theoretical investigation of josephson tunnel junctions with spatially

Journal of Low Temperature Physics, Vol. 70, Nos, 1/2, 1988

Theoretical Investigation of Josephson Tunnel Junctions with Spatially Inhomogeneous

Superconducting Electrodes

A. A. Golubov and M. Yu. Kupriyanov

Institute of Solid State Physics, Academy of Sciences of the USSR, Chernogolovka, Moscow, USSR

(Received May 20, 1987)

The microscopic theory of the Josephson effect in. tunnel structures with electrodes having spatially inhomogeneous superconducting properties is formulated. Two mechanisms of inhomogeneity are considered. The first is associated with the presence of a thin transition normal layer located near the tunnel barrier, which is relevant for junctions based on refractory superconductors. The second case is the trapping of Abrikosov vortices by junction electrodes. The tunnel current components are calculated numerically in the whole temperature range 0 < T < Tc and magnetic field range 0 < H < Hc2. It is shown that the tunnel current is extremely sensitive to the type of smearing of the singularities of the classical tunnel theory at eV = 2A. The results allow experimental determination of the characteristics of real tunnel junctions.

1. INTRODUCTION

Josephson tunnel junctions based on refractory superconductors, such as Nb, NbN, or Nb3Sn, are widely studied and used. ~-9 The behavior of such junctions displays some characteristic features that cannot be described in the framework of the standard tunnel theory of the Josephson effect. ~°'H For instance, the quasiparticle curve of the current-voltage characteristic (CVC) exhibits knee structure near the sum-gap voltage Vg = (A~ + A2)/e, 4'5 leakage current at low voltages eV < A1 + A2, and a supplementary singularity at eV = Aa,2.12'13 In addition, the temperature dependence of the critical current I~ differs from that predicted by the Ambegaokar-Baratoff (AB) theory IA~(T),4'5 with I t (T) significantly suppressed as compared to IAB( T).~2-~5 Attempts have been made to explain these phenomena by taking account of the influence of inelastic relaxation processes in superconducting electrodes and under tunneling. ~6 This theory accounts for the smearing of

83 0022-2291/88/0100-0083506,00/0 ~) 1988 Plenum Publishing Corporation

84 A.A. Golubov and M. Yu. K'upriyanov

the singularities of tunnel theory at eV = m 1 - - [ -m2, but does not explain the other above-mentioned experimental results. From our viewpoint this phenomenon is induced by the spatial inhomogeneity of the superconducting properties of the electrodes in tunnel structures.

One reason for such inhomogeneity is a thin transition layer located near the tunnel barrier, consisting of various niobium oxides, one of them (NbO) possessing the properties of a normal (N) metal. Such an N layer is formed during the fabricating process. 9 As a result, one or both electrodes of the tunnel contact comprise an NS sandwich (see Fig. 1).

The second cause is the trapping of Abrikosov vortices by junction electrodes, the vortices being normal to the plane of the electrodes. Indeed, a simple estimation shows that a magnetic flux of the order of the flux quantum @0 penetrates the junction area of the order of 10 x 10/zm 2 in the earth's magnetic field ( H - 0 . 3 G). In real Josephson junctions trapped vortices may be conserved in electrodes due to pinning even when the external magnetic field is totally switched of/.

The pattern of frozen vortex lines depends essentially on the material of the electrode films. As a rule, they have a granular structure with a characteristic granule size L determined by the technology of film prepar- ation. The value of L may vary in a wide range. For instance, for the Pb-In-Au alloy films 17 widely used in Josephson structures, the parameter L decreases from 4000 to 500 A with the concentration of Au varying from 0 to 10wt%. When the granule size L is small compared to the electrode coherence length ~:, the vortex nucleus efficiently averages the pinning forces acting upon it from the grain boundaries, the curvature of vortex lines being highly improbable and their structure being close to that shown in Figs. 2a and 2b. In the opposite case the magnetic flux penetrates the electrodes

I N|N S S

Fig. 1. (a) SNINS and (b) SNIS junctions.

Theoretical Investigation of Josephson Tunnel Junctions 85

i L

a)

Fig. 2. The patterns of frozen vortex lines in junction electrodes.

mainly along the grain boundaries and leads to the formation of curved vortex lines (Fig. 2c) or to their trapping in one of the electrodes (Fig. 2d).

Furthermore, tunnel junctions with a specially constructed spatial inhomogeneity of the electrodes can be effective tools for studying material properties. For example, structures of the SNIS and SNINS types are used for examination of N-metal properties TM and also for improvement of junction characteristics. 19 The influence of a magnetic field normal to the junction area was studied experimentally 2° and used for the investigation of weak localization effects.

The properties of SNINS- and SNIS-type structures have been described by the method of the tunneling Hamiltonian with modified (as compared to the results obtained by the BCS theory) densities of states at a tunnel barrier. ~-3'5-9'21 These modifications take place due to the proximity

86 A . A . Golubov and M. Yu. Kupriyanov

effect in electrodes, which has been interpreted in terms of the McMillan proximity effect tunneling model. 2: This model assumes the presence of a supplementary potential barrier with low transparency at the SN interface and a small thickness of N and S layers compared to the coherence lengths of these metals. These assumptions make it possible to consider the order parameter as constant in each of the N and S regions and to describe their relationship by the tunneling Hamiltonian method. But in real structures the thickness of the S electrodes generally exceeds its coherence length, and the metals forming the SN sandwich are in good electrical contact (unless precautions were taken in the process of its production). In these conditions the coordinate dependence of the anomalous Green's functions turns out to be essential, 23 the penetrability of the NS interface is close to unity, and the McMillan model is inapplicable.

In ref. 24 an attempt was made to generalize the McMillan model for the case of arbitrary transparency of the supplementary barrier at the NS interface by taking account of higher orders of perturbation theory. However, as shown in ref. 25, the method of the tunneling Hamiltonian yields results coinciding with the predictions of the microscopic theory o n l y in the second order in transparency, and such a generalization thus cannot be considered correct. The effect of tiae space dependence of the order parameter on the properties of the structures under study was taken into account in the temperature range T ~ Tc only. 26 The conditions at the SN interface 27 used in this paper are not correct, 28 and make it possible to estimate the value of Ic at T ~ To.

Finally, in refs. 29-31 a microscopic approach based on calculating the coordinate dependence of the Green's functions was developed. In this case a piecewise model was used effectively for the order parameter that, as in refs. 23 and 28, is valid on lyunder certain relations of the parameters of the materials forming the SN sandwich.

The influence of Abrikosov vortices on the properties of tunnel junctions was studied in a phenomenological approach, 13 with each vortex considered as a normal cylinder having radius ~:. But the real situation is far more complicated. An analysis of the configurations (Fig. 2) allows us to conclude that there exist two mechanisms by which the Abrikosov vortices effect the properties of tunnel junctions. The first one, the "kern" (= nucleus) mechanism, is associated with the modification of the Green's functions of superconducting electrodes in a region of the order of the coherence length in the vicinity of the vortex nucleus. It was shown in ref. 32 that this mechanism is unique in the case of vortices exactly aligned in both electrodes (Fig. 2a). The second, electrodynamical mechanism is based on the bending of the vortex lines, which gives rise to a coordinate dependence of the order-parameter phase difference of the electrodes. Numerical calculations


for the case of SNS sandwiches with a thick interlayer of a normal metal indicate the importance of this mechanism. 33

In sum, no co.qsistent microscopic theory of the Josephson effect in tunnel structures with spatially inhomogeneous superconducting properties of the electrodes has been developed. The aim of the present paper is to construct a microscopic theory of the Josephson effect in such systems.

In formulating the theory, we assume that the dimensions of the tunnel junction satisfy the condition sos<< W~)tj, where hj is the Josephson penetration depth and the electrodes are dirty superconducting films with Ginzburg-Landau parameter ~ >> 1. According to the condition W <~ h j, the current passing through the junction is uniformly distributed over its trans- verse cross section. Suppose the tunnel junction shows low transparency, that is, the effect of currents on the state of electrodes can be neglected. The low transparency of the junction makes it possible to analyze its properties by means of the formulas of the classical tunnel theory. 1°'" According to this theory, the current is determined by the retarded Green's functions F R and G R in the electrodes.

In our case the functions F R and G R are space-dependent. But due to the smallness of the electron mean free path (l << fs) the improbable tunnel processes from the depth of the electrodes may be neglected, and the tunnel current can be expressed via the Green's functions in the vicinity of the junction plane on both sides of the tunnel barrier. In addition, from the condition I<< fs it follows that the relationship of the current with FR(e, p) and GR(e, p) (where e is the energy and p is the coordinate in the junction plane) is local. As a result, the density of the tunnel current may be given by

J(e, p) = Re Jp(e, p) sin q~(p) + Im Jp(e, p) COS q~(p) + Im Jq(e, p) (la)

q~ = ~o+ et, e=2e~"/h

where ~ is the phase difference across the junction; Re Jp, Im Jp, and Im Jq are the amplitudes of the Josephson supercurrent, of the interference current, and of the quasiparticle current densities, respectively. They can be given in the form (together with the total critical current Ic)

Io ~ e' ~ de'th-~--~[ImFRl(e',p) ReF~(e '+e,p) Re Jp( e, ~) = 2eRN

+ R e F~R(e','p) Im F~(e'+e, p)] ( lb)

1 de' th ~ T e - t h Im Jp( e, p) = 2eR-----~

× Im FR(e ', p) Im Fr~(e'+ e, p) (lc)


1 de t h - - ~ - - th ~-~ Im Jq( e, p)=2eRu _

x Re GR(e ', p) Re G R ( e ' + e, p) (Id)

:maxlf R,.,.-o.,.>. where indices 1 and 2 refer to the left and right electrodes, respectively, and RN is normal resistance of the tunnel contact. The integration of relations (1) over the junction area yields the total current passing through the tunnel junction. Equations (1) hold on the condition that the voltage is constant, which is always valid for structures with low transparency.

