PHYSICAL REVIEW B VOLUME 53, NUMBER 10 1 MARCH 1996-II
dc and ac Josephson effect in a superconductor–Luttinger-liquid–superconductor system
Rosario Fazio Institut fu¨r Theoretische Festko¨rperphysik, Universtita¨t Karlsruhe, 76128 Karlsruhe, Germany and Istituto di Fisica, Universita` di Catania, viale A. Doria 6, 95129 Catania, Italy*
F. W. J. Hekking Theoretical Physics Institute, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455
A. A. Odintsov Department of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands and Nuclear Physics Institute, Moscow State University, Moscow 119899 GSP, Russia* ͑Received 21 September 1995͒ We calculate both the dc and the ac Josephson current through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the ͑repulsive͒ Coulomb interaction in the framework of the one-channel, spin-1/2 Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At zero temperature, the critical current is found to decay algebraically with increasing distance d between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures T, lower than the
superconducting transition temperature Tc , there is a crossover from algebraic to exponential decay of the critical current as a function of d, at a distance of the order of បvF /kBT. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circum- ference L, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a junction. These features are robust against thermal fluctuations up to
temperatures on the order of បvF /kBL. For the wire geometry, we have also studied the ac-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron inter- actions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the ac- Josephson effect can be used as a tool for the observation of spin-charge separation.
I. INTRODUCTION connected to two superconductors should be quantized:6 each propagating mode contributes an amount e⌬/ប to the critical Due to the recent development of superconductor- current. semiconductor ͑S–Sc͒ integration technology it has become In the presence of disorder in the normal region, the possible to observe the transport of Cooper pairs through motion of the two electrons will be diffusive. Like in S–Sc mesoscopic interfaces. Examples are the supercurrent disordered metals, the phase coherence between the two through a two-dimensional electron gas ͑2DEG͒ with Nb electrons is limited by the correlation length 1 contacts ͑S–Sc–S junction͒ and excess low-voltage conduc- LcorϭͱបD/max ͕kBT,eV͖, where D is the diffusion con- tance due to Andreev scattering in Nb-InGaAs ͑S–Sc͒ stant. For instance, the excess low-voltage conductance in junctions.2 The transfer of single electrons through the inter- S–Sc junctions2 can be explained in terms of constructive face between a semiconductor and a superconductor with interference occurring over this length scale between the two energy gap ⌬ is exponentially suppressed at low tempera- electrons incident on the S–Sc interface.7,8 tures and bias voltages kBT,eVӶ⌬ (e is the electron In these examples, electron-electron interactions are ne- charge͒. Instead, electrons will be transferred in pairs glected. It is well known, though, that they may have a through the interface, a phenomenon known as Andreev strong influence on the transport properties of mesoscopic reflection.3 It has been realized only recently, that the phase systems. In general the interactions modify the phase- coherence between the two electrons involved in this process coherence length Lcor , which poses limitations on the above- could give rise to distinct signatures in the transport proper- mentioned mesoscopic effects. In specific cases the effects of ties of mesoscopic S–Sc–S and S–Sc systems.4,5 electron-electron interactions will strongly depend on the If the normal ͑Sc͒ region is free of disorder, the propaga- layout of the system under consideration. tion of electrons is ballistic. Phase coherence between the For example, the interactions will modify the critical cur- two electrons is maintained over the length rent Ic through a normal metallic slab sandwiched between 9 LcorϭបvF /max ͕kBT,eV͖, where vF is the Fermi velocity. two superconductors. If the coupling between normal metal In this regime, the critical current Ic through a short and and superconductors is weak ͑tunneling regime͒ and the size narrow constriction in a high-mobility noninteracting 2DEG, of the slab ͑and hence its electric capacitance C) is small, a
0163-1829/96/53͑10͒/6653͑12͒/$10.0053 6653 © 1996 The American Physical Society 6654 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53 phenomenological capacitive model10 can be used to de- scribe the effect of interactions. As a result the critical cur- rent shows strong resonant dependence on the electrochemi- cal potential of the slab dependence has different character 2 for ECϽ⌬ and ECϾ⌬, ECϭe /2C being the charging en- ergy. On the other hand, if the normal metal and the super- conductor are well coupled ͑regime of Andreev reflection͒, electron-electron interactions will modify the results ob- tained in Ref. 9 in quite a different fashion. A perturbative treatment of the interactions12 shows that an additional su- percurrent through the slab arises, whose sign depends on the nature of the interactions in the slab ͑attractive or repulsive͒, and whose phase dependence has period ͑rather than 2 in the noninteracting case͒. If instead of a metal a low-dimensional Sc nanostructure with a small electron concentration is considered, the above- FIG. 1. The geometries discussed in the text: ͑a͒ one- mentioned descriptions of the electron-electron interactions dimensional wire connected to two superconductors by tunnel junc- are no longer sufficient. In one-dimensional ͑1D͒ systems the tions. The distance between the junctions is d. ͑b͒ Ring with cir- Coulomb interaction cannot be treated as a weak perturba- cumference L connected to two superconductors by tunnel tion. As a result a nonperturbative, microscopic treatment of junctions. The distance between the junctions is L/2, the ring is interactions is required. For 1D systems this can be done in threaded by a magnetic flux ⌽. the framework of the Luttinger model.13 Interactions have a phase-coherent propagation of two electrons through a 1D drastic consequence: there are no fermionic quasiparticle ex- 27 citations. Instead, the low-energy excitations of the system normal region. Secondly, various aspects of transport in consist of independent long-wavelength oscillations of the mesoscopic systems ͑parity effects and interference com- charge and spin density, which propagate with different ve- bined with electron-electron interactions͒ and their interplay locities. The density of states has power-law asymptotics at can be enlightened in such a device. Finally, since the Jo- low energies and the transport properties cannot be described sephson effect is a ground-state property, the Josephson cur- in terms of the conventional Fermi-liquid approach. For a rent can be used as a tool to probe the ground state of an quantum wire with an arbitrarily small barrier this leads to a interacting electron system. In particular, for the ring geom- complete supression of transport at low energies.14–17 etry in the presence of an Aharonov-Bohm flux, the various Another interesting feature arises in 1D interacting sys- possible ground-state configurations can be determined by tems of a finite size. For a Luttinger liquid confined to a ring, studying flux and gate voltage dependence of the critical Loss18 found remarkable parity