Blonder, Transition From Metallic to Tunneling Regimes in Superconducting Micro Constrictions

8/6/2019 Blonder, Transition From Metallic to Tunneling Regimes in Superconducting Micro Constrictions

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PHYSICAL REVIEW B VOLUME 25, NUMBER 7 1 APRIL 1982Transition from metallic to tunneling regimes in superconductingmicroconstrictions: Excess current, charge imbalance,and. supercurrent conversion

0 : E. Blonder, M. Tinkham, and T. M. Klapwijk*Physics Department, Harvard University, Cambridge, Massachusetts 02138(Received 19October 1981)

We propose a simple theory for the I- V curves of normal ...superconducting microconstriction contacts which describes the crossover from metallic to tunnel junction behavior.The detailed calculations are performed within a generalized semiconductor model, withthe use of the Bogoliubov equations to treat the transmission and reflection of particles atthe N-S interface. By including a barrier of arbitrary strength at the interface, we havecomputed a family of 1-V curves ranging from the tunnel junction to the metallic limit.Excess current, generated by Andreev reflection, is found to vary smoothly from4a/3eR N in the metallic case to zero for the tunnel junction. Charge-imbalance generation, previously calculated only for tunnel barriers, has been recalculated for an arbitrarybarrier strength, and detailed insight into the conversion of normal current to supercurrent at the interface is obtained. We emphasize that the calculated differential conductance offers a particularly direct experimental test of the predictions of the model.

I. INTRODUCTIONIn recent years there has been much experimental and theoretical work on small-area high

current-density superconducting junctions, sincesuch junctions have advantages for certain practical applications. Although the properties (1-Vcurves, etc.) of classic (high-barrier) tunnel junetionshave long been understood in great detail, 1the properties of metallic (no-barrier) junctionshave been studied only more recently2-4 and inless detail, and the transitional case (small barrier)has received practically no attention at all. Here,we present a new unified treatment which is sufficiently general to embrace both limits, as well asthe intermediate regime.In this paper, we confine our attention primarily

to the N-S interface in the context of a generalizedAndreev reflection model. This avoids the complications of the Josephson effect, while allowingcomplete delineation of 1-V curves including theso-called "excess currrent' at high voltage, as it increases from zero in tunnel junctions to its limitingvalue in clean metallic contacts.

The paper begins with a careful clarification ofthe relation between a general .semiconductormodel of the superconducting energy levels and theexcitations of the BCS model and the Bogoliubov

equations. With this in hand, we are quickly ledto complete J-V curves for the case of arbitrarybarrier strength. We also compute the generalization of the tunnel-injection results for the generation of charge imbalance in the superconductor,and give new insight into the detailed processes involved in the conversion from normal to supercurrent at an N-S interface. Some of the resultswhich we obtain by our simple technique havebeen obtained earlier by other more mathemat.icallycomplex theories, but our method gives greaterphysical insight. Moreover, the simplicity of ourmethod has allowed us to treat a broader range ofproblems having direct experimental implications.We have also used this model in a previous publication" to explain the origin of the excess currentand of the subharmonic gap structure which is observed in .metallic S-S contacts.

II. GENERALIZED SEMICONDUCTORSCHEMEIt has long been conventional to simplify the calculation of tunnel currents by taking advantage ofthe fact that the BCS coherence factors Uk and vdrop out of the computation if excitations with thesame energy Ek , but with k kF,aregrouped together. Then a very simple "semicon-

25 4515 @1982 The American Physical Society


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4516 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWIJK 25

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ductor model" can be established, in which onlythe superconducting density of states Ns(E) distinguishes the superconductor from the situation inthe normal state. However, in combining k < andk ; for each Ei ; information is lost. For instance,the quasiparticle charge imbalance Q* refersspecifically to the imbalance between the twobranches k > kF and k < kF of the quasiparticlespectrum, a distinction which is completelysuppressed in the simplest semiconductor model.Another shortcoming of the elementary semiconductor model is that the density-of-states factormakes no reference to the microscopic nature ofthe states. For example, it is clear that the currentmust be carried toward and away from the junction by waves traveling in the appropriate directions, yet this physical fact is obscured in the usualtunnel-Hamiltonian model.Here, we describe a generalized semiconductormodel specifically designed to avoid this loss of information. The key innovation is the use of theBogoliubov equations" to handle the interface; thistreats all cases from a clean metallic interface to atunnel junction in the same formal way by matching wave functions at the boundary. These resultsdetermine the probability current carried by thevarious excitations characterized by the semiconductor diagram. For simplicity, we restrict attention to metal clean enough to allow useof momentum as a good quantum number for labeling thevarious waves.

If one considers only energies E > Ji., the Bogoliubov equation solutions (seeAppendix for details)for the incident, transmitted, and reflected particlescan be identified with the conventional Bes quasiparticle excitations with energyEk == ( . 2+El )1/2 , (1)

with respect to the electrochemical potential forcondensed pairs, where

fl?k2Ek==---cF 2m

Because only the square of Ek enters the expressionfor E k , there will be a pair of magnitudes of k associated with each energy, namely,

in the absense of spin-flip processes this affectsonly normalizations, and it need not concern ushere.The nature of one type of these Bes quasiparticle states is given by the usual Bogoliubov transformarion" as

YkO=UkCki - V kC - k t , (4)where the Bes coherence factors are given by

uf=l-vf=+(l+ck/Ek). (5)Although, in general, Eq. (5) determines the magnitudes of Uk and Vk (which can have phase factors), for simplicity we take them to be real andpositive. For the N-S case treated here, this involves no loss of generality; only in S-S devicesdoes phase playa role. The excitations describedin Eq. (4)consist partly of electrons created at -s-k(i.e., UkCkt) and partly of holes created at -k (i.e.,V k C - k ~ ' the destruction of an electron at -k) .Since either term increases the system momentumby -s-k, the subscript k in YkO is a well-definedmomentum index suitable for labeling. From Eq.(5) we see that the excitations at k+ are predominantly electronlike, while these at k - are predominantly holelike. The other members of thedegenerate set are described by including the complementary BeS operator Yk 1 and also includingboth senses of k.In preparation for the application of boundaryconditions, we note that the excitation described byEq. (4) is described in the Bogoliubov-equation formalism (seeAppendix for details) as a two-element

column vector,

FIG. 1. Schematic plot of excitation energies Ek vs kalong an axis passing through the center of the Fermisphere in the superconducting state. The scale has beengreatly expanded near kp