An analysis of expressions (1), generalizing the results of the standard tunnel theory 1°'11 for the spatially inhomogeneous case, shows that estimation of the tunnel current in the system depicted in Figs. 1 and 2 can be reduced to the solution of two problems: (1) determination of the energy and coordinate dependence of the Green's functions F R and G R in the superconducting electrodes of the junction; (2) calculation of the phase difference distribution in the junction plane from the solution of the electrodynamic problem on the distribution of fields and currents in the junction.

In Section 2 we consider the method of calculating the densities of states in spatially inhomogeneous superconductors by solving the equations of the microscopic theory. These results are used in Sections 3 and 4 for formulating the theory of the Josephson effect of SNINS- and SNIS-type tunnel structures and for studying the effect of Abrikosov vortices on the properties of tunnel junctions. The results of the calculations are compared with experiment. Section 5 proposes independent methods of determining the parameters of superconductors involved in the theory as well as diagnos- tic methods for tunnel Josephson junctions.

2. METHOD OF CALCULATING THE DENSITIES OF STATES IN SPATIALLY INHOMOGENEOUS SUPERCONDUCTORS

The standard way of finding functions GR(e) and FR(e) is to determine the normal Green's function G~ and the anomalous Green's function F~ in electrodes and to construct their analytical continuation to the real energy variable e = - i w [w = ~rT(2n + 1) are the Matsubara frequencies). In the spatially homogeneous case the dependence of G R and F R on w is found analytically:

G R - - 09 FR _ A (2a) (~o~ + ~)1/~, (o~ + A ~) ~/~


and the functions GR(e) and FR(e) following from Eq. (2a) are determined by the BGS theory formulas:

0, e<A N(e)=ReGR(-ie)= e/(e2-A2) 1/2, e>A

(2b) A

FR(e) = (A2_ 82)1/2

where N(e) is the density of states of quasiparticles normalized to the density of states of a normal metal, and FR(e) is determined by the density of states for the Cooper pairs.

In the spatially inhomogeneous systems under consideration we can calculate analytically the dependences of F R and G R on to for some particular cases only. In the general case the problem of determining FR(e) and GR(e) is reduced to the successive solution of two problems. The first problem is to derive a self-consistent solution of the Usadel equations, 34 which are valid in the dirty limit for the whole temperature range,

t oFR-O [ G R ( v - ~ / A ) 2 F R- FRv2GR] = AG R (3a)

ln--+2~rT ~ - F R =0 (3b) To ~

with the corresponding boundary conditions, and to determine the spatial dependence of the order parameter A(p). The aim of the second problem is to solve the analytically continued equation (3a) (by way of the substitution to = -ie) with the dependence A(p) found from the solution of the first problem and to subsequently determine FR(e, p) and GR(e, p).

2.1. Calculation of the Densit ies of States for a Compound N S Electrode

In calculating F R and G R for the NS sandwich we assume that no potential barrier exists at the interface and T~*N = 0. The condition W ~< Aj

allows us to consider the quantities F and G as dependent only on the coordinate x along the direction normal to the SN boundary. In practice, the most important case occurs when an SN electrode is a sandwich with the parameters

ds >> ~s, IN << dN << ~N

where ds, N are the thicknesses and ~s,N the coherence lengths of the S and N metals, and I N iS the mean free path in an N layer.

90 A.A. Golubov and M. Yu. Kupriyanov

The first condition allows one to neglect a decrease in the critical temperature of the SN sandwich compared to the massive S metal, 35 and the second one makes it possible to consider all the quantities in the N layer as independent of the coordinate along the direction normal to the SN interface and as equal to their values at the SN interface.

The problem of the proximity effect in such an SN sandwich was analyzed in ref. 23. The extent of influence of the proximity effect was determined by the parameter

o-N~s dn YM = - - (4)

~s~N ~N where o'N,s is the normal state conductance of the N and (S) metals. In ref. 23 the dependences A(X) were numerically determined at different values of the parameter yM. These results can be used for the solution of the second p rob lem-- the determination of FR(e, X) and GR(e, X).

Let us bring into coincidence the reference point and the plane of the SN interface; we choose the gauge with zero vector potential and perform the substitution w = -ie in the Usadel equation (3a) describing the proximity effect, which then acquired the form

O"+ie sin 0 + A cos 0 = 0 (5a)

where the quantities A and e are normalized to 7rTc, and the length to ~:s- The function 0(e, X) is related to FR(e, X ) and GR(e, X) by the relations

F R = sin 0, G R = cos 0 (Sb)

The boundary conditions for Eq. (5a) at the SN interface are found from those used in ref. 23 by substituting to = - i e :

O'(e, O) = -ieyM sin 0(e, 0) (5c)

and at X = +oo, that is, deep in the S region,

0(e, oo) -- arctg[ iAo(T)/e] (5d)

where Ao(T) is the unperturbed value of the order parameter in the superconductor.

The calculation of the densities of states at the SN interface is thus reduced to the solution of the boundary value problem (5a), (5c), and (Sd) in the S region. The solution is simplified in the limit e >> ~rTc only. In this energy range Eqs. (5a), (5c), and (5d) are linearized, and the 0 function changes at distances of the order of ~s (~rTc/e)1/2, which are much less than the ~:s scale of variation of the function A(X). This allows us to search for the solution in the form

0(e, X)= iA--+ A exp[ ( - ie / ~rTc)l/EX/ ~s] (6) 8


Determining the constant A from the boundary value condition (5c) for the value 0(e, 0), we get O(e, O) = a + ifl, with

a = TMA0( T)(e/2)'/2/{¢/2 q- [TM e "4- (e /2)1 /2] 2} (7)

/3 = Ao/e - !/MAo[3'Me + (e /2) ' /2 ] /{e /2 + [3~Me + (e/2)1/2] 2} ,

Substitution of the expressions for 0(e, 0) into (5b) gives the energy dependence of the Green's functions FR(e, 0) and GR(e, 0).

At arbitrary values of e and of the parameter ~/M, Eqs. (5a), (5c), and (5d) were solved numerically. The results are depicted in Figs. 3 and 4. Figure 3a shows the dependence of the density of states N ( e ) at the SN interface at T<< Tc for different values of the parameter TM. The sharp peak in the vicinity of the superconductor energy gap is smeared and displaced in the direction of energies e < A0(T), and at e = Ao(T) only a weak singularity in the form of a jump of the derivative d N ( e ) / d e is seen. The width and height of the peak are determined by the value of the parameter TM. At large "YM the peak is almost smeared out and transition to the density of states typical of a normal metal takes place.

The functions Re F(e) and Im F(e ) necessary for calculating the supercurrent components were also determined numerically. The results of calculations for T<< Tc and for a number of values of 3'M are presented in Figs. 3b and 3c. The character of the dependence of Im F(e ) is close to the energy dependence of the densities of states at the SN interface calculated for the same temperature and 3/M. There are two extrema on the dependence of Re F(e ) : one is at e = 12, and the second, which is much less in magnitude and opposite in sign, at e = A0(T ). With increasing temperature or the parameter 1'a4 and with a simultaneous decrease in the height of the peaks, the latter exhibit broadening and the distance between them, Ao(T)-12, increases along the energy axis. At higher energies the functions N ( e ) and F (e ) shown in Figs. 3 and 4 are described by the asymptotic dependence (7). The accuracy of determining N ( e ) was checked against these asymptotics and by verifying the condition of normalization

o [ N ( e ) - l ] de =0 (8)

of the total number of particles following from the conservation law. 36 It should be noted that the density of states is, as in the BCS theory,

zero at small energies up to some quantity e = 12 depending on temperature and on the parameter yu. The quantity 12 is a new energy gap in the spectrum of elementary excitations at the NS boundary (see the inset in Fig. 4). It is always below the energy gap of a bulk S metal and nonzero even for large values of TM. The existence of a gap is due to the fact that at e < 12 the


N{E)/N(O}

Im F(E) 5.0

3.0-

0.!

1,0

0.3

b

I I I !

1.0 2.0


FIEF'((:)

s.a- ir = /

(/~

Fig, 3. Densities of states at the SN interface a t d N << ~:N, ds >> ~:s, T<< T~.

boundary value problem (5a), (5c), (5d) has a single stable solution 0 = ~-/2+ ig(e), leading to the equality N(e)= O. Only at e > f~ is the stability of this solution upset.

The allowance made for the N-film thickness (dN/fN ~ 1) leads to a modification of the type of boundary value condition (5c). The stability of the solution 0 = ~-/2+/f(e) is violated for all energies e > 0 and the energy gap in the spectrum of excitations at the SN interface becomes zero. For instance, in the case of an SN sandwich with dN >> iN the boundary value condition (5c) is replaced by the condition

0'(e, 0) = %(-ie) 1/: sin 0(e, 0) (9)

where the parameter % = ~,Nfs/trsf N determines the proximity effect of a superconductor with a massive (dN >> iN) N metal? s

Calculations of the density of states N(e) in the N and S regions (Fig. 5) imply gapless superconductivity in such a system. However, in what follows we shall not consider in detail the finiteness of the thickness of the N metal (dN ~ iN) and restrict ourselves to the case of a thin N layer (dN << ~N), when the excitation spectrum has a gap.