effects19 for the persistent current. currents. He used the concept of Haldane’s topological The paper is organized as follows. In Sec. II we briefly excitations,20 extending the previous work of Byers and Yang review the properties of the spin-1/2 Luttinger model. In Sec. for noninteracting electrons in a ring.21 Depending on the III the general formalism for the dc-Josephson effect is pre- parity of the total number of electrons on the ring, the ground sented. The dc-Josephson current is obtained by evaluating state is either diamagnetic or paramagnetic. For spin-1/2 the contribution to the free energy which depends on the electrons an additional sensitivity on the electron number difference of the superconducting phases. Starting from the modulo 4 has been found.22 Experimental evidence for general expression ͑12͒ we then consider various interesting Luttinger-liquid behavior in Sc nanostructures has been limiting situations. In Sec. IV, the wire geometry is consid- found recently. The dispersion of separate spin and charge ered ͓see Fig. 1͑a͔͒. The critical current decays as a power of excitations in GaAs/AlGaAs quantum wires has been mea- the distance d between the contacts. The exponent depends sured with resonant inelastic light scattering.23 Transport on the interaction strength. We distinguish two cases in measurements on quantum wires have revealed power-law which the characteristic energy បvF /d for the 1D system is dependence of the conductance as a function of either much smaller or much larger than the superconducting temperature.24 gap ⌬. For the ring geometry ͓Sec. V, see Fig. 1͑b͔͒,we In view of this we expect that electron-electron interac- focus on the dependence of the critical current on the applied tions may well have drastic, observable consequences in sys- gate voltage and/or flux. Both Secs. IV and V contain a dis- tems which consist of low-dimensional Sc nanostructures cussion of the effect of finite temperatures. Section VI is connected to superconductors. In this paper, we will study devoted to the ac-Josephson effect. In this case the imaginary the Josephson current through a Luttinger liquid.25,26 Specifi- time approach of the previous sections is inadequate and we cally, we consider two geometries which can be realized ex- will use a real-time formulation. The amplitude of the ac perimentally: a long wire with contacts to two superconduct- component is found to show oscillations as a function of ors at a distance d ͓see Fig. 1͑a͔͒ or a ring-shaped Luttinger voltage due to spin-charge separation. In the last section we liquid shown in Fig. 1͑b͒. In both cases the 1D electron present the conclusions. liquid is connected to the superconducting electrodes by tun- II. THE SPIN-1/2 LUTTINGER LIQUID nel junctions. This is an interesting system from various points of view. First of all, it enables one to study in a mi- We start the description of the model we use by reviewing croscopic way how the Coulomb interaction influences the the theory of 1D interacting spin-1/2 fermions ͑throughout, 53 dc AND ac JOSEPHSON EFFECT IN A . . . 6655 we use បϭkBϭ1). The long-wavelength behavior of such a spin s in addition to N0 . The number Js is the number of 20 quantum wire of length L is governed by the Hamiltonian current quanta evF /L, carried by electrons with spin s. Here, vFϭ0 /m is the Fermi velocity, with m being the L/2 dx g 2 ˆ j 2 2 electron mass. A net current JevF /L flows through the HLϭ ͚ vj ٌ͑j͒ ϩ ٌ͑j͒ . ͑1͒ ͵ϪL/2 j ͫ 2 g ͬ quantum wire if there is an imbalance between the number j electrons moving to the right and to the left. Using the It is written as a sum of the contributions from the spin boundary condition ͑3͒ one obtains topological constraints ( jϭ) and charge ( jϭ) degrees of freedom. The