-Ilk==(2m )1/2[p(Et - a2) 1/ 2 ] 1/ 2 (3)Moreover, because of the BeS pairing of k and- k, one must consider both signs of k together, sothat there is a fourfold degeneracy of relevantstates for each E, as sketched in Fig. 1. Of coursethere is an additional twofold spin degeneracy, but

I I I- k+ -kF -k - o


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25 TRANSITION FROM METALLIC TO TUNNELING REGIMES IN . . . 4517

=in af -+E'"at ~ (7)

kx

(IOa)

(lOb)

veko

v

required to make an excitation with charge e is

while that to make an excitation with charge - ediffers by 2/-l and is

We can now tabulate (see Table I) the possiblecharge-conserving processes involving subsystems 1and 2, and write down the corresponding conditionset by the conservation of energy. There are fourdistinct cases, each of which conserves energyequally well in .reverse, making eight cases in all.As is evident from Table I, all these energeticallyallowed transitions occur as "horizontal" ones onlyif we plot both Eki symmetrically about each J-Ll'as is motivated by recalling that J-L -E k = -E hk(see Fig. 2). Further, the four cases of Table I include all the combinations of Ek , so that all horizontal transitions in such a "reflected" or semiconductor-model diagram are energetically allowed.There is, however, another constraint on the system; namely, a particle incident from the left canonly produce transmitted particles with positivegroup velocity (dE /dhk) and reflected ones with

negative group velocities. In general, there is a finite probability of all the transfers allowed by theseconsiderations to occur, but in the special case ofzero barrier and no discontinuity in electronic-bandparameters across the N-S interface, it turns outthat there is essentially zero crossover betweenchannels inside and outside the Fermi surface, evennear the gap edge. It is here tha t the Bogoliubovequations prove most useful: The semiconductordiagram shows which transitions are energeticallyallowed; the Bogoliubov equations provide the

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(Sa)(8b)

where S* adds a pair to the condensate and S destroys one. These operators create excitationswhile changing the charge by exactly e, corresponding to the subscripts e and h for electron andhole, respectively. Accordingly, they are betteradapted to describe the energy levels analogous tothose of ordinary electrons. The excitationsdescribed by Eqs. (8a) and (Sb) are identical, and inno sense cancel as one might suppose for electronand hole operators. The systems resulting from(Sa) and (8b) differ only by one condensed pairfrom the reservoir at the chemical potential.Now for any system with fermion excitationssuch as these, the total system energy can be written as

Here fk and gk satisfy the differential equations[ - ; : : V2 - J L ( x )+ V(x)kAg

E = = E G + ~ E k r ~ r k + l t N , (9)k

where EG is the ground-state energy, and the sumruns over all the excitations, which have Ek o.The electrochemical potential It must be introducedexplicitly since we are interested in treating processes which change the number N of electrons ina subsystem at given J-L by 1. In fact, the energy

FIG. 2. Semiconductor version of Fig. 1. Inclusionof the quasiparticle branches in the lower half plane allows all the transitions in Table I to be "horizontal."


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4518 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWIJKTABLE I. Processes.involving transfer of a single electronic charge.

25

Process

1. Electron from 1 2 (or reverse)2. Hole from 1 ~ (or reverse)3. Create electron in 1 and hole in 2 (or destroy both)4. Create hole in 1 and electron in 2 (or destroy both)

Energy condition

J..Ll +Ek 1==JL2+Ek2J-lI-Ekl =J..L2-Ek2/-Ll+Ek1==J.l2-Ek2J..LI-Ek1==J.l2+Ek2

quantitative probability that each will occur.As a final preliminary, we consider the specialcase of the normal metal, obtained by letting a--+oin the expressions for the superconducting case.Here the situation simplifies from the superconducting case shown in Fig. 2 because electronlikeexcitations cannot be made inside the Fermisphere, since the states there are fully occupied inthe ground state, and excitations must be orthogonal to the ground state. Similarly, hole excitationsare possible only outside the Fermi sphere. As aresult, in the normal metal only the branchesshown in Fig. 3 exist.

III. THE N-S BOUNDARYIN EQUILIBRIUM

We start by considering the dynamic equilibriumwhich exists at an N-S interface with no appliedvoltage, as sketched in Fig. 4. In this equilibriumsituation, all relevant quaisparticle states (such asthose labeled 0, 1, 2, 3, 4, and 5 in the figure) areoccupied with the same probability fo(E), whichalso gives the probability of a hole in the state 6 at-E . Viewed kinematically in a semiclassical ap-

proximation, however, particles approach the interface and are transmitted and reflected with certainprobabilities, which are given by the products ofsquared amplitudes of Bogoliubov-equation solutions times their appropriate group velocities. Theself-consistency of the equilibrium state provides auseful check on the computed probability coefficients.To be concrete, consider an electron incident onthe interface from the normal state with energyE > Ii, as indicated by the arrow at the state labeled 0 in Fig. 4. By matching slope and value ofthe wave function across the interface, one findsthe probabilities A (E), B (E), C (E), and D (E ) forthe four processes indicated by those labels in Fig.4, i.e., outgoing particles at points 6, 5, 4, and 2,respectively. In words, C(E) is the probability oftransmission through the interface with a wavevector on the same side of the Fermi surface (i.e.,q+ ~ +, not -k -), while D (E) gives the probability of transmission with crossing through theFermi surface (i.e., q+ -k -). B (E) is the probability of ordinary reflection, while A (E ) is theprobability of Andreev" reflection as a hole on theother side of the Fermi surface. The latter process

o !--- k x N s

FIG. 3. Ek vs k for the normal state.

FIG. 4. Schematic diagram of energy vs momentumat N-8 interface. The open circles denote holes, theclosed circles electrons, and the arrows point in thedirection of the group velocity. This figure describes anincident electron at (0), along with the resultingtransmitted (2,4) and reflected (5,6)particles.


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and similarly

is not included in Table I, since it involves atransfer of a pair carrying 2e across the interface,while the processes in Table I transfer only a single.electronic charge. The energy conservation condi-tion appropriate for Andreev reflection is found byequating the energy of the initial electronlike excitation with that of the final holelike excitation plusa Cooper pair at the chemical potential. That is,Eek =Ehk +2J.L, from which it follows that

(13)(E)+B(E)+C(E)+D(E)= 1 .

It is important to note that the conservation ofprobability requires that

This result is particularly useful in simplifying expressions for energies below the gap, IE I a can be conveniently writtenin terms of Uo and Va, the BCS parameters u and Vevaluated on the branch outside the Fermi surface.The results are as given in Table II. For convenience, in addition to the general results we also listthe limiting forms of the results for zero barrier(Z =0) and for a strong barrier [Z2(u 2_v 2 >1], as well as for a=o (the normal-metalcase). Curves computed using the general resultsare plotted in Fig. 5 for Z =0, 0.3, 1, and 3. Inthe special case of E = a, one notes from Table IIthat A = 1 and B =C =D =0, independent of Z.

