Fazio, Rosario

Schön, Gerd

Wagenblast, Karl-Heinz

Winkelholz, C.

English

KITopen

Tunneling into one-dimensional Josephson chains andLuttinger liquidsRosario Fazio a;b, Karl-Heinz Wagenblast aCarsten Winkelholz a, and Gerd Schon aa Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe, D-76128Karlsruhe, Germanyb Istituto di Fisica, Universita di Catania, viale A. Doria 6, 95129 Catania, ItalyAbstractOne-dimensional arrays of Josephson junctions have been studied because of theirunusual zero-temperature superconductor-insulator phase transitions. Furthermore,their low-energy Hamiltonian coincides with that of a Luttinger liquid. Hence 1-DJosephson junction arrays provide a physical realization of this model system. Inorder to explore these similarities we discuss 1-D arrays with defects, making contactto recent studies of transport in Luttinger liquids. We then study Andreev tunnelingbetween a Josephson chain and a normal metal and between a superconductor anda Luttinger liquid. In both cases the current-voltage characteristics strongly deviatefrom a linear dependence.1 IntroductionIn recent years Josephson junction arrays (JJA) built from small tunnel junc-tions have attracted a growing interest, both from the experimental and thetheoretical side [1,2]. The transport properties of these low-dimensional struc-tures are strongly inuenced by interaction eects. Most of the physics associ-ated with charging eects and Coulomb blockade is described by the so-calledcapacitance model. When the geometrical capacitance of the islands is of theorder of picofarads the associated Coulomb energy may exceed the Josephsonenergy. In this case quantum uctuations destroy the coherence and the arrayis insulating at zero temperature. The phase diagram of the superconductor-insulator transition depends the ratio of Josephson coupling and chargingenergy and on the dimensionality of the system. A one-dimensional Joseph-son chain undergoes a T = 0 transition of the Kosterlitz-Thouless-Berezinskii(KTB) type, as shown by Bradley and Doniach [3]. Dissipation, which mayPreprint submitted to Elsevier Science 8 January 1996be caused by Ohmic shunts, gives rise to additional structure in the phasediagram [4{7].The interest in 1-D systems goes further. Interacting 1-D systems are pro-totypes of non-Fermi liquids, and have been studied for several years theo-retically [8{10] as well as experimentally [11,12]. One-dimensional interactingsystems can be described in terms of the Luttinger liquid (LL) model [8]. Thelow energy excitations of the interacting electron gas in one dimension arelong-wavelength spin and charge oscillations, rather than fermionic quasipar-ticle excitations. Accordingly the transport properties cannot be described interms of the conventional Fermi-liquid approach. The density of states showsa power-law asymptotics at low energies. Depending on the sign of the interac-tion in a quantum wire an arbitrarily weak barrier leads to perfectly reecting(for repulsive interactions) or transmitting behavior at low voltages [9]. Someexperimental evidence for LL behavior has been found in the power-law energydependence of the single-particle density of states in the Bechgaard salts [11].The dispersion of spin and charge excitations in quantum wires has also beenmeasured [12].In this article we point out that a Josephson junction chain is a physicalrealization of a LL. We rst explore the similarities by discussing a Josephsonchain with a defect and show that the problem is similar to that discussed fora LL [9]. We then study Andreev reection between a superconductor and aquantum wire which are coupled via a tunnel barrier, and between a Josephsonchain and a normal island.2 The model for a Josephson chainThe model Hamiltonian for a Josephson junction chain contains the electro-static interaction and the Josephson couplingHchain =12XijQiC 