A theoretical explanation of giant conductance oscillations observed in
normal mesoscopic rings with superconducting ``mirrors" is proposed. The effect
is due to resonant tuning of Andreev levels to the Fermi level, which enhances
the transparency of the system to the normal current. The mechanism is
demonstrated for a one-dimensional model system.Comment: 10 pages, RevTeX, 3 fig. available upon request, Appl. Phys. Report
  94-

Jonson, M.

Kadigrobov, A.

Shekhter, R. I.

Zagoskin, A.

English

arXiv

A theoretical explanation of giant conductance oscillations observed in normal mesoscopic rings with superconducting "mirrors" is proposed. The effect is due to resonant tuning of Andreev levels to the Fermi level, which enhances the transparency of the system to the normal current. The mechanism is demonstrated for a one-dimensional model system

A. Kadigrobov (7171295)

Alexandre Zagoskin (1251504)

R.I. Shekhter (7171298)

M. Jonson (7171301)

Loughborough University Institutional Repository

arXiv:cond-mat/9410080v1  21 Oct 1994Applied Physics Report 94-6Giant Conductance Oscillations in a Normal Mesoscopic RingInduced by an SNS Josephson CurrentA. Kadigrobov(1,2)∗, A. Zagoskin(1)†, R.I. Shekhter(1)‡, and M. Jonson(1)§Department of Applied Physics(1) and Department of Physics(2)Chalmers University of Technology and Go¨teborg University, S-412 96 Go¨teborg, SwedenAbstractA theoretical explanation of giant conductance oscillations observed in normalmesoscopic rings with superconducting “mirrors” is proposed. The effect isdue to resonant tuning of Andreev levels to the Fermi level, which enhancesthe transparency of the system to the normal current. The mechanism isdemonstrated for a one-dimensional model system.PACS: 74.50.+r, 74.80.Fp, 73.20.DxTypeset using REVTEX∗Permanent address: B.I. Verkin Institute for Low Temperature Physics and Engineering, 47Lenin Ave., 310 164 Kharkov, Ukraine†Email: alexz@fy.chalmers.se‡Email: shekhter@fy.chalmers.se§Email: jonson@fy.chalmers.se1Recently, in a series of experiments by Petrashov et al. [1], new unusual properties ofmesoscopic silver rings in contact with superconducting islands (“mirrors”) were observed.These “mirrors” could be placed across the current leads (L-case) as in Fig. 1a, or as shownin Fig. 1b at the stubs connected to the ring perpendicular to the normal current flow (T-case). The three most striking features revealed in experiments are as follows:(i) the amplitude of hc/2e-periodic Aharonov-Bohm (AB) oscillations was at least 100 timeslarger in the L-case than in the ring without “mirrors”; in the T-case the enhancement ofhc/2e-oscillations was about 10 times, and hc/4e-oscillations were observed as well;(ii) in the T-case the normal resistance of the system grows approximately twice, as thesuperconductivity of the “mirrors” is being suppressed by magnetic field (above 500 Gs);(iii) the effects were totally absent in the case when one branch of the ring was completelycrossed by a superconducting strip.The confinement idea proposed in the original paper [1], i.e. that the effect is due toconfinement of quasiparticles in the ring by the “mirrors”, qualitatively explains the L-casebehaviour, but meets difficulties both in obtaining a proper magnitude of the effect in theL-case (the maximum enhancement being of order 10), and in explaning the effect in theT-case, where the quasiparticles are not confined in a longitudinal direction.An alternative explanation suggested by de Vegvar and Glazman [2] is based on a suppo-sition that the “mirrors” induce superconductivity in a significant portion of the ring itself.These superconducting parts play the role of filters, through which only Cooper pairs canpass. However, this mechanism neither accounts for the significant difference between L-and T-configurations, nor for the complete absence of the effect when one branch of the ringwas crossed by a superconductor (as the authors of Ref. [2] point out themselves).In this paper we present some theoretical arguments which seem to explain the physicalnature of the observed phenomena; they are supported by a model calculation. In ouranalysis we assume that the phase breaking length Lφ = (Dτφ)1/2 and the normal metalcoherence length LT = (h¯D/kBT )1/2 are larger than the characteristic sample size L. Here2τ−1φ is the inelastic scattering rate, D is the diffusion constant of quasiparticles, kB is theBoltzmann constant, and T is temperature. We will also assume