previous section. From crent conservationI 15I 25I , where I i is the current whichcrosses junctioni. In the normal state, in the absencecoherence the current through each junction isI i5giVi andthe current conservation constraint determines the voltdrop in each contact,Vi5gi /(g11g2)V. The total currentcorresponds to that obtained adding the resistances in sgseries5g1g2 /(g11g2).

In a similar way, to calculate theI -V curves in the superconducting state we impose current conservationI (V)5I 1(T1 ,V1)5I 2(T2 ,V2). Here I i(Ti ,Vi) is computed ac-cording to the calculations for a superconducting constrictwith a single barrier of transmissionTi5ut i u2.21,22 The volt-age in each junction is numerically determined from the crent conservation equation. Note that subharmonic gap stture ~SGS! will not appear at valuesV52D/n, but at valuesVi52D/n. Except at voltages very large compared to the g(V!D), in which caseI -V curves are linear, the high nonlinearity of the curves prevents the voltage drop at eachrier of being proportional to the normal transmission, cotrary to what happens in the normal state.

Examples ofI -V curves determined by this method ashown in Figs. 6 and 7. In Fig. 6 several curves, correspoing to the case in which both barriers are equal, are plotSubharmonic gap singularities appear atV54eD/n, as thevoltage in each barrier is equal toV/2. The curves areequivalent to theI -V curve corresponding to the transmissiof the barrier, but for a doubled voltage. Figure 7 shcurves for the case in which both barriers are different. Tposition at which SGS appears depends on the value ofbarriers.

Contrary to the fully coherent double-barrier system,the present case theI -V curves are linear at high voltageand the excess current can be defined in the usual way.ing into account the current conservation equation and

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proportionality of the excess current to the transmission osingle-barrier constriction, it is straightforward to show thboth the slope of theI -V curve and the excess current are tones corresponding to adding the resistances of both barin series. Thus, at asymptotically large voltages, theI -Vcurves are equivalent to the one of a single barrier and cductancegseries.

IV. CONCLUSIONS

We have studied the influence of the structure of the nmal region where the voltage drops in superconducting ctacts. We present detailed expressions to analyze the inteence effects which arise due to the Andreev reflectionsdifferent positions within the contact.

The observedI -V characteristics at low voltages can difer from the usually considered single abrupt barrier casthe size of the contact,L, is comparable to the effective co

FIG. 6. I -V curves corresponding to transport in superconduing constrictions when there are two barriers in the structurecoherence is lost between the barriers. The figure shows securves for the case in which both barriers are equal.

FIG. 7. I -V curves of a superconducting constriction in presenof two different barriers, when there is no phase-coherence betwthe barriers.

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herence lengthAj0L, wherej0 is the coherence length in thclean limit. The influence of inelastic scattering in the normregion, leading to a loss of coherence, has also been invgated.

In particular, we have derived the equations for the caswhich the transmission through the constriction dependsenergy. We have analyzed in detail the case in whichdependence in energy arises from the existence of tworiers in the contact, in between which the phase is matained. TheI -V curves differ from the ones obtained witonly one non-energy-dependent barrier with any transmsion. The effect is strongest when both barriers have simand low, transmission coefficients. In this case, the enedependent transmission oscillates between two valuesdifferent in magnitude. Even when at zero energy the tramission is equal to unity, we find that current at zero voltais suppressed. At large voltages, due to the energy dedence of the transmission the excess current cannot befined.

Quantum interference processes like the ones considhere have shown to be relevant in metallic quantum pocontacts, where the scattering centers are both the constion and impurities close to it.10,11 Long SNS constrictionswith scattering processes at both interfaces have beencently created.5,6 While the way in whichI -V curves aremodified, compared to the single-barrier case, is expectebe relevant in these experiments, a quantitative comparis difficult due to the large number of channels which cotribute to transport in these experiments.

In the case in which propagation between both barrierincoherent, current is derived from the current conservarequirement. The main feature of theI -V curves is the ap-pearance of subharmonic structure at voltages different f2D/n. At voltages much larger than the superconducting gthe I -V curve is linear and equivalent to the one obtainwith a single barrier of conductancegseries.

ACKNOWLEDGMENTS

We appreciate useful discussions with N. Agra¨t,N. Garcıa, G. Rubio-Bollinger, J.J. Sa´enz, H. Suderow, andespecially, S. Vieira. Financial support from CICyT~Spain!through Grant No. PB96-0875, Comunidad de Madthrough FPI, the Welch Foundation, and Grant No. NSDMR0115947 is gratefully acknowledged.

APPENDIX

The coefficients in the wave functions are related byscattering matrices giving a set of equations which allowsto obtain them. For the electronic part these equations a

FBnm

JnmG5F r 1 t1

t1 2r 1* t1

t1*G Fa2n

2mAnm1Jdn0

m0

Gnm G , ~A1!

FGnm

Jnm G5F 0 eikL

eikL 0 GF Jnm

GnmG , ~A2!

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F Gnm

EnmG5F r 2 t2

t2 2r 2* t2

t2*G F Jn

m

a2n112m11Fn

mG , ~A3!

and similar equations for the holes. From it the coefficiein regions I and III are related by

FBnm

EnmG5SelFa2n

2mAnm1Jdn0

m0

a2n112m11Fn

m G , ~A4!

with

Sel51

t1* 1t1r 1* r 2

3F t1* r 1e2 ikL1t1r 2eikL T1t2

T1t2 2~ t1r 1* eikL1t1* r 2* e2 ikL!G

~A5!

and

F Anm

Fn21m21G5ShF a2n

2mBnm

a2n212m21En21

m21G , ~A6!

with Sh5Sel* . From the existence of a unique source teJdn0

m0 , it can be shown thatAnm andBn

m will vanish except forn5m. Thus, the superindex can be dropped and the probis equivalent to one in which there is a single barrier wtransmission given by Eq.~6!. Due to the dependence on thmomentum, the transmission depends on energy.