1ij Qj   EJXicos(i   i+1) ; (1)where Qi is the charge and i the phase of the condensate wave function on theisland i. The charge and phase on equal sites do not commute, [j; Qi] = 2eiij .The interaction is described by the capacitance matrix Cij, which is built bythe nearest neighbor capacitance C and the ground capacitance C0. In generalthe electrostatic interaction has a nite range with a screening length (in unitsof the lattice spacing) of the order of  =qC=C0.2In this article we will restrict ourselves to the case of short range interactions,qC=C0  1, where a typical charging energy is dened by E0 = 4e2=C0. Thismodel at T = 0 can be mapped onto a classical two-dimensional XY model [3],which undergoes a phase transition of the KTB type at a critical value of thecoupling constants EJ=E0  4=2. For large values of the Josephson energythe system is superconducting, while in the opposite limit it is insulating.In the limit of large EJ and low temperatures the system exhibits phase co-herence. The low lying excitations can then be expressed in terms of the uc-tuations of the phase within the harmonic approximation. The action relatedto the Hamiltonian (1) expressed by modes with wavevector k and frequency!n isA = 12Xk;!nnC(k)h2!2n=4e2 + EJk2oj k;!n j2 : (2)Here !n = 2n=h are the Matsubara bosonic frequencies, and C(k) is theFourier transformed of the capacitance matrix. This representation will allowus to establish the connection to the physics of the LL as will become evidentin the following.3 Josephson chain with a defectIn this rst section we discuss a Josephson chain with a weak link in the middle.The system bears similarities to a LL with a barrier, which was discussed byKane and Fisher [9]. It is convenient to describe the system in the path integralformalism. The action is given byA =XiZ0d24E02 did!2  E(i)J cos(i+1   i)35 : (3)The notation is the same as in Eq. (1), except for the site-dependent Josephsoncoupling. A weak link in the chain at site 0 is described by E(0)J  EJ (whileE(i)J = EJ for i 6= 0). If EJ  E0 it is possible to use the harmonic approxi-mation for the junctions of the chain, except for the one at i = 0. Followingthe approach of Ref. [9] we derive in this way an eective action for the weakjunctionAe = 12gZd!2j!jj(!)j2 E0JZ0d cos() : (4)3The interaction parameter is given byg = sEJE0;where  is equal to one in the harmonic approximation for the chain. Its exactvalue can be determined by Monte Carlo simulations (see below). The eectiveaction (4) is identical to the one obtained for a dissipative particle in a periodicpotential [18] and also for a spinless LL with a barrier [9].The harmonic approximation is valid when g > 1, therefore the present modelmaps onto the attractive case described in Ref. [9]. In this case, in the renor-malization group ow, E0J ows to larger values. This means that at largedistances the eect of the weak link decays, and the properties of our systemreduce to those of a chain with all Josephson coupling equal to EJ .We have carried out Monte Carlo simulations of a Josephson chain with onedefect using the techniques of Refs. [19]. This enables us to study the validity ofthe approximations and the renormalization of the weak link in the Josephsonchain. The result for the superuid density, which is a direct measure of thestrength of the weak link, is shown in Fig. 1. The data show the renormaliza-tion of the weak link according to the scaling of Ref. [9], i.e. 0(L)  L(1 1=g).The scaling is cut o by the nite superuid density of the bulk. We obtainthe best t for  = 1:56, independent of the strength of the defect and thebulk coupling. This renormalization of  compared to the value  = 1 inthe harmonic approximation is due to phase slips, which also drive the super-conductor to insulator transition. The value of  implies that g = 1 roughlycoincides with the critical ratio of EJ=E0 where the KTB transition of theuniform chain has been found. It still remains an open question whether thetwo values actually coincide rigorously.The physics of the LL with attractive interaction g > 1 can be studied inJosephson chains with EJ=E0 > 1, which make them an interesting, experi-mentally accessible system for studying models with non-Fermi liquid behav-ior. On the other hand, the LL model with g < 1 cannot be realized in aJosephson chain. The reason is that in the EJ=E0 < 1 (which would cor-respond to g < 1) the chain is insulating and the low-lying excitations areno longer spin-waves with an acoustic branch. As a consequence the eectiveaction (4) does not contain an Ohmic kernel as required to describe LLs.4 Andreev current into a Josephson chainIn this section we study Andreev tunneling between a normal metal and aJosephson chain [13]. Quantum uctuations of the phases in the JJA produce4a nonlinear Andreev current, which is very sensitive to the critical propertiesof the JJA. The Hamiltonian of the system under consideration isH =XZNd3r  y(~r)( ir) (~r) + Thei0 "(0) #(0) + h:c:i+Hchain :(5)The tunneling Hamiltonian is chosen to describe Andreev processes betweenthe normal metal and the rst superconducting island i = 0. The creationoperators  y refer to electrons of the normal metal particle and ( ir) = r22m   is the kinetic energy, measured relative to the Fermi energy.When a voltage V is applied, the Andreev current across the NS boundary iscalculated from the time variation of the electron charge on the metal islandINS = 4e j T j21Z 1dt e 2ieV t i(t) h[Ay(t); A(0)]i ; (6)where A(t) = ei0(t) "(0; t) #(0; t). This expression demonstrates that keyfeatures are contained in the phase{phase correlation function of the super-conducting island which is in contact with the normal metal,g( ) = hT ei0()e i0(0)i :In the limit EJ  E0 the harmonic approximation (2) can be used, whichamounts to neglecting quantum phase slips. The system possesses quasi-longrange order, i.e. for  !1 the correlations decay with a power-law g( )  j!p j 1 , with !p =pEJE0 and  = 1+ (2) 1qE0=EJ . As a result the I{Vcharacteristic at low voltage has a power-law behaviorIA = GAe ( 1) (1 + )!p2e 2e j V j!p!; eV  !p ; (7)where  is the Euler number. In the classical limit,  ! 1 , the ground stateof the chain has the phases aligned and no collective modes are excited. Thenthe chain behaves as a bulk superconductor with a linear Andreev I{V char-acteristic.When E0=EJ increases phase-wave collective modes become active. In partic-ular they are excited by the Andreev tunneling events themselves, and theirback-action leads to the anomalous nonlinear characteristics of Eq. (7). Theexcitations of the chain act like bosonic modes of an electromagnetic envi-ronment coupled to the tunneling pair. Indeed the result (7) resembles that5obtained for single electron tunneling in systems coupled to an electromagneticenvironment [14].In the limitEJ  E0 the one-dimensional JJA acts like a line of linear elementswith the equivalent impedanceZ(!)h=2e2=   1q1  (!=!p)2(!p   !) (8)(see Ref. [13] for more details). Notice that the low voltage behavior is dom-inated by the capacitance to the ground C0, and a small nite C does notchange the behavior of Eq. (7).When quantum phase slips are included the harmonic approximation (2) re-mains a xed point Hamiltonian, but the parameters are renormalized. Ac-cordingly in Eq. (7)  is replaced by a renormalized value, which deviatesfrom the phase-wave value near the superconductor-insulator transition. Forfurther details see Ref. [15].5 Andreev tunneling into a Luttinger liquidIn this section we consider a LL coupled to a superconductor by a tunnelbarrier and study the current through the interface [16,17]. The Hamiltonianin this case is given byH = HS +HT +HL : (9)The superconductor is described by the standard BCS-Hamiltonian HS . Thetunneling part isHT =Xs=1ZdxZdr T (x; r)	yL;s(x)	S;s(r) + h:c: : (10)Finally, HL describes the LL. It can be expressed in terms of boson operatorsHL =ZdxXjvj"gj2(rj)2 + 2gj(rj)2#: (11)It is written as a sum of the contributions from spin (j = ) and charge (j = )degrees of freedom. Here vj = (2=gj)vF are the renormalized, interaction-dependent Fermi velocities. For repulsive interaction we have g < 2, while for6attractive interaction g > 2. The elds  and  are canonically conjugated,[s(x);rs0(x0)] = (i=2)(x0  x)s;s0.The similarity to the Josephson chain becomes manifest if we write the corre-sponding action for a LLAL = 12Xk;!nXjngj!2n=vj + gjvjk2oj k;!n;j j2 : (12)Comparing Eq. (12) and Eq. (2) it is clear that the Josephson chain corre-sponds to a spinless LL. This oers the possibility to study the interactioneects of a LL by studying a one-dimensional Josephson array.In the following we will consider spin-independent interactions. Hence, g = 2and v = vF . For an innitely extended quantumwire the Fermi eld operatorscan be expressed by boson operators as	ys =p0Xm=1eikFmxeims(x)eis(x) (13)where s =1p2( + s) and s =1p2( + s). The density of electrons perspin in the LL is 0 = kF =2. For transparency we will assume in the followingthat the gap of the superconductor is much larger than all other characteristicenergies of the system.We rst evaluate the contribution of Andreev tunneling to the nonlinear sub-gap current. The current is evaluated in standard way in an expansion in thetunneling HamiltonianI=e = h _NL(t)i = hTeiR t 1HTI (t0)dt0 _NLI(t) T e iR t 1HTI (t0)dt0i : (14)The terms contributing to the current can be classied diagrammatically. Theimportant element iss;s0(xi; t) = h	L;s(x3; t3)	L; s(x2; t2)	yL;s0(x1; t1)	yL; s0(x; t)i : (15)As long as all energies (eV , kT ) are small compared to the gap of the supercon-ductor the anomalous correlations in the superconductor can be approximatedby delta-functions. The expression for the current then readsI = 16N(0)22jT j4e Re 1Z0dt+; (t3 = t2 = 0; t1 = t)(e2ieV t   e 2ieV t):(16)7The averages can be evaluated by using Eq. (11). The resulting current atT = 0 isI(V; T = 0) =14G2TRKV2eVEF 2g+g2 2 1 (g2+ 2g) 2g! 2g+g2+14G2TRKV2eVEF 2g 1 1 (1 + 2g) 2g! 2g; (17)where GT is the dimensionless tunneling conductance. For repulsive electron-electron interaction the term I  V 2=g dominates while for attractive electron-electron interaction I  V (2=g)+(g=2) 1 is the relevant one. These power lawsconrm the qualitative expectation, a suppression or increase of the currentby repulsive or attractive interactions, respectively. Accordingly the linearconductance G = @I=@V  V  1 (where  is the leading power in Eq. (17))at V = 0 and T = 0 vanishes for repulsive interaction ( > 1), or becomesinnite for attractive interaction ( < 1).At nite temperatures we haveG =14G2TRK24c1 kTEF!g2+ 2g 2+ c2 kTEF! 1g 135 ; (18)where c1; c2 are constants of order one.The divergence of the conductance in the attractive case reveals the failureof the perturbation expansion in the tunneling Hamiltonian. By means of arenormalization group (RG) approach it is possible to identify the Cooper pairchannel (tunneling of two electrons with opposite spin on dierent branches)as the relevant process for attractive interactions (in the RG sense). Moregenerally, if g 6= 2, the xed point is shifted from g = 2, and in principleone may have a fully transparent barrier for Andreev tunneling even in thepresence of repulsive interaction in the LL.6 DiscussionIn this paper we discussed the possibility of using one-dimensional Josephsonjunction arrays to detect transport properties which are currently investigatedin Luttinger liquids. A direct evidence of this connection was discussed in therst part of the paper when the Josephson chain with a weak link was consid-ered. In the second part of the paper we investigated Andreev-type transport8and showed that the I{V characteristics for the Josephson chain and the LLcoincide. A key point in the discussion is the observation that in the supercon-ducting chain the low-lying excitations are spin-waves, which are the equiva-lent of the boson modes in LLs. The analogy between the two models becomesapparent if we note that the the creation operator for an extra charge in aisland is given by  exp(i), which is very similar to the representation (13) ofFermion creation operators. The chain does not possesses any internal degreeof freedom and accordingly is mapped onto the spinless LL. In principle onemay think to use coupled chain to get also model for spin-1=2 electron in onedimension.AcknowledgementThe work on the Andreev tunneling in a Josephson chain was done in collab-oration with Giuseppe Falci and Arturo Tagliacozzo. This work is supportedby the DFG within the research program of SFB 195.References[1] see e.g. Proceedings of the NATO ARW Coherence in SuperconductingNetworks, Eds. J.E.Mooij and G. Schon, Physica B152, (1988); J.E. Mooijand G. Schon in Single Charge Tunneling, Eds. H. Grabert and M.H. Devoret,NATO ASI series Vol. 294 (Plenum Press, New York 1992); MesoscopicSuperconductivity, Eds. F.W.J. Hekking, G. Schon, and D.V. Averin, (Elsevier1994) and Physica B 203 (1994).[2] L.J. Geerligs, M. Peters, L.E.M. de Groot, A. Verbruggen and J.E. Mooij, Phys.Rev. Lett 63, 326 (1989); H.S.J. van der Zant, L.J. Geerligs and J.E. Mooij,Europhys. Lett. 19, 541 (1992); S. Doniach, Phys. Rev. B 24, 5063 (1981)R. Fazio and G. Schon, Phys. Rev. B 43, 5307 (1991).[3] R.M. Bradley and S. Doniach, Phys. Rev. B 30, 1138 (1984).[4] S.E. Korshunov, Europhys. Lett. 9, 107 (1989).[5] W. Zwerger, Europhys. Lett. 9, 421 (1989).[6] P.A. Bobbert, R. Fazio, G. Schon and G.T. Zimanyi, Phys. Rev. B 41, 4009(1990).[7] P.A. Bobbert, R. Fazio, G. Schon and A.D. Zaikin, Phys. Rev. B 45, 2294(1992).[8] V.J. Emery, in Highly Conducting One-Dimensional Solids, edited byJ.T. Devreese, R.P. Evrard, and V.E. van Doren (Plenum Press, New York1979); J. Solyom, Adv. Phys. 28, 201 (1979).9[9] C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 (1992); K.A. Matveevand L.I. Glazman, Phys. Rev. Lett. 70, 990 (1993).[10] R. Fazio, F.W.J. Hekking, and A.A. Odintsov, Phys. Rev. Lett. 74, 1843 (1995).[11] B. Dardel, D. Malterre, M. Grioni, Y. Baer, J. Voit, and D. Jerôme, Europhys.Lett. 24, 687 (1993).[12] A.R. Go~ni, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, L.N. Pfeier,and K.W. West, Phys. Rev. Lett. 67, 3298 (1991).[13] G. Falci, R. Fazio, A. Tagliacozzo, and G. Giaquinta, Europhys. Lett. 30, 169(1995).[14] G.L. Ingold and Yu.V. Nazarov, in Single Charge Tunneling, Eds. H. Grabertand M.H. Devoret, (Plenum Press, New York 1992) p. 221.[15] J.V. Jose, L.P. Kadano, S. Kirkpatrick and D.R. Nelson, Phys. Rev. B16,1217 (1977).[16] M.P.A. Fisher, Phys. Rev. B 49, 14550 (1994).[17] C. Winkelholz, R. Fazio, F.W.J. Hekking, and G. Schon, (unpublished).[18] A. Schmid, Phys. Rev. Lett. 51, 1506 (1983).[19] E.S. Srensen, M. Wallin, S.M. Girvin, and A.P. Young, Phys. Rev. Lett. 69,828 (1992); A. van Otterlo and K.H. Wagenblast, Phys. Rev. Lett. 72, 3598(1994); A. van Otterlo and K.H. Wagenblast, R. Baltin, C. Bruder, R. Fazio,and G. Schon, to appear in Phys. Rev. B 52 (1995).100.1 1 1010superfluid stiffness  ρ0chain length  LFig.1.Thesuperuiddensityasafunctionofthesystemsizeforvariousvaluesofthecouplingstrength.ThethreedatasetsatthetopcorrespondtoadefectwithE(w)J=E0=0:2andEJ=E0=4;8,and20.ThetwodatasetsatthebottomhaveE(w)J=E0=0:1andEJ=E0=1and2.Thelinesarepowerlawtstothedata.11

Tunneling into one-dimensional Josephson chains and Luttinger liquids

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Tunneling into one-dimensional Josephson chains and Luttinger liquids

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