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(12a)

(12b)

In other words, the hole is generated as far belowJL as the electron was above, as drawn in Fig. 4.Of course, this result is obvious in the excitationrepresentation.It should be clearly understood that the probabilities A, B, C, and D (and the similar set A' , B' , C',and D' for quasiparticles incident from the superconducting side) are those appropriate for probability current, taking account of the cancelling effectsof state density and group velocity. For example,consider the flow of particles across the interfacein the energy range E to E +dE, and includingonly states outside the Fermi surface. From left toright, the current from states at 0 to those at 4 inFig. 4 is proportional to vpC(E)!a(E)dE; thebalancing current from right to left, i.e., fromstates at 1 to those at 5, is proportional tovpC'(E)!o(E)Ns(E)dE, where Ns(E) is the normal-ized density of states in the superconductor. Inequilibrium these flows are exactly equal, so that

governs the branch-crossing transfers. The resultsof Eqs. (12)can be derived by direct computation,or by appeal to time-reversal symmetry. For simplicity, in the remainder of the paper we write allexpressions in terms of C (E ) and D (E), avoidingthe need for explicit inclusion of the density-ofstates factor Ns(E) or the quasiparticle group velocity "e=Ns-1vp in integrations over energy.Recognizing the symmetry of the problem aboutu, we see that A (E ) and B (E ) are even functionsof E. Similarly, C(E) and D(E) are also evenfunctions of E, provided one uses the conventionthat D (E ) describes processes inwhich thetransmission involves branch crossing, while C (E )governs transmission processes in which the particle stays on the same branch.


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4520 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWIJK 2STABLE II. Transmission and reflection coefficients. A gives the probability of Andreev reflection (i.e., reflection

with branch crossing), B of ordinary reflection, C of transmission without branch crossing, and D of transmission withbranch crossing (see text). ( r = [u6+Z2(U6 -V 6 )]2, U6 = I-V6 =+{1+ [ ( E 2 _ ~ 2 ) / E 2 P / 2 J, and Ns (E )=(u6 -V6 )-1.)

A B C DNormal state 0 Z2 I 0.1+Z2 I+Z 2General form

E < ~ I -A 0 0E2+(a2_E 2)( 1+2Z 2)2E>a u6v6 (u5 -v5 )2Z2( 1+Z2) u5(u5-v5)( I +Z2) v5(u5-v6)Z2y r y y2

No barrier (Z =0)E < 0 0 0E>a v6/u6 0 I -A 0

Strong barrier [Z2( u2- v2) >> 1]E u6v5 I- I u6 v6Z4(u6'-:'v6)2 Z2(u6-v6) Z2(u6-v6) Z2(u6 -v6)

FIG. 5. Plots of transmission and reflection coefficients at N-S interface. A gives probability of Andreevreflection, B gives probability of ordinary reflection, Cgives transmission probability without branch crossing,and D gives probability of transmission with branchcrossing. The parameter Z measures the barrierstrength at the interface.

Z=0.3c

z=oA

0.5

0 A(T) E -edT) Ez=1.0 1 .......... Z=3.0.... B B.... ...............................l'

0.5 ........ , ...... 0.5

A

0 6(T) E 0 E

Accordingly, except for Z =0, A (E ) has a sharppeak at the gap edge, although this peak becomesunobservably narrow for large values of Z.Although we have developed this technique quiteindependently, other workers have also employed

the Bogoliubov equations, and along similar lines.Andreev" in 1964 used them to calculate thethermal-boundary resistance at an N-S interface.Thus, he was essentially concerned with the restriction of quasiparticle flow from N to S caused bythe gap. Kiimmel9 expanded the equations to in ..elude the condensate phase, and then used them tocalculate the momentum balance of electricalcurrent flow near a vortex core. He emphasizedthe interchange of current between quasiparticlesand the pairs in geometries where the gap variedwith position. Then, in 1971 Demers and Griffin'"and Griffin and Demers,11 calculated the transmission coefficients in N-S-N and S-N-S geometriesand made the extension to a i)-function barrier atthe interface. Surprisingly, this important workremained essentially unnoticed for a number ofyears but recently (1977) Entin-Wohlman12 hasdrawn on it to develop boundary conditions thatshe used in solving the Gor 'kov gap equat ion at anN-S interface. Thus, use of the Bogoliubov equations to determine the fraction and type of reflect-


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25 TRANSITION FROM METALLIC TO TUNNELING REGIMES IN . . . 4521ed particles is not new, though perhaps the technique remains underutilized. What is new is ourextension of the method to calculations at finitebias voltage (as described in the next section) andour development of a framework that allows thephysical consequences of these processes to emergemore clearly.

IV. TH E N-S BOUNDARYAT FINITE VOLTAGE: /-VCURVES

Since the current must be conserved, it can becalculated in any plane. It is particularly convenient to do so on the N side of the interface,where all current is carried by single particles, andnone as a supercurrent. To find the current in ourID model, we take the difference between!-+(E)and ! ~ ( E ) , the distribution functions at pointssuch as 0 and 5 on Fig. 4, and integrate over E.That is,

! ~ ( E ) = A (E ) [ I - !-+( -E) ]+B(E) !.... (E)+[C(E)+D(E)]fo(E) . (16)

In writing Eq. (16), we have chosen to w r i t e !.... (E )instead of !o(E -eV) in anticipation of later dealing with more complicated problems w h e r e !.... (E )and f ~ ( E ) must be found self-consistently. Evenin such cases, Eq. (16)remains valid as it stands,and serves as a boundary condition for a Boltzmann equation, whose solution would be insertedinto the general expression (14)to find the current.However, in the present simple case, we can simplysubstitute Eqs. (15)and (16)into Eq. (14), obtaining

When a voltage is applied, nonequilibrium quasiparticle populations will be generated, which can ingeneral be found only by obtaining a self-consistentsolution to a suitable Boltzmann equation. Thissolution is greatly simplified i f one can assumeballistic acceleration of the particles withoutscattering. This is a good approximation for ourcase of a small orifice connecting massive electrodes, so long as the diameter of the orifice issmall compared to a mean-free path. Moreover, inthis regime, Bogoliubov-equation solutions neglecting scattering potentials except at the interfaceshould also be usable.P The physical assumptionwhich we use to define the problem is then thatthe distribution functions of all incoming particlesare given by equilibrium Fermi functions, apartfrom the energy shift due to the accelerating potential. It is convenient to choose the electrochemical potential of the pairs in the superconductor asour reference level, since that remains a welldefined quantity even when the quasiparticle populations are far from equilibrium. Moreover, it isthe quantity which defines the location of the"mirror plane" of the semiconductor diagram.With this convention, all incoming electrons fromthe S side have the distribution function !o(E),while those coming in from the N side aredescribed by !o(E -eV).

