Incoherent multiple Andreev reflection in an array of SNS junctions
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Incoherent multiple Andreev reflection in an array of SNS junctions.
N. M. Chtchelkatchev abc , a L. D. Landau Institute for Theoretical Physics RAS, 117940 Moscow, Russia b Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142092, Moscow Region, Russia c Moscow Institute of Physics and Technology, Moscow 141700, Russia
Submitted June 19, 2018
Last years many interesting effects related to incoherent MAR have been experimentally found, but onlyfew of them were theoretically explained. It was shown, for example, that if the voltage at the edges of alinear array is V then subgarmonic structures in the current -voltage characteristics appear not only at usualfor nonstationary Josephson effect positions, V n = 2∆ /n , where n is integer, but also at voltages other than V n . A step towards description of electron transport in a dirty array of SNS junctions is done in this letter.It is shown that subgarmonic structures may indeed appear at “unusual” voltages.PACS: 73.23.-b, 74.45.+c, 74.81.Fa Important role plays Andreev reflection mechanismin the subgap charge transfer through a normal metal(N) – superconductor (S) junction [1]. When an electronquasiparticle in a normal metal with the energy belowthe superconduting gap reflects from the interface of thesuperconductor into a hole, Cooper pair transfers intothe superconductor. If the normal metal is surroundedby superconductors, so we have a SNS junction, a num-ber of Andreev reflections appear at the NS interfaces.In equilibrium this leads to Andreev quasiparticle levelsin the normal metal that carry considerable part of theJosephson current; out of the equilibrium, when super-conductors are voltage biased, quasiparticles Andreevreflect about 2∆ /eV times transferring large quanta ofcharge ( ∼ e · [2∆ /eV ]) from one superconductor to theother. This effect is called Multiple Andreev Reflec-tion (MAR). If the voltage is near V n = 2∆ /n , where n = 1 , , . . . , so-called subgap features in current voltagecharacteristics appear. Then large contribution to thecurrent give quasiparticles that go from the gap edgeof one superconductor to the gap edge of the other su-perconductor (after MAR in the normal region); bulksuperconductor DoS is large at the gap and this is thereason of subgap features.This letter is devoted to investigation of electrontransport in arrays of dirty superconductiong meso-scopic SNS (SFS) junctions. I assume that normalparts of the junctions are “long”. It implies that min-imum distance d between adjacent superconductors ismuch larger than the characteristic scale of anomalousgreen function (Cooper pair wave function) decay ξ N from a superconductor in the normal metal. If thediffusion coefficient of the normal metal is D N than e-mail: [email protected] ξ N ∼ p D N /T ≪ d , where T is the temperature. En-ergy relaxation is not included in the calculations. So itis also assumed that the array is not large, its charac-teristic length does not exceed the characteristic lengthscale of quasiparticle energy relaxation in the normalmetal and in superconductors. [Quasiparticle energyrelaxation was “taken into account” in Ref. [4], whereincoherent MAR in long SNS junction was discussed,by small imaginary part supplied to the energy in re-tarded and advanced greens functions; but collisionalintegrals were not taken into account in kinetic equa-tions. It is not clear why this procedure is correct. I donot follow this way here.] Conditions listed above alsomean that in equillibrium the proximity (Josephson) ef-fect between the superconducors is suppressed. Joseph-son current, for example, is exponentially small with d /ξ N . It is known that out of the equilibrium whenthere is a finite bias between the superconducors theproximity effect restores in some sense: subgap featuresappear due to MAR in current-voltage characteristics.MAR in long Josephson junctions usually is referred toas ‘incoherent” since there is no contribution to electrontransport from effects related to interference of quasi-particle wave functions in normal metals (contrary toJosephson effect in “short” superconducting junctions).Last years many interesting effects related to incoherentMAR have been experimentally found, but only few ofthem were theoretically explained. It was shown, for ex-ample, that if the voltage at the edges of a linear arrayis V then subgarmonic structures in the current -voltagecharacteristics appear not only at usual for nonstation-ary Josephson effect positions, V n = 2∆ /n , where n isinteger, but also at voltages other than V n , see Ref. [2]and refs. therein. A step towards description of elec-1 N. M. Chtchelkatchev