that the ring is weaklycoupled to the normal reservoirs (source and sink of normal current).We start from a qualitative discussion. If an electric current is made to pass througha mesoscopic normal ring, it is well known that its magnitude will oscillate as a functionof the magnetic flux threading the ring. These normal AB oscillations persist if the ringis brought in contact with two superconducting mirrors. However, now a new group ofquasiparticles starts to contribute to the oscillations: the electrons (or holes) that undergoAndreev reflections at the NS-boundary (between the ring and one of the mirrors). We willsee below that their contribution may dominate the AB oscillations under certain conditions.In a ring which is weakly connected to external reservoirs, the quantized energy levelsof quasiparticles are well defined [6]. In our case these levels are formed by reflections bothat NN-interfaces (between the ring and the normal reservoirs) and at NS-boundaries withthe “mirrors”. The energy levels and are therefore sensitive to the magnetic flux threadingthe ring, Φ, and to the superconducting phase difference between the mirrors, ∆φ. Thecorresponding quasiparticle states carry both the normal- and the Josephson [6,4] current.Since the normal current in the linear response limit is carried only by the quasiparticleson the Fermi surface, it is resonantly enhanced each time an energy level — driven bythe magnetic flux and/or the superconducting phase difference — passes the Fermi energy.Simultaneously, the Josephson current changes sign.Therefore the relation can be guessedIq ∼ dIJd∆φ. (1)As a matter of fact, using a 1D model (to be described below) we are able to derive therelation valid in the ballistic case:limǫ→0IqV ǫ=(eL4h¯vFdIJd∆φ+e2πh¯). (2)Here ǫ is the probability for an electron to leave the ring for one of the normal reservoirsthrough the NN-interface (see below).3The second term in (2) is due to the fact that the energy levels in the ring are formed notonly by Andreev reflections at NS-boundaries, but by the normal reflections at NN-interfacesas well. Therefore their shift in a magnetic field (and thus the conductance [3]) can not becompletely accounted for by an expression like (1), where the magnetic flux, Φ, entersin a gauge invariant combination with the phase difference between the superconducting“mirrors”, ∆φ. There must be a contribution to the level shift that depends solely on Φ.The relation (2) shows that if there is a Josephson current in the system, it leads to aresonant contribution to the normal current for certain values of the magnetic flux. Indeed,the phase dependence of the Josephson current in an SNS contact has a sawtooth-like form,which would give rize to δ-function shaped peaks when its derivative is taken with respectto phase [4]. The question is, how can a stationary supercurrent flow between two finite andsmall superconducting “mirrors”?Let us first consider a standard SNS-junction in an external magnetic field, B, parallelto the boundary between the normal layer and superconducting half-spaces (Fig. 2a). Thephase difference between the superconductors is [4]∆φ(y) = φ0 − 2eBLyh¯c≡ φ0 − 2π2Φ(y)Φ0, (3)where y is the coordinate on an axis that lies in the plane of the normal layer and is normal tothe magnetic field, φ0 is an arbitrary phase difference between two isolated superconductors,and L is the width of the normal layer. The quantity Φ(y0) is evidently the magnetic fluxthrough the part of the junction between the lines y = 0 and y = y0, and Φ0 = hc/e. Thedistribution of Josephson current is schematically shown in Fig.2a; each current line encirclesone half flux quantum (Josephson vertex).In a real experimental situation the normal stubs connecting the metal ring with the“mirrors” have a finite width. This creates the situation when equal and opposite Josephsoncurrents flow along the edges of the stubs (see Fig.2b). In order to establish such a currentdistribution, the phase difference between the superconducting “mirrors” tunes to each givenvalue of the penetrating magnetic flux (via changing the constant φ0 in (3)). Therefore the4partial Josephson currents will oscillate, contributing to the AB oscillations in the normalconductance of the ring, though their sum (net Josephson current between the “mirrors”)is always zero.Now we can discuss when this contribution can play a major role. It is known that theamplitude of the Josephson current density is not