! -+(E)=!o(E -eV) ,while

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IN S = 2N(O)evpd 1_000(!o(E - eV) - {A (E)!o(E +eV)+B(E)!o(E -eV)

+ [ I -A (E)-B(E)lfo(E)} )dE=2N(O)evpd 1_0

00[fo(E -ev)-!o(E)][l+A(E)-B(E)]dE . (17)

In obtaining the final form, we have used the properties that A +B + C +D = 1,A (E)=A (-E),and !o ( -E )= 1 -fo(E). The quantity [1+A (E)-B(E)] in Eq. (17) can be referred to as the"transmission coefficient for electrical current."Its form shows that while ordinary reflection

[described by B (E)] reduces the current, Andreevreflection [described by A (E)] increases it by giving up to two transferred electrons (a Cooper pair)for one incident one.If both sides of the interface are normal metal,

A = 0 since there is no Andreev reflection, and


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4522 G. E. BLONDER, M .. TINKHAM, AND T .. M.. KLAPWIJK 25I -B =C = ( 1+Z2)-1 as noted above, so that Eq.(17)reduces to the simple form

I - 2N(O)e 2vpd v=L (18)N N - I+Z 2 - RN

Note that even in the absence of a barrier (Z =0) ,there is still a nonzero normal-state resistance.This is the spreading resistance typical of a twodimensional (2D) or 3D geometry, often referred toas the Sharvin resistance'" in the present case of apoint ...contact geometry with infinite mean freepath relative to orifice size. .For the more interesting N-S case, we have Integrated (17) numerically to obtain J ...V curves for anumber of barrier strengths at T =0. The resultsare shown in Fig .. 6.. The effect of changing the

barrier strength is even more dramatic in the differential conductance dl /dV, plotted versus V inFig .. 7, again at T =0. From Figs. 6 and 7, it isclear that barrier strengths Z > 10 give resultsessentially indistinguishable f;;;m those for classical tunnel junctions, although, as our modeldemonstrates, there is a continuous variation fromthe metallic to the tunneling limit. To facilitatethe interpretation of experimental dI /dV curves, itis useful to note at T =0, Eq, (17) implies thatdI /d V is proportional to the transmission coefficient for electrical current, 1+A (eV)-B(eV),while at T::j:=O, dl /dV is proportional to a thermally smeared version of the same quantity.

v. RELATION TO TRANSFERHAMILTONIAN APPROACHWith the arbitrary barrier case well in hand, we

are in a position to contrast our technique with themore conventional transfer Hamiltonian approach ..If one assumes a thick oxide between two banks, tolowest order one can consider the two sides asnoninteracting systems, containing electrons instanding-wave states labeled by quantum numberskL or kR .. Of course, they are not truly unconnected and there is a small probability that an electronfrom one side can tunnel to the other side; thus thewave function must actually penetrate through thebarrier. Since this penetration will change the en...ergy of the system, many workers have tried tomodel the resulting change by adding a "transfer"term to the Hamiltonian, although this phenomenological model has never been fully justified..15The tunneling transfer matrix element T is trickyto evaluate, since it is a function of the amplitudeof the small tail on the penetrating wave and requires some microscopic knowledge of the system.Nonetheless, many workers have calculated T, andthe result, for eV much smaller than the barrierheight, is an essentially featureless matrix elementmuch like that for an NooN contact ..By treating the tunneling as a perturbation, onecan calculate the current by using the GoldenRule, and find the electron-transfer rate proportional to a squared matrix element times the number density of unoccupied final states.. This is inthe origin of the BCS density-of-states factor timesthe thermal weighting factor [1 - f (E - eV)]

2

T=O

eVT

z=o2

oZ=1.5

1 - -------o l::. 2h.

Z ::0.52

o b. 21::.Z;;;5.0

1...---4--__ eVoFIG .. 6. Current vs voltage for various barrierstrengths Z at T ==O. These curves attain their asympototic limits only for very high voltages. For example,the tunnel junction (Z ==50) curve will be within 1% ofthe normal-state curve (dotted line) only when eV 2 7A.

FIG. 7. Differential conductance vs voltage for various barrier strengths Z at T ==0. This quantity is proportional to the transmission coefficient for electriccurrent for particles at E =eV.


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25 TRANSITION FROM METALLIC TO TUNNELING REGIMES IN . . . 4523which appears in the tunneling current, for particles from L to R. When we integrate over allstates on the left, an additional NL(E)f(E) appears, for a total probability (per unit energy) ofIT 1 2N L (E )N R (E -eV) f (E)[ l - f (E -eV)]

found by extrapolation back to the V =0 axis. Bymanipulation of Eqs. (16) and (17), this I exc can bewritten asI exc== (INS - INN) IeVa1

FIG. 8. Excess current (in units of a/eRN) as afunction of barrier strength Z. The temperature dependence of the curve is entirely contained within a( T).

eRN[l-B( 00)]X fooo[A (E)-B(E)+B( 00 )]dE, (19)

which simply gives the extra current due to thesuperconductivity. Since INN (V) is simply V /RN ,finding I exc(V) is equivalent to giving the entire IV curve, as was done in Fig. 6. However, it isworth noting that for Z =0 and a k T , the calculation can be carried out analytically, with theresult,

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2.0z.0

0.5

I exc( V) = IN s (V)-INN ( V) ,

eI excRN- -1 : : : : . -1.0

43"

where from Table II, B (00 )=Z2 /( 1+Z2) is thereflection coefficient at high energy or in the normal state. Using this expression and those inTable II, we have computed numerically the dependence of I excRN (the "insufficient voltage" referredto by Likharev') upon barrier strength Z. Sincethis high-voltage limit depends on T only throughthe magnitude of ~ the normalized quantityI excRN /11 is a function only of barrier strength.This dependence is plotted in Fig. 8. Note therapid falloff with increasing Z; clearly, excesscurrent will be seen only in relatively barrier-freecontacts.More generally, one can define a voltage-dependent excess current,

NL(E)NR(E - eV)[ f (E - -eV) - f(E)] .

VI. EXCESS CURRENT

The latter tw o factors reflect the view that this is astochastic process involving two independentbanks, so that the joint probability for tunneling isproportional to the probability of the initial statesbeing occupied times the probability of final statesbeing unoccupied (Le., unblocked). When we combine currents in both directions, the cross terms involving products of Fermi functions cancel, and weare left with the usual tunneling result in whichthe current is proportional to an integral over

The Bogoliubov equation approach is quite different in spirit. As we saw above, the currenttransmission coefficient (in contrast to the matrixelement squared in the tunnel calculation) variesrapidly for IE I 11. This is the origin of thedensity-of-states factor in the final result. In nosense are the two banks decoupled; we associate agiven set of wave vectors on the left (q +, - q +,q - )with a specific set on the right (k +,k -), regardlessof the barrier height. q + is not an "initial" statefrom which transfer can be blocked by an occupied"final" state k +. Rather, a wave packet describingan incident particle in q+ evolves continuouslyinto transmitted and reflected wave packets in thecompletely deterministic way described by Liouville's theorem without scattering. Thus, we associate one distribution function with this entire setof wave vectors. Combining currents originatingfromboth sides gives a difference in Fermi functions just as the transfer Hamiltonian did (aftercancellation of cross terms), but the rationale is entirely different in detail. Since we can achieve thesame result with a model that has a larger range ofvalidity and usefulness, we have chosen this latterpoint of view.

Although the high-voltage (eVI l ) portion ofeach 1-V characteristic in Fig. 6 is linear with slopeRN , it does not in general fall on the normal-statecurve V =IRN (as it does in a tunnel junction).Rather, it is displaced by a constant amount referred to as the excess current, I exc, which can be


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As noted in our previous work, this result agrees indetail with that obtained by Zaitsev" for a cleanmetallic NS contact, after correcting a factor of 2in his result.We can also calculate the excess current for an

S-S' contact, by using the N-S result above. As weexplained in a previous publication.' in the highvoltage limit, the 8-8' device can be thought of asa series combination of an S-N and a N-S' microconstriction. The "insufficient voltages" of thetwo simply add, so that the excess current is thesum of the contributions from the two sides. Forexample, in the clean contact with Z =0, we find

I exc==4 ( i l + L \ ' ) / 3 eR N While the excess current has a clear mathematical definition, it may be a difficult quantity to

measure reliably, for at least two reasons. First,for voltages greater than the gap, heating will generally distort the 1-V curve away from its ideal. h 1 16IS0t erma shape. There may, in fact, be nolinear region at high voltages until heating hasdriven the contact fully normal. Secondly, even inthe absence of heating, care must be taken to carrythe measurement to sufficiently high voltages toobtain a true asymptote. For example, the tunneljunct ion curve in Fig. 6 (i.e., Z = 50) appears tohave a significant negative (rather than zero) excesscurrent , even at a voltage as high as eV = 3L\. Ofcourse, if one assumes our computed 1-V curvesare correct and applicable to the contact at hand,one can determine a value of Z by fitting themeasured curves at lower voltages.

VII. ANALYSIS OF CURRENTSAND CHARGE-IMBALANCE GENERATIONIn this section we establish a formalism for

separating the total current I.NS into parts differentiated according to the mechanism of chargetransfer. In particular, we distinguish the part ofthe current I ~ associated with the creation ofquasiparticle charge imbalance Q* in the superconductor from the part that is converted directly intosupercurrent. We also define linear-response coefficients giving the normalized differential conductance Y and its counterpart Y*, describing quasiparticle charge injection, and evaluate them as

functions of Z and T.For tunnel junctions, the rate of generation of

quasiparticle charge branch imbalance has beendiscussed in some detail by Tinkham and Clarke'?(TC), by Tinkham.l" by Pethick and Smith"" (PS)and by Clarke, et al.20 (CESST). The latter au- 'thors reviewed carefully the interrelations of thedefinitions and conventions of the various treatments including the rather different conceptual approach of Schmid and Schon." Using our presentmethod, we can generalize the tunnel results to arbitrary barrier strength, providing a descriptionwhich covers all cases from the tunnel l imit to theideal metallic contact, and which gives very detailed insight into the process.To minimize repetition, we refer the reader toCESST for an index of the earlier work, and only

summarize the essential points. Noting that quasiparticles at k have a charge

the total quasiparticle charge (per unit volume inunits of e) is '(23)

where fk is the actual occupat ion number in anonequilibrium system. To retain electrical neu~ r a l i t y , there must be an equal and opposite changeIn the condensate charge, produced by a shift inJ-ls' [The quantity Q defined earlier by TC differsfromQ* by replacement of qk in Eq. (23) bys ~ n q ~ = 1, and represents the numerical population imbalance between the branches. Since Q* appears to have more general physical significance,we confine our attention to it in this paper.]To illuminate this problem, it is helpful torewrite the expression (17) for the current in aform emphasizing the transmitted (rather than reflected) particles by using the sum rule (13) to replace 1+A -B by 2A +C +D. Thus, we write thetotal current as

IN S =::; 2N(O)eVpdX J_"',)Jo(E - eV ) -Jo(E)]

X[2A(E)+C(E)+D(E)]dE.(24)

This can be decomposed into two parts byrecog-nizing that the term 2A (E ) represents the current


11/18

2S TRANSITION FROM METALLIC TO TUNNELING REGIMES IN . . . 4S2Sl ~ s = 2 N ( 0 ) e v p . f ! 2

X I: ,) fa(E-eV)-fa(E)]X [2A (E)]dE , (25)

due to Andreev reflection, in which the current inthe superconductor is carried entirely by pairs.The remainder of INs is the current1 ~ ~ = 2N(0)evp.f!2

X I:co [fatE - eV) - fatE)]X[C(E)+D(E)]dE, (26)

due to quasiparticle transfer. But even I ~ doesnot represent the rate of generation of quasiparticlecharge at the interface, which we will call I ~ s . Tocompute that, we must distinguish quasiparticles inthe two channels, k + and k - , and also take account of their fractional charge qk= Ns (E) -I.Thus,

l; ';s= 2N(0)eVF.f!2X I_coco [fa(E -eV)-fa(E)]

X [C(E)-D(E)]Ns-I(E)dE .(27)

Finally, since experiments usually involve currentbias rather than voltage bias, it has been useful todetermine the ratio

that, but it is important only for weak barriers.Again taking the strong barrier limit to evaluateI;s, we note from Table II that (C -D)=Z-2 , sothat Eq. (27) becomes

* 1 fooNs = - [fo(E-eV)-fo(E)]eRN -00

(30)again exactly equivalent to the standard tunnelingresult for Q7nj.