param- 18,20 for M s and Js , which lead to the following constraints eters g denote the interaction strengths and v ϭ2v /g the j j F j for M j and J j : velocities of spin and charge excitations.14 The parameters ͑i͒ The topological numbers M j and J j are either simulta- g j can be determined once one defines an appropriate micro- neously even or simultaneously odd; scopic Hamiltonian ͑e.g., the Hubbard model͒; an approxi- ͑ii͒ when N0 is odd the sum M ϮM ϩJϮJ takes val- mate form for the spinless case has been given in Refs. 14 ues ...,Ϫ4,0,4,..., when N0 is even the sum and 15. In this paper we will neglect backscattering ͑via M ϮM ϩJϮJ takes values ...,Ϫ6,Ϫ2,2,6,... . umklapp or impurity scattering͒ and restrict ourselves to re- An Aharonov-Bohm flux ⌽ threading the loop couples to pulsive, spin-independent interactions. As a result we have 14 the net current, characterized by the topological number J gϭ2 and vϭvF . of the field . The flux can be incorporated into the Hamil- We also introduced bosonic fields j and j . They are tonian ͑1͒ by changing related to the fields sϭϩs and sϭϩs for spin up (sϭϩ1) and down (sϭϪ1) fermions. These ٌ ٌ Ϫ͑2/L͒␦ f , ͑6͒ j→ j j, ⌽ fields obey the commutation relation ͓s(x),sЈ(xЈ)͔ ˆ where f ϭ⌽/⌽ is the flux frustration and ⌽ is the flux ϭ(i/2)sign(xЈϪx)␦s,sЈ. The fermionic field operators ⌿ ⌽ 0 0 can be expressed in terms of the spin and charge degrees of quantum h/e. freedom:18,20 Since the LL is brought into the contact with particle res- ervoirs ͑superconductors͒ kept at fixed electrochemical po- ˆ † tential , the number of particles ͑characterized by the topo- ⌿L,s͑x͒ϭͱ0 exp͕inkFx͖ n͚,odd logical number M of the field ) should be coupled to . This can be achieved by replacing ϫexp͕in͓ϩs͔͖exp͕i͓ϩs͔͖, ͑2͒ j ٌ jϪ͑2/L͒␦ j, f ͑7ٌ͒ where kF is the Fermi-wave vector and 0ϵN0 /L is the → average electron density per spin direction. The number N0 in the Hamiltonian ͑1͒. The parameter f ϭ(gL/4v)⌬ determines the linearization point of the original electron is related to the difference ⌬ between and the Fermi 18 2 spectrum, kFϵN0 /L. energy EFϭkF/2m of the quantum wire, corresponding to If the wire is closed to form a loop, the periodic condition the linearization point. Generally, the reference point ⌬ϭ0 is defined from the requirement that for ⌽ϭ0 there ⌿ˆ ͑xϩL͒ϭ⌿ˆ ͑x͒ ͑3͒ L,s L,s are 2N0 electrons in the ground state and the energies to should be imposed on the Fermi operators ͑2͒. The fields add/remove electrons to/from the system are equal. The dif- ference ⌬ can be controlled, e.g., by a gate voltage. and can then be decomposed in terms of bosonic fields ¯ Using Eqs. ͑1͒, ͑4a͒, ͑4b͒, ͑5a͒, and ͑5b͒ one concludes and ¯ and topological excitations:18,20 that the Hamiltonian can be decomposed into nonzero modes ¯ 0 and topological excitations: j͑x͒ϭ j͑x͒ϩ j ϩM j͑x/2L͒, ͑4a͒ v g ¯ 0 ˆ ˆ † ˆ j j 2 j͑x͒ϭ j͑x͒ϩ j ϩJ j͓͑xϩL/2͒/2L͔. ͑4b͒ HLϭ ͚ ͚ v j͉q͉bq, jbq, jϩ ͑J jϪ4␦ j, f ⌽͒ jϭ, ͭ qÞ0 4L ͫ 2 Here, ¯ and ¯ are given by j j 2 2 1/2 ϩ ͑M jϪ4␦ j, f ͒ . ͑8͒ i g j g j ͬͮ ¯ ͑x͒ϭ sign͑q͒eiqx͑bˆ† ϩbˆ ͒, j 2ͱ 2 ͚ qL j,q j,Ϫq qÞ0 ͯ ͯ Since this Hamiltonian is quadratic in the Bose operators, it ͑5a͒ is possible to obtain all the correlation functions exactly. i 2 1/2 ¯ ͑x͒ϭ eiqx͑bˆ† Ϫbˆ ͒, ͑5b͒ III. dc-JOSEPHSON EFFECT j 2ͱg ͚ ͯqLͯ j,q j,Ϫq j qÞ0 Both systems depicted in Fig. 1 can be described by the ˆ ˆ † where b j,q ,b j,q are Bose operators. Hamiltonian The boundary condition ͑3͒ gives rise to the topological ˆ ˆ ˆ ˆ ˆ ˆ ˆ excitations M j and J j for the spin and charge degrees of HϭHS1ϩHS2ϩHLϩHTϵH0ϩHT . ͑9͒ freedom. They are related to the usual topological excitations ˆ ˆ for fermions with spin s: M sϭ(1/2)͓M ϩsM͔ and Here, HS1 , HS2 are the BCS Hamiltonians for the bulk su- Jsϭ(1/2)͓JϩsJ͔. Physically, the number M s denotes the perconductors kept at constant phase difference number of excess electrons in the Luttinger liquid ͑LL͒ with ϭS1ϪS2 . For simplicity we assume equal magnitude 6656 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53
⌬ for both order parameters. The tunneling between the su- The stationary Josephson effect can be obtained by evalu- perconductors and the 1D electron system is described by ating the phase-dependent part of the free energy F (). The ˆ 28 HT . It is assumed to occur through two tunnel junctions at Josephson current is then given by the points xϭ0 and xϭd,
. ͑11͒ץ/ FץIJϭϪ2e ˆ † ˆ ˆ HTϭ T1⌿S1,s͑xϭ0͒⌿L,s͑xϭ0͒ ͚s ˆ † ˆ ϩT2⌿S2,s͑xϭd͒⌿L,s͑xϭd͒ϩ͑H.c.͒. ͑10͒ We expand F ϭϪ(1/)lnZ, where ZϭTr exp͕ϪHˆ ͖ and ϭ1/T, in powers of the tunneling Hamiltonian Hˆ , using The constant tunnel matrix elements T1,2 can be related to T standard imaginary-time perturbation theory.29 The lowest the tunnel conductances G1,2 of the junctions, 2 2 Giϭ4e NL(0)Ni(0)͉Ti͉ , where NL(0)ϭ1/vF , and Ni order phase-dependent contribution arises in fourth order. is the normal density of states in the superconductors Using Eq. ͑10͒ we see that there are 24 contributions to the (iϭ1,2). phase-dependent part of F :
1  1 2 3 † 2 ͑a͒ 2 F ͑͒ϭϪ d1 d2 d3 d4͕FS1͑0;1Ϫ2͒T1⌸L ͑0,d;1 ,...,4͒͑T2*͒ FS2͑0;3Ϫ4͒ ͵0 ͵0 ͵0 ͵0
† 2 ͑b͒ 2 ϩFS1͑0;1Ϫ4͒T1⌸L ͑0,d;1 ,...,4͒͑T2*͒ FS2͑0;2Ϫ3͒ϩ͑H.c.͖͒ϩ22 similar terms. ͑12͒
This result has a clear physical meaning, see Fig. 2. The ˆ ˆ FSi͑0;ϪЈ͒ϵ͗T⌿Si,Ϫ͑d,͒⌿Si,ϩ͑d,Ј͒͘Si Josephson effect consists of processes in which a Cooper pair tunnels from superconductor S2 into the LL with an N͑0͒ ⌬eiSi Ϫi ͑Ϫ ͒ 2 ϭ e n Ј , ͑13͒ amplitude (T*) . After propagation through the LL, it tun- ͚ 2 2 2  n ϩ⌬ 2 ͱ n nels into superconductor S1 with an amplitude (T1) . The Hermitian conjugate terms describe processes in the opposite ˆ direction. The propagation in the superconductors is de- where ͗•••͘Si indicates an average with respect to HSi . Propagation through the LL is determined by the Cooperon scribed by the anomalous Green’s function propagator ⌸L(0,d;1 ,...,4). These 24 terms are obtained by considering all possible time-ordered pairs of tunneling events i , j (iϽ j)atxϭ0 and xϭd together with all possible spin configurations. However, which of these terms are important depends on the relation between the character- istic energy vF /d for the 1D system and the superconducting gap ⌬.30 If the distance between the contacts is large, vF /dӶ⌬,a generic process consists of fast tunneling of two electrons from the superconductor into the 1D system and their slow propagation through the LL. Such a process is illustrated by the first term in Eq. ͑12͒. Here,
͑a͒ ⌸L ͑0,d;1 ,...,4͒ ˆ ˆ ˆ † ˆ † ϭ͗⌿L,ϩ͑0,1͒⌿L,Ϫ͑0,2͒⌿L,ϩ͑d,3͒⌿L,Ϫ͑d,4͒͘, ͑14͒
where ͉1Ϫ3͉ϳd/vFӷ͉1Ϫ2͉ϳ͉3Ϫ4͉ϳ1/⌬͓see Fig. 2͑a͔͒. The average is taken over equilibrium fluctuations in ˆ the LL ͑described by HL). The other relevant processes come from diagrams which are obtained from the one in Fig. 2͑a͒ by means of particle-hole conjugation and by changing the time ordering. FIG. 2. Relevant diagram for Josephson tunneling in the limit- In the opposite limit vF /dӷ⌬, diagrams of the type de- ing cases ͑a͒ vF /dӶ⌬ and ͑b͒ vF /dӷ⌬. The shaded area indicates picted in Fig. 2͑a͒ are no longer relevant. Instead, one should the electron-electron interaction. consider fast and independent propagation of two electrons 53 dc AND ac JOSEPHSON EFFECT IN A . . . 6657 through the LL and slow tunneling between S and LL. This d. The topological excitations play no role in this case ͑their is illustrated by the second term in Eq. ͑12͓͒see also Fig. energy is vanishingly small͒ and the wire is described by the 2͑b͔͒, where nonzero modes only ͓first term in Eq. ͑8͔͒.
͑b͒ ⌸L ͑0,d;1 ,...,4͒ A. The case vF /dӶ⌬ ˆ ˆ † ˆ ˆ † ϭ͗⌿L,ϩ͑0,1͒⌿L,ϩ͑d,2͒⌿L,Ϫ͑0,3͒⌿L,Ϫ͑d,4͒͘, The expression for the phase-dependent part of the free energy contains four terms of the type of the first term in Eq. ͑15͒ ͑12͓͒see Fig. 2͑a͔͒. In this case at low temperatures TӶ⌬ with ͉1Ϫ2͉ϳ͉3Ϫ4͉ϳd/vFӶ͉1Ϫ3͉ϳ1/⌬. Also in this one can approximate the anomalous Green functions ͑13͒ by case the other relevant processes can be obtained from the ␦ functions in time. This fixes equal time arguments one in Fig. 2 b by means of particle-hole conjugation and by (a) ͑ ͒ 1ϭ2 , 3ϭ4 in ⌸L . The remaining integration should changing the time ordering. be performed over the time ϭ1Ϫ3 . The dominant con- The direct evaluation of averages like ͑14͒, ͑15͒ with the tribution to the the integral comes from the terms with help of bosonized field operators like ͑2͒ is tedious but nϭϮ1 in Eq. ͑2͒. As a result, the Josephson current ͑11͒ straightforward. The resulting expressions can be simplified through the quantum wire is given by further in the two limiting cases vF /dӶ⌬ and vF /dӷ⌬, which contain all the important physics of the problem. ͑a͒ ͑a͒ IJ ϭIc ͑T͒sin, ͑16͒
IV. dc-JOSEPHSON CURRENT with a temperature-dependent critical current THROUGH A QUANTUM WIRE 4evF G1G2 I͑a͒͑T͒ϭ F͑a͒͑T͒, ͑17͒ We first turn to the geometry depicted in Fig. 1͑a͒.It c d ͑4e2͒2 w consists of a quantum wire of length L ϱ connected to two superconductors by tunnel junctions separated→ by a distance where
2/g Ϫ1 2 2 1/g vF/2d j ͑a͒ 1 dx 2 d 2 Fw ͑T͒ϭ ͟ 2 2 ͑18͒ ͫ kFdͬ ͵Ϫv /2d2 jϭ, ͫ v  cosh͑2d/v j͒Ϫcos͑2dx/vF͒ͬ F j
͑with gϭ2 and vϭvF). At low temperatures, TӶvF/2d, the critical current is sup- In the noninteracting case, at zero temperature, pressed below its zero temperature value in a power-law (a) Fw (0)ϭ1. The Josephson current decreases as 1/d with in- fashion creasing distance between the tunnel junctions. This is re- 2 lated to the fact that the density of Cooper pairs in the LL 2 Td I͑a͒͑T͒/I͑a͒͑0͒Ӎ1Ϫ . ͑20͒ decays in space away from each junction. Hence the overlap c c 3 ͩ v ͪ F of the macroscopic wave functions of the two superconduct- ors, which is responsible for the Josephson effect, is sup- In the high-temperature regime, TӶvF/2d, the decay is pressed. Repulsive interactions in the wire make the Joseph- exponential son effect vanish more rapidly with the distance between the ͑a͒ superconductors, Ic ͑T͒ ͱ8Td ͑a͒ Ӎ exp͑Ϫ2Td/vF͒. Ic ͑0͒ vF
͑a͒ 2/g Ic ͑0͒ϰ1/͑kFd͒ . It is possible to obtain analytical results also in the inter- acting case. In particular, for weak interaction 2ϪgӶ2 and low temperatures TӶv /2d the critical current behaves as The electron liquid acquires an additional stiffness against F density fluctuations, hence the tunneling between S and LL is ͑a͒ 2/g 2 Ic ͑T͒ 2Ϫg 2Td 2 Td suppressed. This fact provides an a posteriori justification of Ӎ1ϩ Ϫ , ͑21͒ I͑a͒͑0͒ 4 ͩ v ͪ 3 ͩ v ͪ our use of perturbation theory when treating electron tunnel- c F F ing in the presence of repulsive interactions. where we dropped terms O͓(2Ϫg )2ϩ(2Ϫg )(Td/v )2͔. We consider now the temperature dependence of the criti- F This result can be interpreted in terms of a competition be- cal current. For noninteracting electrons, g ϭ2, the critical tween two effects. At low temperatures the dominant depen- current can be calculated explicitly: dence comes from the renormalization of the tunneling am- 14 plitudes T1,2 in the presence of interaction. The critical ͑a͒ current increases with temperature. Above the crossover Ic ͑T͒ 2Td 1 ϭ . ͑19͒ temperature TcrossