Coherent multiple Andreev reflections and current resonances in SNS junctions ˚A. Ingerman∗, G. Johansson, V. S. Shumeiko and G. Wendin Department of Microelectronics and Nanoscience, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden (February 1, 2008) 1 0 We study coherent multiple Andreev reflections in quantum SNS junctions of finite length and 0 arbitrary transparency. The presence of superconducting bound states in these junctions gives rise 2 to great enhancement of the subgap current. The effect is most pronounced in low-transparency n junctions, D ≪ 1, and in the interval of applied voltage ∆/2 < eV < 2∆, where the amplitude a of the current structures is proportional to the first power of the junction transparency D. The J resonant current structures consist of steps and oscillations of the two-particle current and also of 8 multiparticleresonancepeaks. Thepositionsofthetwo-particlecurrentstructureshavepronounced 1 temperaturedependencewhichscaleswith∆(T),whilethepositionsofthemultiparticleresonances have weak temperature dependence, being mostly determined by the junction geometry. Despite ] n thelargeresonanttwo-particlecurrent,theexcesscurrentatlargevoltageissmallandproportional o to D2. c Pacs: 74.50.+r, 74.80.Fp, 74.20.Fg, 73.23.Ad - r p u I. INTRODUCTION s . t a Transport properties of small conducting structures are strongly influenced by size effects. Oscillation of magne- m toresistanceinthinmetallicfilms,andquantizationofconductanceinnarrowwiresandpointcontactsareexamplesof - such effects. Size effects in superconducting tunneling have attractedattention since early experiments by Tomasch1. d In these experiments, oscillations of the tunnel conductance as a function of applied voltage were found for tunneling n from a superconductor to a thin superconducting film of an NS proximity bilayer. The geometric resonance nature o of the effect was clearly indicated by the dependence of the period of oscillations on the thickness of the supercon- c [ ducting film. Similar conductance oscillations for tunneling into a normal metal film of NS bilayers were reported by Rowell and McMillan2. Later on an even more pronounced effect – steps on the current-voltage characteristics 1 of SINS junctions at applied subgap voltages, eV < 2∆ – was observed by Rowell3 (for a review see Ref. 4). In v addition to the dependence on the thickness of the N-film, the period of the current steps also shows temperature 3 dependencewhichscaleswiththetemperaturedependenceofthesuperconductinggap∆(T). Thecurrentstepsoccur 8 2 at applied subgap voltages, eV < 2∆, and they are understood as resonant features due to quasiparticle tunneling 1 throughsuperconductingboundstatesexistinginINSwellsatenergieslyingwithinthesuperconductinggap, E <∆, | | 0 de Gennes—Saint-James levels5. 1 Recently, propertiesofsuperconductingbound stateshaveattractednew attentionin connectionwith observations 0 ofconductanceanomaliesinmesoscopicNSstructures. Resonantoscillationsofthesubgapconductanceinmesoscopic / t quasi-ballisticNSjunctionshavebeenreportedbyMorpurgoetal6. Thezerobiasconductancepeak,arelatedresonant a m phenomenon in diffusive NS junctions, was discovered even earlier7,8 and then intensively studied both theoretically and experimentally (for a review see Ref. 9). - d In these recent studies of mesoscopic junctions, attention was switched from the single-particle current through n superconductingboundstatestothetwo-particle(Andreev)current. Thetraditionalviewofsubgapcurrenttransport o in proximity NINS and SINS structures considers single-particle tunneling into bound states in the normal region of c the INS well4,10,11, implicitly assuming that the normal regionof the INS well plays the role of equilibrium reservoir. : v Such a model is appropriate for low-transmission tunnel junctions with low tunneling rate compared to the inelastic i relaxationrate. However,transparentmesoscopicstructuresareinadifferenttransportregimewheretheboundlevels X arewelldecoupledfromthesuperconductingreservoirs,andwhereinjectedquasiparticlesescapefromtheINSwellvia r Andreev reflection.12,13 Resonant two-particle current in quantum NINS junctions has been theoretically studied in a Refs.14,15. InSNSjunctions,thequasiparticlesmayundergomultipleAndreevreflections(MAR)beforetheyescape into the reservoirs.16 In a number of recent experiments with ballistic SNS devices fabricated with high mobility 2D electron gas (2DEG)17–20 the highly coherent MAR transport regime has been realized. ∗Email: ingerman@fy.chalmers.se 1 The purpose of this paper is to develop a theory of coherent multiple Andreev reflections which will be applicable to2DEG-basedSNSjunctions. In2DEGdevicestheseparationofthesuperconductingelectrodesListypicallylarger than 200 nm, which is of the same order of magnitude as the superconducting coherence length, ξ = h¯v /∆ (v 0 F F is the Fermi velocity of the 2D electrons), and superconducting bound states are formed well inside the energy gap. The presence of bound states in the junctions of finite length gives rise to resonances in the MAR transport, which dramatically affects the subgap current. Furthermore, it is possible in 2DEG devices to use electrostatic gates to reach the quantum transport regime with a small number of electron modes and variable transmissivity. A theory of coherent MAR has earlier been developed for short superconducting junctions,21–25 L ξ , where superconducting 0 ≪ bound states do not play any significant role.26 Such a theory is consistent with the physical situation in atomic-size superconducting point contacts,27–30 where quantization of conduction modes has turned out to be very helpful for detailedcomparisonbetweentheoryandexperiment. Thepurposeofourstudyistoinvestigatetheinterplaybetween superconductingboundstate resonancesandcoherentMARinlongSNSjunctions,L>ξ , inthe quantumtransport 0 regime. ∼ In a number of publications, the coherent MAR approach has been applied to long SNS junctions31,32. However, these studies were restricted to fully transparent junctions where the bound states are strongly washed out and the resonances are not pronounced (in fact, as we will show, at zero temperature the current in such junctions does not show any structures). We will study junctions with arbitrary transmissivity, 0∆ =a(E)= − − | | . (6) u E i√∆2 E2 /∆, E <∆ (cid:26)(cid:0) − − (cid:1) | | Since the time dependencies of the wave fun(cid:0)ctions in the two(cid:1)reservoirs are different, the quasiparticle scattering by the junction is inelastic, and one has to consider a superposition of plane waves with different energies in order to construct scattering states. III. CALCULATION OF CURRENT USING SCATTERING STATES A. Recursion relations for MAR amplitudes We will now proceed with the construction of recurrences for the scattering amplitudes following the method suggested by Johanssonet al.39. To this end we introduce the wave functions in the left/right normal region (l/r) of the junction with respect to the position of the impurity. A particular scattering state, labeled with the energy E of the incoming quasiparticle,will consist of a superposition of plane waves with energies E =E+neV, where n is an n integer, 0 cˆ =[Mˆ ]−1cˆ , n<0 (13) n− n0 0+ n+ 0n 0− Mˆ =Tˆ Uˆ ...Uˆ Tˆ , n>m, (14) nm n−1 n−1 m+1 m whereTˆ =Tˆ−1 andTˆ =Tˆ fortheinjectionfromtheleft(fortheinjectionfromtheright,the evenandoddside 2k 2k+1 band indices are interchanged). For paths within the superconducting gap, E ,E < ∆, the matrix Mˆ satisfies n m n0 | | the standard transfer matrix equation, Mˆ σ Mˆ† = σ , which provides conservation of probability current along nm z nm z the energy axis, jp =cˆ† σ cˆ , jp =jp , E <∆. (15) n± n± z n± n± m± | n,m| 5 An important consequence of the coherence of MAR is the possibility of transmission resonances in energy space. From the form of the Mˆ-matrix, Mˆ =...Tˆ−1eiσzϕkTˆ..., (16) n0 itisevidentthatwhenϕ =mπ,thetwomatricesTˆ−1andTˆwillcanceleachother,andtheprobabilityoftransmission k through this part will be unity, which leads to resonant enhancement of MAR. The solutions E(m) of the resonance equation, 2E L E (m) k l,r k ϕ =ϕ mπ = arccos mπ =0, (17) k k− ∆ξ − ∆ − 0 coincide with the spectrum of the de Gennes–Saint-Jameslevels localizedin INS quantum wells5. The corresponding bound states are located either on the left or the right side of the junction. Without loss of generality, the calculations can be performed with a real matrix Tˆ. The transformation to such a real matrix is given by Tˆ Vˆ TˆVˆ , with diagonal unitary matrices Vˆ whose elements are constructed with the 1 2 1,2 → scatteringphases,whichareenergy-independent. ItisclearfromEq.(16)thatsincetheseenergy-independentmatrices commutewiththe matricesUˆ ,theycanceleachother,andthematrixMˆ undergoesasimilartransformation. This n n0 will lead to an overallphase shift of the scattering state which does not affect the current. It is interesting to consider the special case of fully transparent junctions, D = 1, which has been studied in the literature31,32. In this case, all matrices Tˆ in Eq. (14) are equal to the unity matrix, and the Mˆ-matrix takes the n simple form Mˆ = exp(iσ n−1 ϕ ). The length of the junction then enters only through the phase of the MAR n0 z m=1 m amplitudes, which drops out of the side band current. Thus the dc current of fully transparent SNS junctions is independent of length and eqPual to the current in quantum constrictions23,40. In particular, at zero temperature this current does not show any structures in the subgap current. It is also worth mentioning that in this particular case of fully transparent SNS junctions, the Mˆ-matrix is diagonal and therefore a closed set of recursive relations can be derived for the MAR probabilities (not just for the MAR amplitudes, as in the general case), equivalent to the equations for distribution functions derived in the original paper by Klapwijk, Blonder and Tinkham16. Equation(13)describes “source-free”propagationalong the MAR ladder. To complete the set ofequations for the MAR amplitudes we need to take into account quasi-particle injection, which introduces a source term in Eq. (13). To this end, let us consider a quasiparticle incoming from the left superconducting electrode with energy E, having a wave function of the form ΨS(E)=e−iEt/h¯−iσzeVt/2h¯ δ eik˜ex u +δ e−ik˜hx v . (18) l νe v νh u (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) Thetwotermsinthisequationrefertoelectron-(ν =e)andhole-like(ν =h)injectedquasiparticles. Wenowinclude this wavefunction into the continuity conditionatthe NS interfaceatenergy E, whichgivesus the followingrelation between the MAR amplitudes cˆ and cˆ , 0+ 0− cˆ =Uˆ cˆ +Yˆ (19) 0+ 0 0− δ /u Yˆ(E) = (u2 v2) νe . − δνhe−2iELl/∆ξ0/v (cid:18) − (cid:19) For quasiparticles injected from the right, a similar equation holds with the substitutions e h, and L L . l r → → Equations (13) and (19) give a complete set of equations for the MAR amplitudes with the boundary conditions cˆ =0 at infinity. ±∞ B. Calculation of MAR amplitudes AformalsolutionofEqs.(13)and(19),whichisusefulbothfornumericalcalculationsandanalyticalinvestigations, can be constructed by reducing this infinite set of recursionrelations to a finite set by representing the MAR process aboveE andbelowE byboundaryconditionsinvolvingreflectionamplitudesr andr ,definedasc↓ =c↑ r n 0 n+ 0− n+ n+ n+ and c↑ =c↓ r . This gives the following representation for the vectors in Eq. (9), 0− 0− 0− 1 r cˆ =c↑ , cˆ =c↓ 0− . (20) n+ n+ r 0− 0− 1 n+ (cid:18) (cid:19) (cid:18) (cid:19) 6 The reflection amplitudes r and r are independent of the injection, in contrast to the coefficients c↑ , c↓ . n+ 0− n+ 0− Furthermore, they are determined by the boundary conditions cˆ =0 and can be expressed in terms of the matrix ±∞ elements of Mˆ and Mˆ , where N , Nn 0(−N) →∞ 1 r lim Mˆ =0, lim [Mˆ ]−1 0− =0. (21) N→∞ Nn rn+ N→∞ 0(−N) 1 (cid:18) (cid:19) (cid:18) (cid:19) In other words, the vectors in Eq. (21) are equal to the asymptotical values of the eigenvectors of Mˆ-matrices correspondingto the eigenvalues whichdecreasewhen N goesto infinity. The advantageof introducing the reflection amplitudes r and r is that although they have to be calculated numerically, the recurrences which they obey n+ 0− do not contain resonances and converge rather quickly. This is in contrast to the matrix Mˆ , which does possess n0 resonances, but which can be calculated analytically in a straight forwardway for any given n. The solutions of the recursion equations Eqs. (10) and (12) can now be explicitly written down. For any given energy E we get four different sets of solutions for four scattering states including electron/hole injection from the m m left and the right. Using the formal expression in Eq. (20) and the matrix elements of M = 11 12 , the n0 m m 21 22 (cid:18) (cid:19) solutions for injection from the left (n>0) have the form, cˆ = u(1−a20)eiϕn[δνe+a0r0−δνh] 1 . (22) n+ m22+m21r0−e2iϕ0 −m12rn+e2iϕn −m11r0−rn+e2iϕ0+2iϕn (cid:18)rn+ (cid:19) The solutions for injection from the right can be found through interchanging e h and calculating all quantities ↔ with respect to injection from the right. The solutions for n<0 are calculated in a similar manner. C. Calculation of current Now turning our attention to the current, we calculate it in the normal region next to the barrier, using the wave function in this region, Ψ, and assuming quasiparticle equilibrium within the electrodes. The current then takes the form, e −∆ ∂ E I(t)= dE (u2 v2)−1 Im Ψ† Ψ tanh | | , (23) hk − ∂x 2k T F Z−∞ e/Xh,l/r (cid:26) (cid:27) B where (u2 v2)−1 = E /√E2 ∆2 = E /ξ is the superconducting density of states, and the sum is over the four − | | − | | scatteringstatesata givenenergyE associatedwiththe electron-andhole-likequasiparticles(e/h)injectedfromthe left and right (l/r). The current can be divided into parts with different time dependence and expressed as a sum over harmonics: I(t)= I e2iNeVt/h¯ (24) N N X Focusing onthe dc (N =0)component, andcalculating the contributionof eachscattering state at the injection side of the junction, we express the current spectral density J(E) through the probability currents of electrons and holes at energies E (Fig. 3), 2n e −∆ I = dE J(E), dc h Z−∞ ∞ E J(E)= | | cˆ† σ cˆ +cˆ† σ cˆ . (25) ξ 2n− z 2n− 2n+ z 2n+ e/Xh,l/r n=X−∞(cid:16) (cid:17) These currents coinside with the probability currents jp , Eq.(15), flowing along the energy axis. n± It is convenient to introduce a leakage current J , defined as the difference of the probability currents before and n after Andreev reflection, E J = | | jp jp . (26) n ξ n−− n+ e/Xh,l/r (cid:0) (cid:1) 7 J represents the amount of probability current from all the scattering states injected at energy E and leaking out n of the junction at energy E (Fig. 3). The leakage current is zero inside the energy gap due to complete Andreev n reflection, J =0, E <∆ [cf. Eq. (15]. n n | | The explicit expressionfor the leakage currentfor n=0 followsfrom Eq.(26)after insertionof Eqs.(22)and(10), 6 (1 a 2)(1 a 2)(1+ r a 2)(1+ r a 2) 0 n 0− 0 n+ n J = −| | −| | | | | | . (27) n Xl/r |m22+e2iϕ0r0−m21−e2iϕnrn+m12−e2iϕ0e2iϕnr0−rn+m11|2 It follows from Eq. (27) that the leakage currents are positive for all n = 0, J 0. One can also show that they n 6 ≥ satisfy the inequality J 4, which is a consequence of the conservation of probability current: the leakage n6=0 n ≤ current of all side bands except of the side band n = 0 does not exceed the probability current injected into four scattering states. FurtPhermore, the leakage current satisfies the important detailed balance equation39, J (E)=J (E ) (28) −n n −n i.e. theleakageatenergyE duetotheinjectionatenergyE isthesameastheleakageatenergyE duetoinjection −n at energy E . −n Using the continuity of current across the barrier, jp = jp , guaranteed by the transfer matrix Tˆ, we can n+ (n+1)− express the probability currents in Eq. (25) through the leakage current, Eq. (26), ∞ E | |jp = J , n>0 (29) ξ n− k e/Xh,l/r kX=n ∞ E | |jp = J , n < 0, ξ n+ − −k e/Xh,l/r kX=−n by adding and subtracting consecutive terms in the sum. The spectral density of the dc charge current Eq. (25) can then be written on the form J(E)= nJ (E), (30) n n X since J appearsin n probabilitycurrents. This formulahas a clear physicalmeaning: the contributionto the charge n current of the n-th side band is proportional to the leakage current of the side band times the effective transferred charge ne. The detailed balance of the leakage currents, Eq. (28), allows us explicitly to prove that at zero temperature the scattering processes between (occupied) states with negative energies, E,E ∆ do not contribute to the current, n ≤− in agreement with the Pauli exclusion principle. Indeed, by separating the contributions from side bands with n<0 and remembering that the leakage current is zero within the gap, we get for zero temperature, e −∆ ne −∆−eV −∆ −∆ I = dE nJ (E)= dEJ (E)+ dEJ (E) dEJ (E) , (31) dc n n n −n hZ−∞ n6=0 n>0 h Z−∞ Z∆−eV −Z−∞ ! X X where the first and the third terms cancel each other by virtue of Eq. (28). At finite temperature, these two terms produce current of thermal excitations while the second term gives the current of real excitations created by the voltage source. Keeping only this term which dominates at low temperature, we finally get, ne −∆ I = I , I = θ(neV 2∆) dEJ (E)tanh(E /2k T). (32) dc n n n B h − | | n>0 Z∆−neV X We end this section by noting a technically useful symmetry in the current density, namely J (E)=J ( E neV), n n − − seen from the explicit form of the Mˆ -matrix. This allows us to reduce the integration interval in Eq. (32) to n0 neV/2∆, seen as oscillations in the DOS. | | 3 2.5 2 E) L=0 (1.5 l,rN 1 L=ξ L=πξ 0 0.5 0 0 1 1.5 2 2.5 3 E/∆ FIG. 5. Density of states in the N region for different lengths L of the region. Singular behavior of the DOS for short junctions, L = 0 (equal to the DOS in a superconductor), is suppressed for finite length junctions. The amplitude of the first oscillation increases as the length increases, indicating accumulation of the spectral weight at the energy gap edge and formation of a boundstate for L=πξ . 0 The single particle current in Eq. (34) is written as the integral over the product of the DOS at the entrance energy E and the exit energy E+eV. The latter depends on the applied voltage, as well as the integration interval, and therefore the DOS oscillations produce oscillations of the current I as a function of voltage (Rowell-McMillan 1 oscillations2,3). The oscillations become more pronounced when the junction is sufficiently long, and the differential conductance may even become negative. It is also clear that the DOS oscillates as a function of the length of the junction, which give rise to oscillations also in I . 1 In short junctions, L ξ , the current onset at eV =2∆ is very steep, see Fig. 4. In junctions with finite length, 0 ≪ the current onset is smeared and replaced with a smooth oscillating behavior. This can be directly related to the smearing of the singularityin the DOS at the gap edge. The length where the crossoverbetween these two behaviors occurs can be taken as a measure of when finite length effects become important. To estimate this length, we write equation (27) for small lengths L ξ , near the threshold, eV =2∆+Ω, Ω ∆, keeping the first order terms in D 0 ≪ ≪ in the denominator. For a symmetric junction, L =L =L/2, we get l r e∆tanh(∆/k T) π Dsin2z dz B I = . (36) 1 h Z0 sinz+ D∆ 2+ L2∆ 1+ D∆ 4Ω ξ2Ω 4Ω (cid:18) (cid:19) 0 (cid:18) (cid:19) From this formula it is clear that for short junctions (L = 0), the current onset has the width Ω ∆D/4. If L is ∼ of the order of ξ √D/2, the size of the onset has substantially diminished, and there is no visible onset at eV =2∆ 0 when L ξ √D/2. This crossoverbetween steep onset and smooth behavior, which happens already for quite short 0 ≫ lengths if D is small, can be interpreted in terms of a bound state which is situated exactly at the gap edge in short junctions (L = 0), and which moves down into the gap when L > 0, the effect becoming fully pronounced when the distance from the gap edge, h¯v /L, exceeds the dispersion of the Andreev state, √D∆, in symmetric junctions42. F When L approaches πξ /2, the lowest quasi-bound state in the continuum spectrum approaches the gap edge. l,r 0 This leads to anaccumulationofthe spectralweightatthe gapedgeandreappearanceofthe singularityinthe DOS, which results in the reappearance of a sharp current onset at eV = 2∆, but with smaller magnitude; see Fig. 4 (L =πξ ). l,r 0 It is of interest to note that in our calculations, based on the scattering theory approach, the bound states are not directly involvedin the single-particle transport, which therefore is non-resonantand shows no subgap resonance peaks. Withinthetunnelmodelapproachthesituationisqualitativelydifferent: theDOSinEq.(34)usuallyincludes 10

Coherent multiple Andreev reflections and current resonances in SNS junctions

Similar Coherent multiple Andreev reflections and current resonances in SNS junctions