N((] 'N(O)

' , O i

0 2 u 4 0 6 0 III

T/Ze ~

X =

{ i ~ l I I I 0 .8 o.g 1.0 1.1

E/A

Fig. 4. Density of states N(e) in the S region of the SN proximity sandwich at various distances x from the SN boundary at YM = 0.1, dN << ~:N, ds >> ~s, and T<< T,. Inset: temperature dependence of the energy gap 1) at the SN boundary.

The behavior of the densities of states in moving off the SN boundary (see Fig. 5) appears to be interesting: the singularity in N(e) becomes more pronounced, and at X >> ~s a transition occurs to the BCS density of states. The nonmonotonic dependence of the maximum in the density of states on the coordinate X can be accounted for by the nonmonotonic dependence O(X) resulting from the structure of Eqs. (5a), (5c), and (5d).

With increasing temperature the peak broadens and is shifted to the region of lower energies and less height. In this case the energy gap is reduced, since when the temperature approaches Tc the influence of the proximity effect is defined by the parameter yM/(1-T/T c) 1/2"28 which increases as T approaches To.

The dependence N(e) obtained differs qualitatively from the density of states in the McMillan tunnel model, a curve with two peaks. As in the density of states calculated in this paper, the first peak of the tunnel model is observed at values of the energy e less than Ao(T). The second peak, at


N(E)/ {OI

S

3

1

-- X / ~ s , u - 0 3 Ii I5

t.0 e / (-r )

Fig. 5. Density of states N(e) in the SN proximity sandwich at various distances x from the SN boundary at YM =0.1, d N >>¢N, ds >> ~:s, and T<< T c. Positive (negative) x values correspond to the S (N) region.

the location of the former BCS singularity at e = Ao(T), is the result of the assumption of constancy of As, N in each metal. It can be seen that taking into account the spatial dependence of As, N(X) causes the second singularity to vanish.

Below we apply the results for calculating the Josephson properties of SNINS- and SNIS-type contacts.

2.2. Calculation of the Density of States for a Lattice of Abrikosov Vortices in Electrodes

If an external magnetic field normal to the junction plane is sufficiently large and vortices penetrate the junction in the form of a regular lattice, the Wigner-Seitz method 37 is convenient for finding the coordinate dependence of the Green's functions. According to this method, a hexagonal elementary cell of the vortex lattice is replaced by a circular one with the radius

Ps = P ~ ( H ~ z / H ) ~/2 (10)


The temperature dependence of the second critical field Hc2(T) and of the critical radius pc(T) are determined by 3s

T /1 T~ ¢_~'~ pc=(dPo/~Hc2) 1/2, ln-~c+d/~+'-~ pZc]-~O(½)=O (11)

where ~O(z) is the psi function qbo is the magnetic flux quantum, and ~s is the coherence length of a superconducting electrode. It was shown in ref. 37 that, while calculating cell-average values, such a dependence introduces an error not exceeding 0.2%. In such an approach the functions F R and G R depend on the coordinate p only, measured from the center of an isolated vortex, and satisfy the equations

d20 1 dO dp2+p dp-wsinO-Q2(p)sinOcosO+AcosO=O (12a)

A ln-~-+ 2-~- ~ - s i n 0 =0 (12b)

where the function 0(~o, p) is connected with FR(~o, P) and GR(w, p) by relations (5b), and the gradient-invariant vector potential Q(p) in the limit x >> 1 is 39

Q(p)= p-l-p/p2s (13)

The system of equations (12) should be supplemented with the boundary conditions at the center and at the boundary of the cell:

A(0) = 0(to, 0) = 0, ~pA(ps)=~pO(w, ps)=O (14)

Here A(p) and O(to, p) are normalized to ~'Tc, the length to ~s, the vector potential Q(p) to the quantity ~bo/2zr#s, and the magnetic field to ~bo/2~-s ¢2. The calculation is significantly simplified for the limiting case of large magnetic fields.

2.2.1. Large Fields H ~ H,2 In fields near H~2 the functions A(p) and 0(to, p) are small, which

makes it possible to cancel nonlinear terms in Eqs. (12). The solution of the linearized equations has the form

A(p) = cop exp(-p2/2p2s) (15)

O(a,,p)=A(p)/(o~+a), a=2/p=s The constant Co should be found from the solution of the nonlinear equations (12) and at T = T~ is equal to 4°

,24(1-2/e){ 2 8 Co2 = V/o ~ _ ~ k 1-'~2), ~0o = 7---~ (1- '~') (16)


The explicit dependence of the function 0 on the frequency allows us to construct its analytical continuation to the energy axis by the substitution w = - i e in expression (15):

O(e, p ) = A ( p ) / ( a - ie) (17)

To an accuracy of the terms of the first order in A2(p) for the Green's functions entering into (1) we get

N ( e ) = Re G R --- Re cos 0 ~- 1 + - - A 2 ( p ) e 2 - a 2

2 (e2"+- a2) 2 (18a)

Re F R = Re sin 0 --~ A ( p ) a (18b) e 2 + a 2

Im F a = Im sin 0 ~ A(p)e (18c) e 2 + a 2 ,

N(Q/N(O) ~"r"

d

C

b

I

~'/A° Fig. 6. Density of states averaged over an elementary unit cell as a function

of energy at H/H~2 = (a) 0.5; (b) 0.2; (c) 0.05; (d) 0.


The function N(e) is the density of states at the Fermi level, normalized to that of a normal metal N(0) . This expression coincides with that obtained in ref. 38 by another method and exhibits the fact that superconductivity in the vicinity of He2 is gapless.

2.2.2. Arbitrary Magnetic Fields

At an arbitrary value of the external field the problem was solved numerically. Figure 6 depicts the results of a calculation of the cell-average density of states N(e) at T<< Tc and at different values of the field H. As can be seen, in the whole range of fields 0 < H < mc2 superconductivity is gapless, which agrees with the results of a calculation performed by Watts- Tobin et aL 41 for the case of x = 1. The states with the energy e < A0(T) correspond to elementary excitations localized near the vortex nucleus in the region of a radius of the order of several coherence lengths. With increasing field the extent of smearing of the singularity in the density of states increases. In the range of fields 0.9H~2 ~< H ~< He2 the dependence N(e)/N(O) is described with an accuracy of 10% by expression (18a). The accuracy of the calculation was checked by satisfying the normalization condition (8).

3. C A L C U L A T I O N OF C U R R E N T S IN S N I N S - AND SNIS-TYPE TUNNEL STRUCTURES

Since the Green's functions obtained in Section 2.1 and the phase difference ~ in the case under consideration are independent of the coordinates in the contact plane, expression ( la) is valid for the total current, too. It can be expressed in the form

I(e) = Re Ip(e) sin ~o + Im Ip(e) cos ~ + Im Iq(e) (19)

where the amplitudes of the current components Re I,, Im Ip, and Im Iq should be calculated by substituting the values of F~a(e) and G~E(e) at the SN interface (for the contact SNINS) into formulas ( l b ) - ( l d ) . For the SNIS contact the values of F~(e) and G~(e) are taken at the SN interface of the left-hand electrode, and for F~(e) and G~(e) we apply (2b) for a homogeneous superconductor.

3.1. Calculation of the Tunnel Current

Figure 7 presents the results of calculations of the quasiparticle current Im Iq(V) for symmetric (SNINS) and nonsymmetric (SNIS) junctions at different temperatures and values of the suppression parameter TM. In the absence of suppression (yM = 0) the current coincides with that calculated by the BCS model for SIS structures. With increasing y~, the CVC for SNINS and SNIS structures change in different ways. Unlike the results


obtained in terms of the standard tunnel theory, in symmetric junctions at sufficiently low temperatures and at values of ~/M ~< 0.5, instead of a current jump at e V = 2Ao, there appears an oblique section starting with the voltage 212/e and terminating with a sharp singularity at e V < 2 a o , leading to a kneelike CVC structure. With further growth of the parameter 3'M the singularity is smoothed out and the dependence Im Iq(V) becomes linear.

With increasing temperature at a fixed value of 7M = 0.5 (Figs. 7C, 7d) the kneelike singularity on the CVC is smeared, and the values of the current increase at voltages below 212/e, which finally leads to an almost linear dependence Im Iq(V) in the region T ~ To.

The kneelike singularity on the CVC of nonsymmetric SNIS junctions at e V >~ f~ + Ao is smoothed compared with the case of symmetric junctions at the same values of T and YM. In addition, on the CVC of the structures there is a peak in the region of voltages corresponding to the difference in gaps (Ao-12)/e. The height of the peak increases with 7M, which results from an increase in the difference Ao-f l . It should be noted that these singularities observed on the CVC of the structures under consideration are associated with the shape of the peak in the density of states at e = 12, and not with a singularity existing in it at e--Ao in the McMillan model 22 or with a bound state at e < Ao for the Arnold model. 3°

Figure 8 shows the voltage dependence of the amplitude of the supercurrent component in SNINS structures at T<< Tc and at different values of 3'M. The proximity effect in electrodes leads to the smearing of the Riedel singularity observed at voltages V = 212/e, which are less than 2no/e. With increasing YM, the singularity shifts to the region of lower voltage, the height of the peak is reduced, and its half-width increases. A rise in the temperature at a fixed YM leads to analogous changes in Re Ip (V) .