Fig.1. a) An array of SNS junctions, like in experi-ments [2]. b) A sketch of the normal layer connection tothe superconductors in experimental SNS arrays. Blackarea is insulating. tron transport in a dirty array of SNS junctions is donein this letter. It is shown that subgarmonic structuresmay indeed appear at “unusual” voltages. Investigationof electron transport is based on Usadel equations:[ ˇ H, ˇ G ] = iD ∇ · ˇJ , ˇJ = ˇ G ∇ ˇ G, ˇ G = ˇ1 , (1)ˇ H = ˇ1( i ˆ σ z ∂ t − eφ + ˆ∆) , (2)ˆ∆ = − ∆ ∗ ! , (3) I ( t, x ) = π ~ σ N e Tr ˆ σ z ˆ J K ( t, t ; x ) , (4)where ˇ G = ˆ R ˆ K A ! . (5)Here ˆ R , ˆ K and ˆ A denote retarded, Keldysh and ad-vanced quasiclassical green functions. The hat remindsthat greens function are in turn matrices in Nambuspace. ˆ σ are Pauli matrices that act in Nambu space; D is a diffusion coefficient that equals D N in a normalmetal and D S in a superconductor; ∆ is the supercon-ducting gap; φ is electrical potential; σ N is the conduc-tivity of a normal metal. The boundary conditions forthe Usadel equations at NS interfaces are: σ N ˇ J · n = G G , ˇ G ] − , (6)where 1 , G is the surface con-ductance and n is the unit normal to the interfacepointing to the second half-space.The problem is to calculate the current in a long SNSarray using Eqs.(1)-(6). Solving these equations directly Fig.2. The simplest array of SNS junctions: SNSNS. is a hard task because they are nonlinear, nonuniformand essentially time dependent (relative phases of su-perconductors rotate with biases). So the main task isdeveloping an approach for the problem in hand thatallows to perform significant part of transport calcula-tions analytically and that is applicable in rather widerange of system parameters. There is no universal ap-proach that helps to solve Usadel equations analytically,so any analytical method of Usadel equations solutionis usually specific to the given class of the physical sys-tems.Normal layers in experimental SNS arrays [2] con-nect with superconductors like it is shown in Fig.1. SNSjunctions of this type are usually referred to as “weak-links” [3]. Boundary conditions, Eq.(6), can be simpli-fied in this case: retarded and advanced Greens func-tions at superconducting sides of NS boundaries can besubstituted by Greens function from the bulk of the su-perconductors. These “rigid” boundary conditions ap-proximation are reasonable because a) the magnitudeof the current is much smaller than the critical cur-rent of the superconductor (this is assumed), b) thecurrent entering the superconductor from narrow nor-mal metal wire [the width . ξ = p D S /T c , where T c isthe critical temperature of the superconductor] spreadsnearly at the NS interface over the whole superconduc-tor. There are also other cases when rigid boundaryconditions are correct, for example, if the NS bound-ary has small transparency due to an insulator layer orother reasons.The Keldysh greens function has the following gen-eral parametrization: ˆ K = ˆ R ◦ ˆ f − ˆ f ◦ ˆ A , where ˆ f is the distribution function. It was shown in Ref. [4]that the current in a long SNS junction can be foundfrom investigation of the current distribution in an effec-tive network where the role of voltages play distributionfunctions made from components of ˆ f , the role of resis-tances play NS resistances renormalized by proximityeffect and normal layer resistances. This idea can beapplied to an SNS array.It is convenient to write ˆ f = ˆ1 f + + ˆ σ z f − . Phasesof the superconducting order parameters rotate with ncoherent MAR in an array of SNS junctions. f ( ǫ, ǫ ′ ) = P n = { n i ; i =1 , ... } ˆ f n ( ǫ − ǫ ′ ) δ ( ǫ − ǫ ′ + n i V i ), where n i is in-teger, V i is the bias at the i ’s superconductor. (Similarconsideration apply for retarded and advanced greensfunctions.) However due to the absence of Josephson-type interference effects nonzero are only that ˆ f n whichprovide time independent (dissipative) component ofthe current as it is well explained in Ref. [4]... One canwrite boundary conditions Eq.(6) at an NS interface forˆ f n as σ N D + ∇ f (2)+ = − G + ( f (2)+ − f (1)+ ) , (7) σ N D − ∇ f (2) − = − G − ( f (2) − − f (1) − ) . (8)Here the label 2 corresponds to the normal metal. Forexample, D + = 14 Tr (cid:16) ˆ1 − ˆ R ˆ A (cid:17) , (9) D − = 14 Tr (cid:16) ˆ1 − ˆ R ˆ σ z ˆ A ˆ σ z (cid:17) . (10)Definitions of G ± are similar and can be found, e.g., inRef. [4].It is useful to go from f ± to n e ( h ) [4] that are relatedas follows: f + = 1 − ( n e + n h ), f − = ( n h − n e ). Thenthe boundary conditions, Eqs.