In the following, transmission is characterized by a sctering matrix

S r ~e! t~e!

t~e! 2r * ~e!t~e!

t* ~e!D . ~A7!

As usual, transmission and reflection coefficients satisfy

R~e!1T~e!51, ~A8!

whereR(e)5ur (e)u2 and T(e)5ut(e)u2. In the two-barriercase, the dependence in energy comes from the dependin momentum. We assume that momentum does not chawhen the particle crosses the barrier; thus the reflectiontransmission coefficients depend on the energy of the ornally incident particle. However, in the following we includboth the possibility that transmission and reflection coecients depend, as in our case, only on the energy oforiginal quasiparticle, and then we define

r n~e!5r ~e!, ~A9!

tn~e!5t~e!, ~A10!

and the case in which both depend on the energy ofquasiparticle,e12neV, which is crossing the barrier and haemerged after 2n Andreev reflections. Then

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r n~e!5r ~e12neV!, ~A11!

tn~e!5t~e12neV!, ~A12!

ne-

n

md

apa

r-

n

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andRn(e) andTn(e) are analogously defined.As before we can find the relations between the coe

cients of the wave functions. After some algebra the eqtions which relate theAn andBn coefficients are

tn* ~e!An112t2~n11!* ~2e!a2n11a2nAn5tn* ~e!r n11* ~2e!a2n12Bn112t2~n11!

* ~2e!a2n11r n* ~e!tn* Bn

1t2~n11!* ~2e!a2n11Jdn0 ~A13!

and

Tn~e!T2(n11)~2e!a2n12a2n11LnBn112$a2n2 Ln11~e!@ tn21~e!tn* ~2e!a2n21

2 2r n21~e!r 2n* ~2e!tn21* ~e!tn~2e!#1Ln~e!

3@ t2(n11)~2e!tn* ~e!2r 2(n11)~2e!tn11* ~2e!r n* ~e!tn~e!a2n112 #%Bn1Tn21~e!T2n~2e!a2n21a2nLn11Bn21

52Ln~e!Ln11~e!Jdn0 , ~A14!

with

Ln~e!52r n~2e!tn* ~2e!tn21~e!a2n212 2r n21~e!t2n~2e!tn21* ~e!. ~A15!

In the casetn(e)5t(e)5t(2e), Eq. ~A14! can be simplified to

T~e!a2n11a2n12~12a2n212 !Bn111T~e!a2n21a2n~12a2n11

2 !Bn212$a2n2 ~12a2n11

2 !@a2n212 2R~e!#

1@12R~e!a2n112 #~12a2n21

2 !%Bn52r ~12a2n212 !~12a2n11!Jdn0 , ~A16!

which is simpler than Eq.~A14! and reduces to the expresioobtained in Ref. 21, but including explicitly the energy dpendence.

Including the contributions from quasiparticles incideon the constriction from both superconductors@electronsfrom the left superconductor—with momentumk and prob-ability f (e)—and holes from the right one—with momentu2k and probability (12 f (e)!# the dc current, as calculatein zone I, is given by

I 052e

p\E2m2eV

m

de sgn~e!FJ21Ja0* A0* 1Ja0A0

1(n

~11ua2nu2!~ uAnu22uBnu2!GUm→`

. ~A17!

This expression was defined in Ref. 21 and takes intocount the contribution of all scattering processes of quasiticles with both spins. One must solve Eq.~A14! or ~A16!and obtain theBn coefficients and, using this solution, detemine An from Eq. ~A13!.

It only rests now to discuss how Eq.~A14! or ~A16! canbe solved. Let us write these equations in the form31

VnBn111UnBn1HnBn215Fndn0 , ~A18!

whereVn , Un , Hn , andFn depend on energy and are giveby the coefficients of the corresponding Eq.~A14! or ~A16!.ConsidernÞ0. Then,31

t

c-r-

Vn

Bn11

Bn1Un1Hn

Bn21

Bn50. ~A19!

We define

Sn.05Bn

Bn21, ~A20!

Sn,05Bn

Bn11. ~A21!

In terms of these factorsSn ,

Bn.05)n

Sn.0•••S1B0 , ~A22!

Bn,05)n

Sn,0•••S21B0 . ~A23!

If n.0, substituting the definition ofSn in Eq. ~A19!,

VnSn111Un1Hn

Sn50, ~A24!

and from it

Sn.052Hn

Un1VnSn11~A25!

or, equivalently,

5-7


Sn.052Hn

Un2VnHn11

Un112Vn11Hn12

Un122Vn12Hn13

Un131•••

. ~A26!

Analogously, forn,0,

Vn

Sn1Un1HnSn2150, ~A27!

andSn,0 is given from

Sn,052Vn

Un1HnSn21, ~A28!

which in terms of recurrent fractions is written

re

.na

ira

riss

.

e

17450

Sn,052Vn

Un2HnVn21

Un212Hn21Vn22

Un222Hn22Vn23

Un232•••

. ~A29!

Thus, from Eqs.~A26! and~A29!, the termsSn can be easilyevaluated. To calculateB0, we write Eq.~A24! for n50 as

@V0S11U01H0S21#B05F0 . ~A30!

Thus

B05F0

V0S11U01H0S21~A31!

and the rest of coefficents are determined from Eqs.~A22!and ~A23!.

,

. B

.

es

din,

din,

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5-8

dc current through a superconducting two-barrier system

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