Note that the mechanisms of broadening of the Riedel peak considered in ref. 15 affect its height and half-width only; its position and the value of the critical current Ic = Re Ip( V = 0) remain the same. In our model the supercurrent amplitude is suppressed with increasing parameter YM at all values of voltage, and, in particul~ir, at V = 0, that is, the critical current is suppressed. The dependences Ic (T) at different values of YM is depicted in Fig. 9. In the absence of the proximity effect (TM = 0) the temperature dependence of the critical current is determined by the well-known formula of Ambegaokar and Baratoff (AB). 42 With increasing parameter 7M the critical current Ic is suppressed in the whole temperature range, and this suppression is stronger in SNINS structures than in SNIS junctions. In the vicinity of the critical temperature the critical current is proportional to (T~ - T) 2 for a symmetric junction and to (T~ - T) 3/2 for a nonsymmetric one, whereas in the absence of suppression the linear dependence I~oc (T~ - T) follows from the AB formula. This can be explained by the fact

100 A. A. Golubov and M. Yu. Kupriyanov

ImI, .eRN/a

0

Im Iq,eR./6

1

T . = 0.01

°, l

I

eV/6 2

fJ jdl jf

/

~,11.

iiii/1111//1111111111

/ / !

1 evlA


Im I, .eR N/A

-- / //7

ZZ/Z Z'5"

f

1 eV/~

I 0 2

,l

Im I¢ .e R N/A

d

/ J /

f /

f / f X

/ / / / / / / ,S,I ~'f

~ I I I I 0 1 2

evlA

//]

Fig. 7. Quasiparticle c o m p o n e n t o f the tunnel current as a function of vol tage for various YM at T<< T c in (a) S N I N S and (b) S N I S junctions; and at T = 0 .6T c in (c) S N I N S and (d) S N I S junct ions.


ReIgeRN/A 5F

~N = 0

0.1

0.3

1 2 eV/A

Fig. 8. Supercurrent component of the tunnel current of the SNINS junct ion as a function of voltage for various YM at T<< Tc.

that at T ~ Tc the critical current is proportional to the product of the electrode order parameters, which are temperature-dependent as Ao(T)oc ( Tc - T) 1/2 in the absence of suppression (TM = 0), or as (To - T) at 7M ~ 0 and T ~ Tc.23'28

Figure 10 shows the amplitude of the interference current component Im Ip(V) calculated for the SNINS junction at T = 0.6 T~ for different values of 7M. Here, instead of an abrupt jump to the region of negative values (at 3'~ = 0), we observe a smooth transition, starting at V = 2f~/e, from small, positive values to the negative region. With increasing ~/~ the negative peak in Im Iv(V) is flattened and diminished in amplitude, the gap 12 tending to zero, and the sign may change at voltages much less than 2Ao(T)/e.

The question of the sign of I m l p ( V ) has repeatedly been dis- c u s s e d . 16"43-45 It was shown that a number of mechanisms responsible for broadening the Riedel singularity in the supercurrent component should cause a change of the sign of Im Ip(V) in the range of low voltages. The

z CC

Theoretical Investigation of Josephson Tunnel Junctions

~Y

c5

t ~ c5

t ~ O

• ~

~ N

~ . ~ Z

• = , N

@,.1~

• a ~ ,1

o II = o

~ '~ . ~ .

103


I m I ,'ORN/fl

-1

-2

~ 3 1.

eV/A Fig, 10. Voltage dependence of the interference current amplitude of the SNINS

junction for various YM and T= 0.6T c.

proximity effect in electrodes considered in this paper is one such mechanism, but it gives rise to a change in the sign at )'M ~> 1 only (Fig. 10). The reason is that in our model the functions Im F(e) are zero at e < fl, whereas the mechanisms considered in ref. 16 induce a gapless state.*

3.2. Comparison with Experiment and Discussion of the Results

The results obtained are compared with the experimental data for tunnel junctions on the basis of niobium. 4"5 Figure 9 shows the temperature dependence of the critical current for symmetrical Nb-NbOx-NB, 4 and nonsymmetrical, Nb-NbOx-Pb, s junctions. The best agreement with the calculated data is achieved at "I'M ~ 0.05 for the junctions considered in ref.

*As shown above, the gapless state at the SN interface may be due to the proximity effect if the N metal has a finite thickness, that is, d N ~ ~N. Apparently, taking account of the finiteness of the N-layer thickness makes it possible to describe quantitatively the dependence Im Ip (V) at small voltages.


4 and at YM = 0.1 for those from paper ref. 5. Different estimates obtained for the parameter 7M for different structures can be accounted for by the fact that its value depends on the thickness and conducting properties of the transition layer. In calculating the temperature dependence of the order parameter of the lead electrode Apb(T) we used the formula of the strong coupling theory

a(T) = th/' T~ a( T)~, (20) A(0) /

where Apb(0) = 1.33 MeV. Figure 11 gives experimental CVCs for the same junctions and calcu-

lated curves for selected values of 7M. The theoretical curves well describe the kneelike singularity in the region of the sum gap voltage. Note that in refs. 4 and 5 the resistance of a junction at V > V g = 2 ~ + ~ , exceeding

Iml ,eRN/A Nb(O) / i

/ / /

/ /

/ i / / / / I

/ / / /

/ I / / / Q

///

/ ] /I i / / / / I

i / I" // i! i~ I • / . / . > f '

/ . . . . . . . __o.j_.1 1 ..... I I

0 I eVlANb(0) 2

Fig. 11. (--) Experimental I - V curves of NB-NBOx-Pb tunnel junction 5 at T~ Tc = (a) 0.65 and (c) 0.23 and of N b - N b O x - N b tunnel junction 4 at T~ T c =

0.68 and (d) 0.23. ( - - ) Calculated curves for (b, d) SNIS (YM =0.05) and (a, c) SNISa (YM = 0.1) junctions.


somehow the real value Of RN, served as RN. Thus, the CVCs of the junctions in these papers [depicted as the dependence I(V)RN] did not fall below the straight line corresponding to Ohm's law. In our theory (as well as in the BCS theory) the CVCs fall below the straight line I = V/RN even at V >> Vg, and approach the line only asymptotically. For this reason, when processing the experimental data we renormalized the product IRN in such a way that the calculated and experimental curves coincided directly outside the kneelike singularity.

In comparing the calculated data with the CVCs of the junctions from ref. 4 we brought the curves into coincidence according to the position of the current jump on the voltage axis. Use of the dependence ANb(T ) following from the BCS theory did not provide a coincidence: the calculated CVCs turned out to be shifted to the range of high voltages as compared to the measured ones. This suggests that the method of determining the energy gap ANb(T) by the position of the current jump on the CVC in ref. 4 underestimated the value of ANb(T ) by approximately 5-10%. In the region of low voltages V < Vg the experimental values of the current exceed the calculated ones. Such an "excess" current, usually referred to as a leakage current, appears only when a significant low-energy density of states exists in electrodes, that is, the energy gap is zero. Such a situation takes place when the thickness of the N metal is finite (dN >-¢N) and also in the case of Abrikosov vortices trapping in junction electrodes (see Fig. 12).

Further experimental evidence of tunneling in proximity effect systems is found in a recent study of superconducting ceramics BaPb~_xBixO3.46 The experimental dependence Ic(T) is quantitatively described by the results of our calculations for SNINS junctions. This fact supports the model of BaPbl_xBixO3 as a granular superconductor in which neighboring grains are coupled by Josephson tunnel weak links with normal transition layers near the tunnel barriers.

Therefore, the introduction of one parameter YM describing the proximity effect in electrodes makes it possible to account for the run of experimental curves I(V) at eV=2A(T) and of It(T). Note that the value of the parameter TM can be estimated independently by measuring the dependence of the T* of an SN sandwich on the thickness of the S layer 35 and also by tunnel experiments on NSIN- and SNIN-type structures and according to the dependence of the second critical field Hc2(T) for the SN sandwich. These methods are considered below in Section 5.

4. INFLUENCE OF ABRIKOSOV VORTICES ON THE PROPERTIES OF TUNNEL JUNCTIONS

A low transparency of a tunnel junction suggests that the Lorentz force acting on Abrikosov vortices frozen in the electrodes on the side of the


current passing through the electrode is small compared to the pinning forces confining the vortices in a static state. In consequence, the tunnel current can be determined by the formulas of the tunnel theory (1) and the result depends significantly on the configuration of vortex lines in the electrodes.

4.1. Lattice of Abrikosov Vortices in the Junction

When the pinning forces in electrodes are not large and Abrikosov vortices are linear, then the phase of the order parameter X in each electrode changes by the quantity 27r in tracing a closed contour around a single vortex. Nevertheless, the phase difference q~ = X2-X1 is independent of the space coordinate in the junction plane. The Green's functions FxR2 and G1R2 depend on p, and to find the tunnel current mean density the expressions for the current components ( lb)- ( ld) should be averaged over the junction area. However, since the vortex lattice is assumed to be regular, it possesses translational symmetry, and the average taken over the whole area is reduced to averaging over an elementary cell. The calculation of the tunnel current is significantly simplified for the limiting cases of large and small magnetic fields.