(7), can be written as I e = G T ( n (1) e − n (2) e ) −− G A (( n (2) e − n (2) h ) − ( n (1) e − n (1) h )) , (11) I h = G T ( n (1) h − n (2) h )++ G A (( n (2) e − n (2) h ) − ( n (1) e − n (1) h )) . (12)Here G T = G + , G A = ( G − − G + ) / I e ( h ) = − σ N ( D + ∂ ( n e + n h ) ± D − ∂ ( n e − n h )). If the supercon-ducting bank labelled by index “1” is in equilibrium, so n (1) e = n (1) h = n F , then we arrive at boundary conditionswritten in Eq.(21) of Ref. [4]. If one writes Eqs.(11)-(12)for NS and SN interfaces of a superconducting islandand uses conditions of electron and heat currents con-servation in the island then quasiparticle distributionfunctions corresponding to superconducting islands canbe excluded from boundary conditions: I e = A + ( n (2) e − n (1) e ) + A − ( n (2) h − n (1) h ) , (13) I h = A + ( n (2) h − n (1) h ) + A − ( n (2) e − n (1) e ) . (14)It is worth noting that Eqs.(13)-(14) are derived for alinear array of SNS junctions; generalization of theseequations is straightforward for more complicated ar-rays. In the last pair of equations labels 1 and 2 corre- Fig.3. a) An illustration of the boundary conditionsEqs.(13)-(12) in terms of a circuit is given in Fig.3.Electron and hole currents entering the left side of thepyramid flow in one normal layer, the right currentsflow in the other normal layer. − A − resistance de-scribes Andreev reflection, A + — quasiparticle normaltransmission through the superconductor and A − is An-dreev transmission characteristics. b) An illustrationof the boundary conditions Eqs.(13)-(12) (developed inRef. [4]) at the surface of the superconductor connectedto an electron reservoir , i.e., when n (1) e = n (1) h = n F inthe superconductor. Here R A = 1 /G A , R T = 1 /G T . spond to normal layers surrounding the superconduct-ing island. Information that the island is superconduct-ing is included in A ± definition through G ± : A ± = 12 ( µ + ± µ − ) , (15) µ ± = G (1) ± G (2) ± G (1) ± + G (2) ± . (16)A circuit illustration of this boundary condition is givenin Fig.3a. Electron and hole currents entering the leftside of the pyramid flow in one normal layer, the rightcurrents flow in the other normal layer. − A − resis-tance describes Andreev reflection, A + — quasiparticlenormal transmission through the superconductor and A − is Andreev transmission characteristics. It was as-sumed deriving Eqs.(13)-(12) that electrochemical po- N. M. Chtchelkatchev tential within the superconductor is constant. This iscorrect if the characteristic distance between NS inter-faces of the superconducting island is smaller than theelectron disbalance characteristic length λ Q [5]. It isimplied that nonequillibrium quasiparticles above thegap do not have enough time for energy relaxation attheir fly through the superconductor, so electrochemi-cal potential within the superconductor is constant. Orif characteristic bias value δV between adjacent super-conducting islands is much smaller than the gap and thetemperature is much below T c then most part of the cur-rent carry Cooper pairs through superconductors ratherthan quasiparticles above the gap. Then correction tothe total current in the array from quasiparticle energyrelaxation in superconductors is expected to be small as δV / ∆ ≪ λ Q is not relevant.Next important step is expressing all boundary con-ditions in terms of the distribution functions ¯ n e ( h ) [4],where n ± = n ± − I ± σ N Z ∞ dx ′ (cid:18) D ± ( x ′ ) − (cid:19) ≡ n ± − m ± I ± . Here n ± = n e ± n h and ¯ n ± = ¯ n e ± ¯ n h , D ± ∂f ± ≡ I ± /σ N . The integral here goes from the NS boundary( x = 0) into the depth of the normal metal. Bound-ary conditions Eqs.(13)-(12) written in terms of ¯ n e ( h ) will have the same form if one replaces G ± by ¯ G ± = G ± / (1 + G ± m ± ). Then¯ µ ± = µ ± m (2) ± − m (1) ± ) µ ± , (17)where indices 1,2 correspond to two normal layers con-tacting with the superconducting island. It is usefulto work with ¯ n e ( h ) because then the proximity effectrenormalization of the boundary resistances, m ± , is au-tomatically taken into account.The next step toward current calculation is to drawan effective network that describes MAR in the array ofthe junctions using Eqs.(13)-(12) and evaluate partialcurrents in this network using to Kirchhoff’s laws. Im-portant task is to find voltages at the superconductingislands. But this can be found easily in several cases.For example, when the array consists of equal SNS junc-tions, or if most of the voltage drops at normal layers.Below these situations will be discussed. More compli-cated cases I leave for extended paper. If currents inthe MAR network are found then electric current canbe evaluated, e.g., as follows: I = 12 e Z dE ( I e ( E ) − I h ( E )) , (18) Fig.4. MAR in a SNSNS array with equal SNS junc-tions. The graph shows the effective circuit for quasi-particle currents in energy space. The role of voltageshere play quasiparticle distribution functions. For ex-ample n n is the quasiparticle distribution function de-pending from E + nV / U n = n F ( E + nV / U ’s correspond tothe first superconductor of the array, the lower raw – tothe last superconductor. The thick line in the center ofthe graph represents the superconducting island. where I e ( h ) correspond to the normal layer at the edge ofthe array contacting with the superconducting reservoirwith zero voltage.The format of the letter does not allow to describecomplicated arrays here, so one of the simplest SNS ar-rays, the SNSNS junction (Fig.2), will be consideredbelow as an example. It will be shown how to constructan effective network that helps to describe its transportproperties. Transport in more complicated arrays, likein Fig.1 can be described in a similar manner as forSNSNS; it will be demonstrated in an extended versionof this paper.Effective MAR network for a SNSNS junction likein Fig.2 is shown in Fig.4. The bias between the su-percondcutors at the edges of the array is V . Then thebias of the central superconductor is V /
2; it follows fromsymmetry reasons. Currents I n and J n correspond I e for lines beginning from U n and ending at U n with n > n and − I h vice-versa. The role of voltages playquasiparticle distribution functions. For example n n isthe quasiparticle distribution function depending from E + nV / U n = n F ( E + nV / ncoherent MAR in an array of SNS junctions. U ’s correspond to the first superconductorof the array, the lower raw – to the last superconductor.The thick line in the center of the graph represents thesuperconducting island.Lets find recurrence relations for the currents I n and J n . It follows from Fig.4 that I n = ¯ G n − T ( n n − − U n − ) + ¯ G n − A ( n n − − n n − ) ,I n = G N ( n n − − n n − ) ,I n = A n − ( n n − − n n − ) + A n − − ( n n − − n n − ) ,I n = G N ( n n − n n − ) ,I n = ¯ G nT ( U n − n n ) + ¯ G nA ( n n − n n ) ,J n − = ¯ G n − A ( n n − − n n − ) + ¯ G n − T ( U n − − n n − ) ,J n +2 = ¯ G nA ( n n − n n ) + ¯ G nT ( n n − U n ); J n = ¯ G n − T ( n n − − U n − ) + ¯ G n − A ( n n − − n n − ) ,J n = G N ( n n − − n n − ) ,J n = A n − ( n n − − n n − ) + A n − − ( n n − − n n − ) ,J n = G N ( n n − n n − ) ,J n = ¯ G nT ( U n − n n ) + ¯ G nA ( n n − n n ) ,I n − = ¯ G n − A ( n n − − n n − ) + ¯ G n − T ( U n − − n n − ) ,I n +2 = ¯ G nA ( n n − n n ) + ¯ G nT ( n n − U n ) . Getting rid of the distribution functions in these set ofequations one can get: I n A + n − − A − n − + ( a n + a n − ) ! −− I n − b n − − I n +2 b n = ( U n − U n − ) . (19)Same equation satisfies J n . Here a n = 1 G N + 1¯ G nT −
12 ¯ G nA + ¯ G nT ¯ G nA ¯ G nT , (20) b n = 12 ¯ G nA + ¯ G nT ¯ G nA ¯ G nT . (21)Eq.(19) coincides with recurrence relation Eq.(52) fromRef. [4] derived for a single SNS junction if I replace inEq.(52) G N by g G N = G N G − G N + ¯ G − . (22)It means that SNSNS array behaves similarly as a singleSNS junction but with energy dependent resistance ofthe normal layer. ¯ G − has singularities at energy corre-sponding to the gap edges of the superconducting is-land in the center of the SNSNS array. This is the R N d I / d V Fig.5. Subharmonic structure in differential conductiv-ity of SNSNS junction with the ratio r of SN boundaryresistance to normal layer resistance R N , r = 0 .
2. Peaksat half-integer 2∆ /V do not appear in dI/dV of SNSjunctions. reason of subharmonic singularities in current voltagecharacteristics if 2∆ /V = n/ n = 1 , , . . . instead of“usual” positions,2∆ /V = n ; Fig.5 illustrates it. FromEq.(22) follows that unusual position of subharmonicsingularities disappear if ˜ G N → G N if the resistanceof the normal layer exceeds the resistance of SN inter-faces. Then the central superconducting island of theSNSNS array effectively “disappear”. It was checked ifthe exchange field in SFSFS junction splits subharmonicstructure; there was found no exchange field splitting ef-fects because configurations of the ferromagnet and thesuperconductor like in Fig.1 does not lead to enoughexchange field deformation of superconductor DoS nearSF boundary that is necessary for the splitting effectobservation. This paragraph is conclusion of the paper.I’m grateful to T.Baturina and I.S. Burmistrovfor stimulating discussions. I especially thankI. Drebushchak for helpful discussions and the idea todraw Fig.3. I also thank RFBR 03-02-16677, the Rus-sian Ministry of Science, the Netherlands Organizationfor Scientific Research (NWO), CRDF and RussianScience Support foundation.
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