sensitive to an increase in the area ofan SNS junction [4]. On the other hand, the amplitude of the conductance oscillations ina normal metal ring decreases by a factor√N⊥ as its cross-section grows [5] [the numberof channels in a ring with the cross-section area A is N⊥ ∼ A/λ2F ]. The reason is that inthe former case it is the effective momentum of the electron-hole excitations that carry theJosephson current, which is quantized:p(e)‖ − p(h)‖ =√2m(EF + E)− p2⊥ −√2m(EF −E)− p2⊥ ≃2mE√EF − p2⊥, (4)while in the latter case it is the electronic momentum itself:p‖ ≃√2mEF − p2⊥ +mE√2mEF − p2⊥. (5)The first term in (5), which is absent in (4), causes fast oscillations when the partial currentis integrated over the transverse momentum p⊥. This is the reason of drastical reduction ofthe “normal” AB oscillations’ amplitude in a ring with large cross-section. In this case the“Josephson” contribution will be dominating.The above considerations lead us to the conclusion that the ratio of the amplitude of ABoscillations in a ring with superconducting “mirrors” to the amplitude of AB oscillationsitsin a “normal” ring is of the order of√N⊥.Let us make some numerical estimates. Taking for the cross-section area the valueA ≈ 5 · 10−11cm2, qhich is consistent with the experiment [1], and λF ∼ 10−8cm, we findthat the enhancement of the AB conductance oscillations due to the proposed mechanismis ≃ 700 times compared to the ring without superconducting “mirrors”. (The actual valueof Ref. [1] was up to 400 times).5The above qualitative speculations can be corroborated by a model calculation. Themodel system is shown in Fig.3. It contains all the physically significant features of theexperimental setup (Fig.1); the normal part of the current from reservoir I to reservoir IIthrough leads 3, 1, and 4 is controlled by the Josephson current between the “mirrors”through leads 1 and 2.All the normal scattering in our model is confined to the nodes A and B. These aredescribed by identical 6 × 6 S-matrices , which relate the incoming and outgoing waveamplitudes of the quasiparticles in the 1D wires. We use real matrices parametrized by areal number 0 ≤ ǫ ≤ 1/2 [6], which in the limit ǫ≪ 1 (weak coupling to the reservoirs, i.e.small transition probability from lead 1 to leads 3 and 4) have the formS =−ǫ/2 · 1ˆ (1− ǫ/2) · 1ˆ √ǫ · 1ˆ(1− ǫ/2) · 1ˆ −ǫ/2 · 1ˆ √ǫ · 1ˆ√ǫ · 1ˆ √ǫ · 1ˆ (−1 + ǫ) · 1ˆ. (6)Here 1ˆ is the unit 2×2-matrix, which reflects the fact that in the presense of an NS-boundarywe have to use the two-component wave function of the quasiparticle even in the normalleads, in order to account for the electron-hole correlations created by Andreev reflections[4]. Eq.(6) reflects the fact that the electron- and holelike excitations are not mixed in thenodes A and B. On the other hand, we assume that at the interfaces of leads 1, 2 and thesuperconducting islands only Andreev reflection takes place, and the normal reflection isabsent.The calculations to be described are straightforward. We are interested in the linearresponse value of the normal conductance. Therefore, the problem is reduced to the cal-culation of the transition probability of a quasiparticle from, say, reservoir I to reservoirII. The initial scattering matrices can be replaced by effective matrices of dimensionality4× 4, SA and SB. They only relate the quasiparticles in leads 3, 4, and the portion of lead1 between A and B to each other and include the effects of Andreev reflections (see [7] fordetails). We have a standard Landauer configuration with two reservoirs connected by a 1D6wire with scatterers A and B, but with two-component wave functions of the quasiparticles;the components are mixed by Andreev reflections at NS interfaces. The conductance is thenobtained as [8]G =2e2h· 2∫ ∞0dξ (T>0 +R>a )(−∂nF (ξ)∂ξ)+ η, (7)where T>0 (R>a ) is the normal transition (Andreev reflection) probability for an electronincident from the left normal reservoir; nF (ξ) is the Fermi distribution, and ξ is energymeasured from the Fermi level. The term η in (7) quickly oscillates as a function of theelectron momentum (as ∼ exp 2pFL, L being the length of lead 1) [7] and is exactly zero inthe case of time reversal symmetry [8].After averaging over fast spatial oscillations on the scale of λF ≪ L, the coefficients in(7) take the following values [7]:T>0 (ξ) ≈ R>a (ξ) ≈ ǫ2(|a+(ξ)|−2 + |a−(ξ)|−2). (8)The resonant denominators |a±(ξ)|2 vanish close to the energies ξ±n of the Andreev