4.1.1. Large Fields H ~ Hc2

Substituting expressions (18) into the formulas for the tunnel current (1) and calculating the integrals, we get the following equations for the tunnel current components at T<< To:

RN(ImJq(V))~- V(1 (A2(P))~ ~2-~-5a2] (21a)

a(A2(p)) RN(ReJp(V))~-~[In(I+~2) +2(2a+V~arctgV]\ V a] (21b)

RN(ImJp(V))= V(4aZ+V2 ) In 1+-~- 3a2+ V2 arctg (21c)

and the CVC for the SIN junction has the form

' V2 (22)

where (A2(p)) is the mean square of the order parameter taken over the Wigner-Seitz lattice cell.

It follows from (21) that at H ~ Hc2, and even at T<< To, the principal contribution to the current comes from the quasiparticle component through the tunnel junction. The CVC of the junction is close to Ohm's law in the whole range of voltages. Supercurrent and interference current components


are noticeably suppressed. Formula (21b) shows how the Riedel singularity is smeared in the supercurrent component. The quantity (Re Jp(V)) has a maximum at the voltage V~ a/e, which is approximately one-fourth of 2h (T) / e , and the shape of the smearing cannot be described by Lorentz or by Gaussian laws and has a half-width equal to a.

The interference current component at T<< Tc is nonzero at voltages V < 2A(T)/e , which is related to the fact that at low energies the functions Im FxR, z(e) are nonzero. Instead of the jump in the tunnel theory at V = 2A(T)/e, a smooth maximum is observed at V~- a/e in our case. It should be noted that at T<< Te the quantity

I-(A2)V3/4a 4, V<< a (23) Ru(ImJp(V))=[-(A2)V-' ln(V/a), V>>a

is negative in the range of low voltages. This mechanism thus leads to a change of the sign of the interference component at sufficiently low temperatures in the range of low voltages V<< MT)/e. This is in agreement with the general phenomenological analysis made in ref. 16 and results from the low-energy "tails" of the functions Im F~2(e) [see formula (18c)].

The qualitative features of the voltage dependence of the current mentioned above are also found at arbitrary values of the magnetic field.

4.1.2. Small Magnetic Fields H<< Hc2 In small magnetic fields Abrikosov vortices are isolated, i.e., in practice

they do not affect one another and their effect on the current passing through the junction is small. In our case it follows from relations (1) that the existence of vortices brings about corrections to the tunnel current components proportional to the ratio H/Hc2<< 1. In particular, for the quasiparticle current and for the critical value of the supercurrent we have from ( ld) and ( le)

Im Iq = Im I~cs+(H/Hc2)F1(T, V) (24a)

Ic = i~cs _ (H/H~z)F2(T) (24b)

where the values of currents Im l Bcs and lBcs -q -c are determined in the framework of the classical tunnel theory. 1°'1~

The functions FI(T, V) and F2(T) as determined numerically by the above method are displayed in Figs. 12a and 12b. As shown in Fig. 12a, the function FI(T, V) is negative at sufficiently high temperatures T ~ 0.4 T~ for small voltages, that is, the density of the current Im Iq(V) is less than

BCS Im Iq (V). This is due to the fact that Abrikosov vortices in electrodes lead to the absence of the logarithmic singularity in conductivity at V ~ 0 that exists in the classical tunnel theory l°'H and is associated with a root singularity in the density of states. The quasiparticle current due to Abrikosov vortices in electrodes (leakage current) increases with voltage.


.3

.I

'2 o . . .k :6 ;8

T/T c

T/Tc=

0.6 I.O /

O.

p

/ ,¢

e V/~o

1 .5

0 .5

b

i i I i i i i i i

O. 0.2 0.4 0,6 0.8 I.O

T/T o

Fig. 12 (a) Leakage current function F,(T, V) versus voltage V across the junction. Inset: F 1 ( T, V(T)) versus temperature T at V(T) = 0.5A( T)/e. Curve a is the function 0.1 dFl/dV for T=0.6T, . ; circles are the experimental data from ref. 13. (b) Tem- perature dependence of the function F2, which determines the critical current suppression.


A comparison of the values of these currents at voltages V = 0.5A( T)/e (see the insert in Fig. 12a) may help to evaluate the extent of the effect of Abrikosov vortices on the quasiparticle current at different temperatures. The dependence FI(T, V(T)) is nonmonotonic and achieves a maximum in the range of temperatures T~ Tc ~ 0.6; the leakage current determined by the function FI(T, V) is approximately twice as large in the temperature range 0.4~ < 0.7 as at T<< To. Therefore, Abrikosov vortices have a maximal effect on the quasiparticle current in the temperature range 0.4 ~< T~ T~ ~ 0.7.

Figure 12 compares the surplus differential conductivity of the tunnel junction calculated theoretically for T = 0.6To as a function of the voltage (curve a) with the experimental data of ref. 13, obtained in a magnetic field H =21 G at T=0.6Tc =4.2 K. As a fitting parameter we used the value of the second critical magnetic field of the electrodes He2, which was not given in ref. 13, and was defined by having the calculated curve coincide with one of the experimental points (V = 1.3 MeV). This allowed us to find the value of He2 ~ 1.3 kG. Such a processing of the experimental data makes it possible to evaluate both the parameter Hc2(T) of the electrodes and their coherence length from Eq. (11), as follows.

The function Fz(T), according to (24b), defines the suppression of the critical current of the junction in a weak magnetic field. As is seen from Fig. 12b, the function F2(T) is practically constant at T<~O.5Tc and diminishes monotonically with increasing temperature, becoming zero at T-+To.

4.1.3. Arbitrary Magnetic Fields In the whole range of fields 0 ~ < H ~ He2 the dependence, FR(e, p) and

GR(e, p) obtained numerically from the solution of the analytically continued equations (12) and (13) with boundary conditions (14) were applied for calculating the tunnel current by Eq. (1). The calculated quasiparticle, superconducting, and interference components of the tunnel current are given as Im Iq(V), Re Ip(V), and Im Ip(V), respectively, in Figs. 13-15 for different temperatures and values of the magnetic field.

As follows from the CVCs given in Fig. 13, at voltages V<2A(T)/e there is a significant quasiparticle current, increasing with field. It is also important that even at low temperatures T<< Tc all the curves in the range of low voltages have finite slopes, which exhibit nonzero conductivity (see also Fig. 12a). This property is well illustrated by the curves of the voltage dependences of differential conductivity, shown in the insert of Fig. 13a. These curves also exhibit singularities at the voltage V-~ A( T)/e. It is known from ref. 47 that the singularities on the CVC at V ~ A(T)/e are usually associated with the two-particle tunneling processes. In the present paper we perform calculations in the framework of the tunnel Hamiltonian

Theoretical Investigation of Josephson Tunnel Junctions l l l

C,C" qJ

i,.~1 .-, 0"£

o

q..l

O.)

D

rn

S'~ O'E S' ! 0"! S" O"

0"£ S'~ 0"~ S'! 0"! S" O"

E

¢)

fl)

O ¸

~E

Ii


method, taking account o f one-particle processes only. Therefore, these singularities arise exclusively due to the presence o f Abrikosov vortices in the junct ion, and their posi t ion and scale depend on the values o f temperature and magnet ic field. With increasing temperature at T ~ 0.3 T~ the singularities are smoothed out and they almost vanish at T~-0.6Tc. In

Re lp .eRw /'~ Tc a

0

0

I I

I 1 I I

eV/~o

b

0 0.5

H/FIc~

Fig. 14. (a) Smearing of the Riedel singularity at T<< T C with increase of magnetic field. (b) Critical current as a function of magnetic field; (- -) asymptotic solution of (24b).


I ~ (N

o"

+i ~ ~ o

PO

C~

o

o' d d d I t J

. - ! . . . . . . . . . I . . . . . . . . . + . . . . . . . . . ~ . . . . . . . . .

~ ~ QO o

d ~ d d T I I I I

< 1 )

o

qJ

II

I+

¥

_ + .=


addition, their amplitude, unlike the two-particle tunneling effects, is first order in the tunnel barrier transparency of the Josephson junction, and not second order. In large magnetic fields H ~ 0.9/-/~2 the CVC at T<< Tc is adequately described by expression (21a).

As can be seen in Fig. 14a, the superconducting component of the tunnel current is suppressed by a magnetic field at all voltages, the extent of suppression increasing with magnetic field. At H ~> 0.9Hc2 and at low temperatures T<< Tc the dependence Re Ip(V) is found from Eq. (21b). With increasing field the Riedel logarithmic singularity undergoes significant broadening at V=2A(T)/e and the critical current of the junction Ic= Re lp(V=O) is suppressed. The dependence /~(H) is given at different temperatures in Fig. 14b; Ic(H) is shown to decrease monotonically down to zero at H=Hc:, and in small fields H<< H~2, It(H) reaches linear asymptotics, (24b).

The voltage dependence of the interference current component is shown in Fig. 15. As shown above, at low temperatures T<< T~, the sign of the function Imlp(V) becomes negative in the range of small voltages V<< A(T)/e associated with the negative contribution to the current from low- energy "tails" of the function FR(e), whereas the positive temperature contribution to the current at T<< T~ is exponentially small. With increasing temperature the region of changing sign is displaced toward the direction of higher voltages. The change of sign of Im Ip(V) has been observed experimentally (see references in ref. 16) at small voltages even in the range of sufficiently high temperature ( T = 0.5 To). To explain this effect quantitatively one needs to apply other physical models, not associated with the presence of Abrikosov vortices in the junction.

With increasing magnetic field the current jump at V=2A(T)/e is smoothed out and its amplitude is suppressed, and at H ~ 0.9Hoe and T<< Tc the interference component amplitude of the tunnel current Im Ip(V) is determined by expression (21c).

4.2. Single Abrikosov Vortices in the Electrodes of the Contact

Once the pinning in electrodes is significant and the external magnetic field is small, the vortex lines are likely to bend at the interface between two electrodes, their configurations being dependent on the magnitude and distribution of pinning centers in superconducting electrodes. This case is characterized by the presence of a magnetic field component in the junction plane. Therefore, in order to determine the contribution of the single vortices shown in Fig. 2b-2d to the tunnel current, one should analyze the problem of the distribution of fields and currents in the junction in addition to the calculated functions F~2(e, p) and G~.2(e, p) in the junction plane, that is, one needs to determine the spatial dependence of the phase difference q~(p).


Below we consider the situations when the characteristic dimension of the bent vortex a (see Fig. 2c) is much less or greater than ~:1,2. In the first case the vortex line is actually linear, and its effect on the properties of a junction was considered in the previous section. In the second case (a >> s¢1,2) the principle of superposition can be applied for calculating the tunnel current. The distribution of fields and currents in the junction is presented as a sum of fields and currents of two vortices localized in the upper and lower electrodes (Fig. ld), respectively.

The ensuing simplification takes account of the inequality ¢1,2<< A1,2 (where A~,2 is the depth of the magnetic field penetration for the upper and lower electrodes, respectively), which makes it possible to calculate the electromagnetic and nucleus regions of the vortex separately.

4.2.1. Electrodynamics of a Single Abrikosov Vortex in a Tunnel Junction

In the electromagnetic region of the vortex, that is, at the distance p ~> ~:1,2 from its axis, the Usadel functions reach their equilibrium values FR(e) and GR(e). Due to the condition W~<Aj, the Josephson currents passing through the tunnel barrier can be neglected compared to vortex currents, and the equation determining the gradient-invariant vector potential

Q1,2 = VXl,z - (2"/T/t:I)o)A1,2 = (0, Q1,2, 0)

in the polar coordinate system related to the vortex axis (see Fig. 2) acquires the form

0 2 + 0 [ 1 0 ] Q1.2 (25)

In the given coordinate system the relationship between the supercurrent j = (O,j~,2, O) and the vector potential has the form

j,,2 = ( Cebo/87r2,k ~,z) Q,,2

The boundary conditions for Eq. (25) deep in the electrodes are found from the condition that Q(p, z) has a well-known solution, obtained in ref. 48 for a single vortex in a homogeneous superconductor:

Q1,2(p, +o:3) = A ~,I KI(p / A1,2) (26) \

for the case shown in Fig. 2b and

(AT'K~(p/A,), z>>A, Q1,2 = [0, Z << --A2 (27)

for the case given in Fig. 2d. The boundary-value conditions in the junction plane (z = 0) consist in the continuity of the magnetic field components Hp


and/-/g:

~p[pQ(p, +0] = +[pQ(p, -0 ) ]

0 0 -~z Q(p, +0)=~zQ(p, -0)

(28)

The solutions of the set of equations (25), (26), and (28) describing the electrodynamic structure of the vortex (Fig. 2b) can be given as follows:

fO ~ 2 -1 -1 --1 Q1,2 = ZF,~K,(p/hl,2) - Y JI(YP) exp(-oq,2[z[)al,2(al,2- a2,1) dyl (29)

~_ ), -2,~1/2 where al,2= (~2--,~1,2] and JI (X) is the first-order Bessel function. It follows from (29) that the spatial inhomogeneity along the z axis in the normal direction has no effect on the vector-potential behavior (and, consequently, on the current) in the area of the vortex nucleus:

Q(p, z) ~ l /p , p ~ ~:1,2 (30)

Further, though the magnetic field component Hp is nonzero in the junction plane, there is no dependence of the phase difference of the electrode order parameter ~ on the coordinate p. To prove this, it is sufficient, in calculating the ¢ value, to retreat from the junction plane z = 0 deep into the electrodes for a distance of ]zl>>A,.2. In these regions Eq. (29) defining Q(p, z) becomes (26). As a result, the order parameter phases X~ and 2(2 are determined for each of the electrodes by the polar angle 0: X~ = 0 and X2 = 0 + ~o, and the phase difference ~ = X2- X1 is equal to a constant 9~0 whose value is determined by the supercurrent passing through the junction.

The electrodynamic structure of the vortex, shown in Fig. 2d, follows from the solution of the boundary-value problem (25), (27), and (28) and can be described by

Ql(p, 2 > 0 ) AI1KI(fl)+A12 f ~Ol2Jl(~p)exp(-OllZ)d~/ -- 2 , (31a)

f ~ JI( YP) exp(c~2z) dy Q2(p, z < O)= -h ~2 [ c~l(a, + old) do

(31b)

We find from (31a) and (31b) that at p >> A1,2 fields and currents are localized in the region -A2< z < A1 in the vicinity of the contact plane. It is easy to check that the magnetic flux passing away from the vortex in the radial


direction is equal to the quantum qbo. In the other limiting case p << A1,2 we have from (31a) and (31b)

QI(p, +0)-~ 1/p- (p/4A~)ln(A1/p) (32a)

Q2(p, - 0 ) ~ - (p /gA 12) ln(Al/p) (32b)

It follows from (32a) and (32b) that the profound deformation of the electromagnetic region of the Abrikosov vortex in our case does not give rise to any additional features in the behavior of the vector potential in the region of the vortex nucleus at z > 0, and the dependence of the vector potential QI(p, z) at p ~< E1 is again determined by (30). In the lower electrode there are no vortices, and a supercurrent flows in the direction opposite to that of the current at z > 0. This circumstance gives rise to the spatial dependence of the order parameter phase difference of the electrodes ~ (p). Indeed, in the depth of the upper electrode (at z >> A1) the phase is still equal to the polar angle 0, and in the lower electrode at z << -A2 the phase X2 is constant, since in this region Q2 -~ 0. Thus, the gradient-invariant phase difference is

q~=~o+0 (33)

For the vortex indicated in Fig. 2c, the distribution of fields and currents at a >> scl,2 can be obtained, as has been shown above, by linear superposition of the solutions of (31) for two vortices localized in the upper and lower electrodes at a distance a from one another.

Suppose the upper electrode is a thin superconducting film of thickness d~<< A1; then Eq. (25) must be solved simultaneously with the equation rot rot Q = 0, predetermining the vector-potential distribution in the space over the film. Using the condition of field continuity (28) at the second boundary of the electrode (at z = dl) and also the boundary conditions (27) and (28) for Q~a(p, z), we have

O,(p, O < z < d,) = A T' Kl(-ff~l)

+ f o ( B , choqz+B2sheqz)J~(yp)dy (34a)

02(0, z < O) = CJ~(yp) exp(ot2z) dy (34b)

BI=A120172 1 ol2+yj, B2=dlA~-za71 1 ol2+T , 0~2 q" 'y

(34c)


The electromagnetic structure of the nucleus (p ~< £1) is the same as that for the massive (dl >> hi) film, and at a large distance from the vortex axis p >>/~2

the gradient-invariant vector potential has the form

Ol = p- l ( 1 -h2 /h l± ) , 0 < z < dl (35a)

Q2 = - p - ' exp(z/h2) (/~2//~t 1±), Z < 0 (35b)

where Air = h~/dl is the effective penetration depth of the field into the upper film.

It follows from (35a) and (35b) that in this case the current drops more slowly with increasing distance from the vortex axis p than in the case of an isolated thin film at p > h . , 48 Then, from the condition of fluxoid quantization it follows that the external magnetic flux passing through the thin film in the direction paralled to the vortex axis differs noticeably from

~o:

qb = (h2/h~.)qbo<< qb o (36)

Another result of the slow decrease of current is the logarithmic dependence of the interaction energy U(p) of a pair of vortices a distance p apart:

U ( p ) = i((I)02/8q'/'2/~ 1±) In(p/sol), P "~ E1 (37)

Here the plus (minus) sign corresponds to vortices of opposite (similar) sign. The analysis shows that under any deformation of the electromagnetic

region of the vortex the behavior of the vector potential in the nucleus area, that is, at distances p <~ ~:a,2 from the vortex line center, is determined by expression (30). Further calculation of the quasiparticle current may thus be performed by the method described in Section 4.1, using the values of the functions GR(e, p) and FR(e, p) for nucleus area of a single vortex.

4.2.2. Current- Voltage Characteristic ( CVC) Suppose a vortex is localized in one of the electrodes; then, in the

second electrode [see Eq. (32b)] only weak screening currents flow, not suppressing the order parameter. So, the function Re G R in formula ( ld) can be regarded as independent of the coordinate p, and equal to zero at e < A(T), and we have

ReGR2(e)=e/(e2-A2) 1/2, e > A ( T ) (38)

Numerical calculation shows that in this case there is a singularity on the CVC at V = A( T)/e in the whole temperature range, which is particularly well exhibited in the dependence of the differential conductivity d(Im Iq)/dV on voltage V (see Fig. 16). This singularity is due to the fact that the electron tunneling in the vortex nucleus area proceeds from the gapless region to the superconductor with a gap in the density of states. If

I Theoretical Investigation of Josephson Tunnel Junctions 119

R .d !, /,tV

d_ -g

o ! o t eVlA ~v/4

:, 3

0 1.0

eVM° Fig. 16. (1-5) Theoretical and (a-d, a'-e') experimental 12 differential conductance curves; the vertical scale is arbitrary. T/T c = (1) 0.8, (2) 0.7, (3) 0.6, (4) 0.4, (5) 0.1.

a deformed vortex penetrating both electrodes is trapped in the junction, then, based on the principle of superposition, it will produce a contribution to Im Iq(V) at V = A / e twice as large as that of a vortex localized in one of the electrodes.

Singularities of differential conductivity analogous to those shown in Fig. 16 were observed experimentally in a number of papers (see, for instance, refs. 12 and 13). Figure 16 depicts the experimental dependence of d ( I m Iq ) /dV on voltage V obtained in ref. 12 for Sn- I -Sn (a-d) and P b - I - P b (a ' -e ' ) junctions of 25 and 16/zm 2 area, respectively, at T = 1.5 K in a normal magnetic field. Curves a -d and a ' -e ' define different values of the field, with the corresponding magnetic fluxes passing through the junction differing by the order of magnitude of the flux quantum qb 0. The authors of ref. 12 attributed the discrete changes in conductivity to the entrance of


misaligned vortices into the junction. This assumption is confirmed by the presence of peaks on experimental curves of d[Im Iq( V ) ] / d V at V = A( T ) / e for the junctions Sn-I -Sn and Pb-I-Pb, which agree qualitatively in shape with theoretical curves 4 and 5, respectively, for T~ Tc = 0.4 and 0.1.

4.2.3. Critical Current Ic

If the vortices trapped in the junction are linear (see Figs. 2a and 2b), then from Eq. (le) and the condition of constancy of the phase difference ~o = X2- X1 = q~o = const obtained above, it follows that their effect on Ic is determined by the "kern" mechanism only. In this case, in the most interesting experimental case of (~ = ¢2 = ~:, corrections to the Ic associated with the presence of Abrikosov vortices in the electrodes are proportional to their density n:

Ic(T) =/co(T) - F2( T)27rsC2n (39)

Here F2(T) is the function plotted in Fig. 12b. The correction to the critical current associated with the contribution of a single vortex is, to an order of magnitude,

A I~/ Ico = ( I~o - Ic)/ I~o oc (~/ W) 2

that is, it turns out to be proportional to the area where superconductivity is suppressed. Here /~ and /co are, respectively, the critical currents of the junction in the presence and absence of vortices in the electrodes.

When the trapped vortices are not linear (Figs. 2c, 2d), a change of the critical current is mainly caused by an electromagnetic mechanism. Indeed, in the case shown in Fig. 2c, it follows from (33) and the superposition principle that the phase difference ¢ depends on the polar angles Q1 and 02 defining the directions from the axes of vortices A and B to the reference point C (Fig. 17). A decrease in the critical current can be obtained from geometrical relations and the definition of Ic, Eq. (le):

A/~ 1 ( ( p 2 _ a 2 / 4 ) p d p d 0 I~o - S J {[(a/Z)2+ p212 - a2p2 cos2 0}1/2 (40)

~ C

A 9

Fig. 17. D e p i c t i o n o f vor tex axes A a n d B in u p p e r a n d l o w e r e lec t rodes , respec t ive ly ; p h a s e d i f fe rence ~p = 0 2 - O~.


Integration in (40) is performed over the whole junction area S. For a2<< S we have from (40)

AI~ ~ a2[ 4S 1.4) (41) / ~ o ~ " -~-~ln 7ra-- 5 -

that is, AIc/IcoOC ( a / W ) 2, which exceeds significantly the contribution of the "kern" mechanism for a >> ~:.

If the vortex is trapped in one electrode only, a change in the critical current AIc may even be of the order of the critical current itself,/co. For instance, in the case of a vortex trapped in the center of a junction with circular electrodes, from (1) "and (33) we have Ic = 0, i.e., AIc =/co. With broken symmetry, that is, when the vortex is displaced from the center of a circular junction of radius R by a distance b, we get, taking into account the effect of the junction boundaries by the method of images,

Ic 110xcsin/3+ Ip'max (p2- 1)p dp Ico - 2~/32- dO (42a) ~. [ (p2+ i )2_4p2 COS 2 0]I/2

/3_ ( _s in 2 0~ 1/2 b / R Pmi~×=-~+++cosOqZ2fl_ 1 4/32+] ; f l ± - l + ( b / R ) 2 (42b)

Numerical calculation using (42) (see Fig. 18) shows that at b ~< 0.SR the critical current /~ grows linearly with increasing parameter b: Ic/I~o ~ 1.3b/ R and I~ ~ I~o at b-~ R.

In the foregoing we estimated the contribution to the suppression of the critical current provided by the vortices of the different configurations shown in Fig. 2. These results and the superposition principle make it

Fig. 18. The total critical current I c as a function of distance b between the vortex axis and the center of a circular junction having radius R. (- -) The asymptotic dependence Ic/ Ico = 1.3b/ R.

~o/io° I .o

O.S

I i I

. . . . I i i i I

a~ 1.0 g/~


possible to calculate the critical current at any given arrangement of vortices in the junction. Let us consider some interesting cases.

1. Suppose that the external magnetic field is sufficiently small, so that one vortex is trapped in the junction, all its positions being assumed to be equally probable. If a vortex penetrates both electrodes, then in a characteristic case a << S 1/2 the suppression of the critical current is independent of its coordinate in the junction and is determined by expression (41). If the vortex is localized in one of the electrodes, the mean statistical critical current can be calculated by averaging the ratio (42a) over the parameter b:

="-~ fo Ic(b) (43)

The result shows that when the vortex is trapped in one of the electrodes of the tunnel junction, its critical current loses on the average one-fourth of its value.

2. Suppose a finite number of vortices is trapped in the junction, but the external field H is sufficiently small (H <</-/~2) that the vortices may be assumed to be isolated. In this case the critical current suppression is proportional to the number of vortices. The proportionality coefficient is determined by relation (39) for bent vortex lines. In a field H~ Hcz >~ (~/a) 2, where a is the characteristic size of a bent vortex (Fig. 2c), the electromagnetic regions of vortices overlap and the variation of /~ depends on the vortex ordering in the electrodes.

Let us consider the particular case when vortices are located regularly in each electrode. The maximum possible suppression of the critical current by the electrodynamic mechanism is achieved under maximum relative displacement of two vortex lattices in both electrodes (Fig. 19). The plot in Fig. 19 suggests that the phase difference ~ =X2-X~ at a given point of the junction p and consequently the supercurrent, is determined only by the distribution of fields and currents produced by two adjacent vortices in the corresponding region of overlap of their Wigner-Seitz cells. Thus, the critical current is given by

I~olC Io/2 dO . . . . (1-p2)pdp [ f~/2 fo~ )-~ [(pZ+a)Z_4p2cosZO]UZ~J ° dO odp =0.25 (44)

where ~ = (3 + cos 2 0) ~/2- cos 0. In the case under consideration the critical

= ~, Ico . current Ic 1 When vortices are randomly located in the junction, the phase difference

at a given point p is determined by the expression

~o(p) = q~o+ Xl (p ) - X2(P) (45)


\

Fig. 19. The scheme of vortex lattices in (O) upper and (O) lower electrodes. Shaded region is the superposition of adjacent Wigner-Seitz cells corresponding to the vortices O and O' in the upper and lower electrodes, respectively.

in which the phases in the first and second electrodes X1 and X2 are random functions of the coordinate p in the plane of the junction and their values are distributed uniformly in the range from 0 to 2~'. Therefore, the phase difference is also a random function uniformly distributed in the range from 0 to 2~- and the integration over the space coordinates in ( lb) and ( le) leads directly to the equality Ic -- O.

4.3. Topological Phase Transition in Josephson Tunnel Structures

Suppose one of the junction electrodes is a thin (dl << h~) superconducting film. In the absence of an external magnetic field, vortex pairs, which do not penetrate the lower film (that is, with the electromagnetic structure of the type shownjn Fig. 2d), are formed fluctuationally in this film. The equilibrium concentration of vortex pairs is determined by the Boltzmann factor:

n ~ ~7 2 e x p ( - 2 l z / k T ) (46)

2 2 where/x ~ ( H c / 8 ) £ i d ~ is the energy associated with the formation of normal nuclei of the vortex pair. Taking account of expression (37) for the potential energy of the vortex-antivortex interaction, one finds the Hamiltonian of the vortex system under consideration in the form

fit°--= Z [4"(qb2o/8~2Al±)ln(lpi-pjl/~l)q-2/~] (47) i ~ j

where p; are the coordinates of the vortices in the electrode, and the plus (minus), is chosen by the same rule as in formula (37). The Hamiltonian


(47) does not include the energy of vortex pinning on the film inhomogeneities, which is assumed to be negligible compared to the energy of vortex interaction. A most significant property of the Hamiltonian (47) is the logarithmic dependence on the intervortex separation I Pi - P~l. Accord- ing to refs. 49 and 50, a topological phase transition of Berezinskii-Koster- litz-Thouless (BKT) type takes place in such a system. It consists in dissociation of vortex-antivortex pairs with the subsequent formation of a free vortex plasma. The temperature of this transition is determined by the implicit relation

kT2D = dgo2/32~-2)t 11(T2D) (48)

It should be noted that in an isolated thin film the BKT transition does not, strictly speaking, take place, since, according to ref. 48, the logarithmic divergence of the vortex energy in such a film is cut off on the scale A± = .~2/d (where d is the film thickness) and at any finite temperature the free vortex density is nonzero. Thus, the results of the BKT model can be applied to an isolated superconducting film only when its dimensions are W<< ;t±.51-53 However, in such a bounded system the vortex interaction is more complicated, due to the influence of boundaries, which give rise, for instance, to an additional phase transition. 54

The Josephson system considered here is of particular interest, since the BKT transition can be detected in it by the temperature dependence of the Josephson critical current. At T < T2D the thin-film electrode of the junction is occupied by vortex-antivortex pairs having the mean dimension (a2), 5° with the concentration n in Eq. (46):

(a2(T)) = ~:~(q2_ kT)/(q2_ 2kT), q = qbo/4~-A~l (49)

Each pair of vortices contributes to the suppression of the critical current of the junction due to the electrodynamic mechanism considered above. At temperatures not too close to T2D, the mean dimension of such a pair is small compared to the mean distance between the pairs (that is, (a2)<< n -1) and the total suppression of the critical current is the sum of contributions of all separate pairs, the proportionality coefficient being determined by expression (41). The estimate for corrections to the critical current can be easily obtained, to an order of magnitude:

Ic(T)=lco(T)[1-n(T)<a2(T))], na2<< 1 (50)

Here/c0(T) is the temperature dependence of the critical current calculated in ref. 56 near Tc, including the effect of fluctuations of the order parameter modulus and without taking account of the formation of vortex pairs. With increasing temperature the dependence (50) turns out to be invalid only for a narrow range near T2D of the order of r = ( Tc - TZD). In this temperature


range ( a 2 ) ~ rt -1 and at T o TRD the quantity Ic falls abruptly (practically by a jump), becoming zero at T2D--at the point of transition of the thin-film electrode into a resistive state, and further at T > T2D, Ic = 0.

To our knowledge experiments have not been performed on Josephson junctions with one or both thin-film electrodes (the thickness of the film should be of the order of 10 -7 cm). Note that the temperature T2D approaches T~ of the film with decreasing parameter R J R c , where R~ is the film sheet resistance and Rc = h/e 2 is the maximum metal resistance. Estimations show that T2D differs noticeably from Tc (that is, T2D/T~ < 0.999) if R~ ~> 10-3Rc . Here Tc is the transition temperature of a massive sample suppressed by fluctuations of the order parameter modulus, 55'56 that is, the temperature when local superconductivity sets in.

5. METHODS OF DETERMINING THE PARAMETERS OF SUPERCONDUCTORS IN TUNNEL EXPERIMENTS

A microscopic theory of the Josephson effect in tunnel structures with electrodes having spatially inhomogeneous superconducting properties has been formulated in this paper. The results allow experimental determination of characteristics of real tunnel junctions.

First, the experimental study of CVCs provides information on the structure of Abrikosov vortices trapped in electrodes. Indeed, the analysis carried out in Section 4 shows that the existence of linear Abrikosov vortices (Fig. 2a) in electrodes leads to weak singularities of the differential conductivity d(Im Iq)/dV at voltage eV= A(T), which smear out with increasing temperature and practically vanish at T > 0.5 Tc (see Fig. 13). However, if bent vortices (Figs. 2c, 2d) are trapped in the junction, the singularity on the curve d(Im Iq)/dV at eV = A(T) is more pronounced and has the shape of a peak in the whole temperature range, as shown in Fig. 16. Such a difference in the CVC structure is associated with the fact that in the first case in the vortex nucleus region the electron tunneling proceeds from the gapless region to the gapless one, whereas in the second case it goes from the gapless region to the superconducting electrode with a gap in the density of states.

One should bear in mind that a quantitative comparison with the developed theory requires separation of the singularity formation mechanisms from the contribution of two-particle tunneling processes with a thresh- old at e V = A ( T ) , 47 and also from the one-particle tunneling processes in the nucleus region of a linear vortex, provided the coherence lengths of the electrode materials are essentially different. Separation of the contributions of the above-mentioned mechanisms is possible due to different singularity behaviors at eV= A(T) due to increasing external magnetic field. In the


last of the above cases the effect is proportional to the number of vortices in the junction, that is, for H << He2 the vortex contribution grows linearly with field. The two-particle processes are not affected by the magnetic field. But an increase in the number of bent vortices in electrodes with increasing field may lead to their effective "straightening" as a result of the saturation of pinning centers and of the vortex interaction. That is, the singularity amplitude on the d(Im Iq)/dV curve at V= A/e per vortex decreases in this case. Therefore, the total contribution of vortices to Im Iq may either grow or decrease with increasing field. Nevertheless, the presence of a nonlinear dependence of the singularity on the magnetic field means that bent vortices are trapped in the junction. The quantitative comparison with experimental results 12 in Section 4.2 (see Fig. 16) shows that such a situation is likely to have been realized in the experimentJ 2

An important parameter of superconducting electrodes is the upper critical field He2. A comparison of calculated and experimental CVCs allows us to determine the value of/-/c2 in the case of identical electrodes (see Fig. 12a), and, consequently, the electrode coherence lengths £s.

If the tunnel junction is based on refractory superconductors (e.g., Nb, NbN), the experimental determination of the bulk value of the electrode energy gaps is difficult because of the presence of a surface transition layer having properties different from the bulk ones. Tunneling experiments give information on the surface properties of the electrodes, and for extracting bulk characteristics from such measurements it is necessary to know the corresponding value of the paraneter y~. So it is obvious that an independent determination of TM is an important problem. This parameter can be obtained by three methods.

The first one is based on comparing the critical temperature Tc of the S N electrode of a tunnel junction and the theoretical dependences Tc (ds/£s) calculated for a series of TM, 35 a s depicted in Fig. 20.

The second method deals with the temperature dependence of the critical magnetic field Hc2( T, ds) perpendicular to the SN electrode:

Hc2(T, ds)=Hs 1- T -Hv, Hs=27r¢ 2=--~Tc (51) T = T c

"Hss) tg L~-s ~ , ~ s ) j = 1.17+~-~ ln(1 (52)

Taking formally T = 0 in Eq. (51), one obtains

= 1 dgc2 - H~2(0, ds) (53) l i p T~ dr r=Tc

where H~2(O, ds) is the value of the magnetic field following from the linear


O.6 i....,~

i-..a

4 O.8 ~-

~'. = o.ol

/ ' 0 . 4 -

0.~1 1 [ I I I 1 l l l l l I t 10 -4 I I0

d,/F," Fig. 20. Critical temperature T* of SN sandwich as a function of the thickness of the S layer d s for a series Ys~; Tc is the critical temperature of a bulk S metal. ( 0 ) experimental points from ref. 57.

extrapolation of experimental data from the Tc region to the T = 0 point. The parameter yM is determined by substitution of tip into Eq. (52).

The third method is based on a comparison of the experimental differential conductivity of the NSIN-type junction for T<< Tc with the densities of states at the interface of a compound SN electrode and tunnel barrier calculated for a series of YM. A calculation with finite thickness ds ~ ~s was carried out by the method described in Section 2.1, involving the substitution of condition (5d) by the condition dO(x)/dx = 0 at the free boundary. Figure 21 shows that the results of calculations made in the framework of the McMillan model 22 differ qualitatively from the experimental data obtained in ref. 57. At the same time the densities of states calculated in the present paper are in good agreement with experimental ones. This comparison, in combination with a comparison of the T~(ds/~s) dependences for the NS electrode make it possible to estimate the coherence length of the S metal (Pb), ~:s = 110/~, and the corresponding parameter yM---0,5. Furthermore, the comparison shows the inadequacy of the McMillan tunnel model for the given situation, since in this model a potential barrier at the NS interface is assumed.


5

4

~ 3

Z

2

1

0

i0

2 i i sZ.' I "~. \ • i ~3 k. 4

0.5 I E/A.

Fig. 21. Density of states at the outer boundary of the S layer of an NS sandwich at 3,M=0.5, T<< To. (--) represent N(e) obtained experimentally 57 for AI-AI203-Pb/Cd junctions. (-. -) N(e) calculated in the framework of the McMillan tunnel model.

These me thods were recent ly app l i ed for the se l f -consis tent exper i - menta l de t e rmina t i on o f the supe rconduc t ing pa rame te r s o f Z r N films 5s us ing a Z r N - I - P b J0sephson junc t ion .

6. C O N C L U S I O N

Analys i s o f the concre te phys ica l mechan i sms respons ib le for the smearing o f the s ingular i t ies o f the classical tunne l theory at eV = 2A(T) shows tha t the tunne l j unc t ion p roper t i e s are very sensi t ive to the shape o f this


smearing. In the case of the proximity effect in the junction electrodes, modifications in the densities of states shown in Fig. 3" lead to the "knee" structure of the CVC and to variations of the I t (T) dependence as compared with the Ambegaokar-Baratoff result. 42 The presence of vortex lines in junction electrodes leads to the gapless state (see Fig. 9) as opposed to the proximity effect. As a result, there is no "knee" structure in this case, but significant leakage current appears at low voltages e V < 2A(T) for T<< Tc. It is interesting that both mechanisms considered do not allow one to describe quantitatively the experimentally observed change of the sign of the cos ~ term. At the same time, the phenomenological approach 16 accounts for this effect, but it does not describe the other phenomena mentioned above. All this means that the Josephson tunnel current is extremely sensitive to the various physical mechanisms of modification of the electrode properties.

ACKNOWLEDGMENTS

Useful discussions of this work or parts of it with D. E. Khmel'nitskii, A. I. Larkin, K. K. Likharev, and V. K. Semenov are gratefully acknowl- edged.

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theoretical investigation of josephson tunnel junctions with spatially

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