For comparison, we now consider the samequantities in the other limiting case of no barrier,Z =0. Then, as noted in Table II, B =D =0 andC = I -A. In this case, the total current is1 fooIN S = -R [!o(E - eV ) - !o(E)]e N -00

X[I+A(E)]dE', Z=O.(31)

The Andreev portion of the current I ~ is stillgiven by Eq. (25), but I ~ can be rewritten interms of A (E ) asIi!s= -R f 00 [fo(E -eV)- fo(E)]e N -00

X[ I -A (E)]dE, Z =0F*=.I* /1 , (28)giving the charge imbalance generated as a fractionof the injected current. These quantities have beenevaluated in the past for tunnel junctions, but nowwe can evaluate them for arbitrary barrier strength,as a function of V and T.To make contact with the earlier work on tunnelinjection, we note that for strong barriers the results in Table II show that A ex:Z - 4


12/18

4526 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWIJK 25In the important special case of very low voltages across the contact (eV


13/18

25 TRANSITION FROM METALLIC TO TUNNELING REGIMES IN . . . 4527Z=o2.0-- - - -=-- - - - - -Y(Z,T)

1.0,-----------------F*0.9

Z=O.3

0.4

0.7

0.5

0.6

0.8

.............r-- -'"........ AVZ-DIRTYZ=0.55 ........

Z=O.9

1.5

1.01--==-----,4----1

0.5

1.0r---------------

FIG. 9. Y, the zero-voltage differential conductancein units of the normal-state conductance, vs T, for various barrier strengths Z. The curve AVZ-dirty isdescribed in the text.Green's-function method.

OUf charge-imbalance results may also provide auseful perspective on the work of Hsiang andClarke22 on resistance due to charge-imbalance relaxation near an N-S interface. By a physical argument, they reasoned that the appropriate F* fora clean metallic interface could be approximated bywhat we call here y* (Z >> 1) as computed for atunnel junction. OUf calculation would insteadgive F* = Y*(Z =O)/Y(Z =0). As just notedabove, Y* is almost independent of barrier strengthZ, so Y*(Z 1);,:::: Y*(Z =0). But the denomina-

2=10Z=0.9

o0.3 0.4 0.5 0.6 0.7

0.2

0.3

0.8 0.9 1.0T/Te

FIG-. 11. F*, the ratio of quasiparticle charge injected to total injected current, as a function of T for various barrier strengths Z.

Zaitsev - CLEAN-Z

ex> 2 1.5 1 0.5

tor factor Y(Z =0) varies from 1 at T; to 2 atT =0, so the resulting T dependence of our F* issomewhat different from that used by Hsiang andClarke, although it has the same qualitative shape.Because the interfaces studied by Hsiang andClarke were not point contacts of the sort we havebeen treating, however, it is not clear that our results should be expected to describe their data indetail.

1.0Z=10

0.5

V*(Z,T)1.0

0.5

Z=10

AVZ-DIRTY

0.5

z=o

FIG. 10. Y*, the normalized differential charge imbalance conductance at zero voltage, vs T, for variousbarrier strengths Z.

o 0.5 1.0 012(Z,T=O)

FIG. 12. e l e x c R N / ~ vs Y(T=O) as the barrierstrength parameter Z ranges from 00 to O. The pointsAVZ-dirty and Zaitsev-clean are discussed in Sec. VIII.


14/18

4528 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWUK 25As noted above, the total current IN S can be expressed as the sum of three terms: Its, the An ...dreev term with no quasiparticle transfer; I ~ s , theinjection rate of quasiparticle charge; and ( I ~ ~

-I;;s), the current carried by the pairs that mustflow to compensate for the fractional charges ofthe quasiparticles. The current I ~ relaxes to supercurrent over a distance AQ*= (DTQ* )1/ 2, a dif-fusion length determined by TQ*' the chargeimbalance relaxation time.

The picture described in the previous paragraphcorresponds to the viewpoint of TC and PS, inwhich the current (I -1*) is viewed as being converted discontinuously to supercurrent at the interface.23 If one examines the problem more deeply,however, one can obtain a more microscopic viewof the conversion process. The key point is thatthere are solutions of the Bogoliubov equationseven for IE I < ti., but they are evanescent waveswhich decay in a distance

(zvF I 2(Ji.2_E 2 )1/2 -...s( T) .(See the Appendix for details.) Thus, even the Andreev current Its is carried for ~ as a quasiparticle current before decaying into a pair current, andthe interference of k + and k - waves causes asimilar effect also for the current I ~ - I ~ abovethe gap. This two-step conversion process isshown schematically in Fig. 13. This point ofview is reminiscent of that of Schmid and Schon,who treat the entire injected current as a source of

charge imbalance, but combine "conversion" termswith rQ* relaxation to find a shorter effective re-laxation time TR < TQ*' which reproduces the samesteady-state Q* as found by the two-step analysisof Tinkham and Clarke17t18 and of Pethick andSmith.!" There seems to be a gain in physical insight, however, in distinguishing the two types ofprocesses.Recognition of the existence of a penetrationdepth of order g for the evanescent quasiparticlewaves in the gap also provides a more clear-cut ex-.planation for one aspect of the observations of Octavio et al.16 on subharmonic gap structure (SGS)in tin variable-thickness bridges (VTB's). They observed voltage-dependent shifts in the positions ofthe SGS which they attributed to heating. Usingthe gap shift as a thermometer, with Ji.BCS(T) asthe calibration curve, they could infer temperaturerises. These temperature increases were roughlyproportional to the dissipated power P, but much

0.3r------r------- ,- - -- . ,- - - - ,- - ---- . ,

lI.J(f)it: 0.21::l

WQ.

o 0.1-w0::0::lI.Ju,0.0 opec I0.0 0.5

N s o.--Isr*

FIG. 13. Schematic diagram of conversion of normalcurrent to supercurrent at N-S interface. The evanescent waves due to Andreev reflection die out over a distance of order 6, while the injected quasiparticle chargeimbalance relaxes over a charge diffusion length AQ*.

A 0"1, #11 n=l, #4o 0=2, #4

Q1 i-0;:0;:

4 6PIe (Tb) ("W/JIom)

FIG. 14. Temperature rise in tin variable-thicknessbridge inferred from depression of energy gap as foundfrom subharmonic gap structure (after Octavio et al.i.(a) depicts temperature rise vs power dissipated, P. (b)depicts temperature rise vs P Is( T). The superior linearfit in (b) is consistent with observed gap being sensed ata distance roovS from center of bridge.