levels inlead 1:|a±(ξ)|−2 ≈∑n(2Lh¯vF)2·(ξ − ξ±n )2 + ǫ2(h¯vF4L)2−1; (9)ξ±n =πh¯vFnL+h¯vF2L(π ∓ 2π2ΦΦ0). (10)Here vF is the Fermi velocity and Φ0 = hc/e is the magnetic flux quantum. In place of thesuperconducting phase difference, ∆φ, only the (dimensionless) magnetic flux Φ/Φ0 throughthe loop formed by the normal wires 1, 2 and the superconducting mirrors enters expression(10) for the Andreev energies. This is due to the fact that the Josephson current in lead1 must be exactly cancelled by the one in lead 2. This condition fixes the phase differencebetween the superconducting mirrors.Provided that the Andreev level separation exceeds the level width ∆E = ǫh¯vF/4L, acondition which is satisfied for small enough ǫ, we can calculate the integral in (7) using thePoisson summation formula. At zero temperature one finds the expression7G = ǫe2h(1 + 2∞∑n=1(−1)ne−2|n|ǫ cos(n · 2π2ΦΦ0)). (11)If we compare this result to the well known expression for the Josephson current in a planarSNS-junction [4],IJ(∆φ) =8evFπL∞∑n=1(−1)n+1 sinn∆φn, (12)we see that indeed, in the weak coupling limit (ǫ → 0) we have the relation between thenormal conductance and the Josephson current given by equation (2).The above calculations are directly generalized to the case of N⊥ > 1 transverse modesin the normal wire, provided that they are not mixed by scattering. Then the conductance(11) should be simply multiplied by N⊥. The fact that the amplitude of its oscillations nowcan exceed the conductance quantum, 2e2/h, reflects the ballistic character of the systemunder consideration.In conclusion, we have demonstrated that the normal conductivity of a mesoscopic ringwith superconducting “mirrors” is sensitive to the Josephson current between them. Thecorresponding contribution to the conductance is resonant and in a many-channel case givesrize to greatly enhanced Aharonov-Bohm conductivity oscillations in the system. The resultsprovide an explanation to recent experimental results by Petrashov et al. [1].ACKNOWLEDGMENTSThe authors are grateful to V. Antonov, C.W.J. Beenakker, T. Claeson, P. Delsing, Yu.Galperin, L. Gorelik, V. Petrashov, S. Rashkeev, V. Shumejko, and A. Slutskin for manyfruitful discussions.This work was supported by the Royal Swedish Academy of Sciences (KVA), the SwedishNatural Science Research Council (NFR), and by the Swedish Board for Industrial andTechnical Development (NUTEK). One of us (A.K.) gratefully acknowledges the hospitalityof the Department of Applied Physics and Department of Physics, Chalmers University ofTechnology and Go¨teborg University.8REFERENCES[1] V.T. Petrashov, V.N. Antonov, P. Delsing, and T. Claeson, Phys. Rev. Lett. 70, 347(1993).[2] P.G.N. de Vegvar and L. Glazman, Phys. Rev. Lett. 71, 2351 (1993).[3] W. Kohn, Phys. Rev. 133, A171 (1964).[4] A.V. Swidzinskij, Spatially Nonuniform Problems of the Theory of Superconductivity (inRussian), Ch.IV; ”Nauka”, Moscow, 1982.[5] J. Imry, in Directions in Condensed Matter Physics, ed. by G.Grinstein and G.Mazenko;World Scientific Publ., Singapore, 1986, p.101.[6] M. Bu¨ttiker, Y. Imry, and M. Ya. Azbel, Phys. Rev. A 30, 1982 (1985).[7] A. Zagoskin, A. Kadigrobov, R. Shekhter, and M. Jonson (unpublished).[8] C.J. Lambert, J. Phys.: Condens. Matter 3, 6579 (1991); P.M.A. Cook, V.C. Hui, andC.J. Lambert (unpublished).9FIGURESFIG. 1. Sketch of the experimental setup of a mesoscopic ring with superconducting “mirrors”used in Ref. [1]. I, II are normal reservoirs, S - “mirrors” (small superconducting islands). Themagnetic field is normal to the picture plane. (a) L-configuration. (b) T-configuration.FIG. 2. (a) Distribution of Josephson current in an SNS-junction in the presence of a magneticfield; (b) Josephson current in a mesoscopic ring with superconducting “mirrors”FIG. 3. Model system where 1, 2, 3, and 4 are ideal normal-conducting 1D leads. Nodes A, Bare described by real scattering matrices (see text). The directions of normal (Iq) and Josephson(IJ) currents are schematically shown. The magnetic field B is normal to the plane of the picture,and the distance between the “mirrors” equals L.10

Giant conductance oscillations in a normal mesoscopic ring induced by an SNS Josephson current

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.305.3144

Giant conductance oscillations in a normal mesoscopic ring induced by an SNS Josephson current

Abstract

Similar works

Full text

Available Versions

Loughborough University Institutional Repository