15/18

25 TRANSITION FROM METALLIC TO TUNNELING REGIMES IN. 4529

a [ -fiv2 1ii a = - 2m - /L(x)+ V(x) g(x,t)

There are two roots for E. We are concerned withexcitations above the ground state (E 0), and inthis case

fi2= 1.[1+ (E 2_a2)1/ 2 ] = l-v2 (A4)2 - E 'Ilk =v1'm [JL(E2_a 2) 1/ 2 ] 1/ 2 . (AS)

We also define

(A3)

(A2a)

(A2b)

(Ala)

(Alb)d(x)!(x, t) ,

E -- 1,,2k2 1- A-U = 2m -f.-l U +L.lU ,E- [fPk2 1- A-U= - 2m -JL v+L.lU

Solving Eqs. (A2a) and (A2b) for E we obtainE 2= [ ~ : 2 ~ / r+a2.

where d(x) is the energy gap and JL(x) is thechemical potential (specifically that of the condensate, in case of nonequilibrium conditions where JLis not uniquely defined). In the normal metal[a(x)=O], Eq. (Ala) is the Schrodinger equationfor electrons, while Eq. (Alb) is the time-reversedSchrodinger equation for electrons. Since an electron satisfying the time-reversed Schrodinger equation behaves in many ways like a hole, we willadopt that terminology here.

For a:#:o, the electron and hole wave functionscouple together, with two major consequences.First , a gap appears in theE vs k relation, just asit does in elementary BCS theory. To see this result more clearly, we solve Eq. (AI) for a specificgeometry. The simplest example is one wheref.-l(x), a(x), and V(x) are all constant, and thesolutions to Eq. (AI) are time-independent planewaves. Taking as our trial solutions! =iie ikx -iEt/1i and g =fJe ikx -iEt/li, we find forv=o

more precisely proportional to P / s( T), as shownby the comparison of Fig. 14(a) and 14(b). Giventhe 3D cooling of these VTB's, in which the temperature rise is proportional to P /r (where r is thedistance from the center of the contact), this observation was interpreted as showing that the energygap giving the SOS was that in the superconductorat a distance ~ from the center. In a recent paper,s we have argued that the SOS arises frommultiple Andreev reflections together with thesharp change in A (E ) at the gap edge. Given thepenetration of the evanescent waves to a depth( T), it is plausible that the gap reflected in theSOS should be that at a similar depth into thebank, as assumed by Octavio et ale on the basis ofsomewhat different reasoning.

The Bogoliubov equations may be written

ACKNOWLEDGMENTS

IX. CONCLUSION

APPENDIX

We have proposed a simple theory for the N-Smicroconstriction or point contact that is generalenough to describe the complete crossover frommetallic to small-area tunnel-junction behavior.From this model, a deeper understanding of thesemiconductor picture, excess currents, currentconversion, and charge-imbalance processes haveemerged, in addition to calculation of 1-V curvesfor arbitrary barrier strength. The latter results aredirectly accessible to experiment, and should behelpful in interpreting the wide variety of 1-Vcurve shapes observed in the laboratory. The application of this model to the S-N-S case has beenpresented elsewhere.i where it has been shown tooffer an explanation of the SGS observed in metallic weak links.

One of us (M.T.) would like to acknowledgestimulating discussions at the initial stage of thiswork with J. Clarke, U. Eckern, A. Schmid, G.Schon, and A. F. Volkov. The work was supported in part by the National Science Foundation, theOffice of Naval Research, and the Joint ServicesElectronics Program. One of us (T.M.K.) wouldlike to thank the Nederlandse Organisatie voorZuiver Wetenshappelijk Onderzoek for a grant.


16/18


(A9)

Q=e( If 12 - 1g 12 ) . Using Eq. (AI), it is easy toshow thatoQ +V.jQ= 4ea Im(j *g )at Ii 'where-- efzJQ=-[Im(f*Vf)+Im(g*Vg)] . (AIO)m

The term on the right in Eq. (A9) is a source (ordrain) term connecting the quasiparticles with thecondensate, and we will see an explicit interchangeof currents when treating the N-S case below.However, for now we concentrate on Eq. (AIO).Both the electron and hole contribution to thequasiparticle current enter with the same sign, anda moment's thought shows this to be reasonable.An electron ( f a: eik+x ), moves in the positivedirection and carries a posit ive current. A hole,(g cce ik +x ), moves in the negative direction, bu talso carries a positive current due to the sign of itscharge. Thus, while Jp a: vg =0 at the gap edge,JQ==evp, so one can view the charge current astraveling with the Fermi velocity. Of course, forthe normal metal nothing so dramatic occurs, andwe always find simply JQ =eJpoStarting from the basic Eq. (AI), it is a s implematter to work out the boundary conditions onsteady-state plane-wave solutions at an N-S interface. To model the elastic scattering that usuallyoccurs in the orifice, we include a B-function potential in Eq. (Al), i.e., V(x)=H8(x).The appropriate boundary conditions, for particles traveling from N to S are as follows.(i) Continuity of 1/1 at x =0, so tPS(O)=1/JN(O)

=tf!(O)e(ii) (1i/2m)( tP's - tPN )=H t/J( 0), the derivativeboundary condition appropriate for B functions.(iii) Incoming (incident), reflected, and transmit

ted wave directions are defined by their group velocities. We assume the incoming electron producesonly outgoing particles.Thus,

(A7)

(A6b)and

tP+k-= [vo ]eik-X .- Uo

ap +v.lp =oat 'where

..... 'IiJ p=-[Im(f*Vj)-Im(g*Vg)] . (AS)m

The 1//s are the position space representation of therkO=UkCZt --vkc-k! since Ck t produces the wavee ikx , while c - k destroys an electron at - k, leaving the system with a net posit ive momentumrepresented by the creation of a hole wave e ikxA second consequence, which follows from the

E(k) relation (A3), is that the group velocityvg=dE /dhk goes to zero at the gap edge, evenwhile the phase velocity remains near the normalstate value. This has some interesting consequences, as can best be illustrated by deriving con..servation laws for probability and charge.We define Pt.x) to be the probability density forf inding either an electron or a hole at a par ticular

time and place. Thus , for a solu tion to Eq. (AI),P(x,t)= i f 12+ Ig 12 Using Eq. (AI) to evaluateap fat, we find

For IE I< t1, it and vwill be complex conjugates.This definition differs from the usual BeS convent ion in which Uk and Uk are not defined for E < a.Using the notation

_ [ f (x)]t/J- g(x) ,the four types of quasiparticle waves for a givenenergy take the forms

1/1 = [uo ]eik+X (A6a)k+ Vo

Note that the hole current enters Eq. (AS) wi th asign opposite to that of the electron part, just asone expects for a "time-reversed" particle. Onecan also show that Jp a: vg = aE/afik. Thus, at thegap edge, it is zero.We can also derive a conservation law for quasiparticle charge. Assigning a unit charge +e to theelectron and -e to the hole, the net quasiparticlecharge density in one of these excitation waves is

..I. - [I ] iq+x t n - A ;;:;--2 ;--:-:;:::;E' f ' inc- 0 e , ' "1 -y an -y J . L ~ ,,,1. [0] iq-x b [I] - iq+x' f ' ref l=a I e + 0 e ,"I, =c [uo ]eik+X+d [vol : .f'trans uo Uo

Applying the boundary conditions, and carryingout an algebraic reduction, we find


17/18


2 (. 2 2 )Z2r =uo+ uo-Vo For simplicity, we have let k+=k-=q+=q=kF , whenever that substitution did not lead to aqualitative charge. Note that in the absence of abarrier (i.e., Z =0), b =d =0. Physically speaking,this means that all reflection is Andreev reflectionand all transmission occurs without branch crossing.The quantities A (E), B(E), C(E), and D(E)used in the main body of the paper are actually theprobability currents, for the particle, measured inunits of VF. For example, A =Jp/vF=a*a, andD =d*d(u5-v6). Since plane-wave currents arespatially uniform, we need not specify the positionat which they are evaluated. However, forE >O. Explicitly,Ja=(efzlm)Im[( ceik+x)*V(ceik+x)

X ( u ~ u o + v ~ v o ) ] .Letting k + kF+i 11 /liVF' wheren= (6.2- E 2)1/2,we have

J.c I Uo 1 2+ 1 Vo 1 2 -2T/x/fwpQ= IUo /2 eVFe Below the gap, I Uo 12= I Vo 12, so

TC 2 -2T/x/fwpJQ= eVFeThe "disappearing current" reappears as currentcarried by the condensate. Rewriting the drainterm in Eq. (A9) as

-V-l s =:(4eA/ Ii )Im(f *g ) , (A13)

(AI4)2 . (1 -2T/x/wp)- eVF -e .

then

(AI2)

(AIle)

(AlIa)

(Allb)

(Alld)

fl [ [ 121 1/ 2A= 2 : 1 -

mHZ = _.-_ . =H /flVF ,~ k

UOVoa=--,r(u5 -v5)(Z2+iZ)b=- - - - - - - -ruo( l - iZ)r

ivoZd=--,rwhere

and

Although right at the gap edge the length diverges,the characteristic length is flvF /2A= 1.22g(T).Thus, in order of magnitude terms, one can saythat the particles penetrate a depth s( T) beforethe current is converted to a supercurrent carried

This is the desired result, explicitly showing the supercurrent increasing to an asymptotic value asx 00 , at the same rate as the quasiparticlecurrent dies away.

Permanent address: Laboratorium voorTechnischeNatuurkunde, Technische Hogeschool Delft, Delft,The Netherlands.ISee, for example, W. L. McMillan and J. M. Rowell, inSuperconductivity, edited by R. D. Parks (MarcelDekker, New York, 1969),Vol. 1, p. 561.

2K. K. Likharev and L. A. Yakobson, Zh. Eksp. Teor.Fiz. 68, 1150 (1975) [Sov. Phys.-JETP 41, 2301(1975)].3S.N. Artemenko, A. F. Volkov, and A. V. Zaitsev,Pis 'ma Zh. Eksp. Teor. Fiz. 28, 637 (1978) [JETPLett. 28, 589 (1978)]; Zh. Eksp. Teor. Fiz.76, 1816


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4532 G. E. BLONDER, M. TINKHAM, AND T. M. KLAPWIJK 25(1979) [Sov. Phys.-JETP 49, 924 (1979)]; Solid StateCommun. 30,771 (1979).

4A. V. Zai tsev, Zh. Eksp. Teor. Fiz. 78, 221 (1980)[Sov. Phys.-JETP l l , 111 (1980)].

sT. M. Klapwijk, G. E. Blonder, and M. Tinkham, inProceedings of the 16th International Conference onLow Temperature Physics, [Physica B + C (in pressl].

6p. G. DeGennes, Superconductivity ofMetals and Alloys (Benjamin, New York, 1966), Chap. 5.

7See, for example, M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1975), Chap.2.8A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964)[Sov. Phys.-JETP 19, 1228 (1964)].

9R. Kiimmel, Z. Phys, 218, 472 (1969).10J.Demers and A. Gri ff in , Can. J. Phys. 49, 285(1970).

11A. Griffin and J. Demers, Phys. Rev. B 4, 2202(1971).120 . Ent in-Wohlman, J. Low Temp. Phys. 27, 777(1977).

13Scattering in the bulk near an S-N boundary introduces some interesting complications based on thecoherence on the Andreev reflection process. Thesehave been discussed qualitatively by A. B. Pippard, inNonequilibrium Superconductivity, Phonons, and Ka-

pitza Boundaries, edited by K. Gray (Plenum, NewYork, 1981), Chap. 12, and references therein.14yu. V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965)[Sov. Phys.-JETP 21, 655 (1965)].

15C. B. Duke, Tunneling in Solids (Academic, NewYork, 1969).16M.Octavio, W. J. Skocpol, and M. Tinkham, IEEETrans . Magn. MAG-13, 739 (977); M. Octavio,Technical Report No. 13, Div. of Appl. Sciences,Harvard University (unpublished).

17M.Tinkham and J. Clarke, Phys. Rev. Let t. 28, 1366(1972).18M.Tinkham, Phys. Rev. B fi, 1747 (1972).19C. J. Pethick and H. Smith, Ann. Phys, (N.Y.) i l l ,133 (1979).20J. Clarke, U. Eckem, A. Schmid, G. Schon, and M.Tinkham, Phys. Rev. B 20, 3933 (1979).

21A. Schmid and G. Schon, J. Low Temp. Phys. 20,207 (1975).22T. Y. Hsiang and J. Clarke, Phys. Rev. B 21, 945(1980).

231n the notat ion of Pethick and Smith, 1 corresponds-Q *o our 1*, and their J s corresponds to our I- I .

Similarly, their 1n corresponds to our JQ as definedin th e Appendix.

Documents

Blonder, Transition From Metallic to Tunneling Regimes in Superconducting Micro Constrictions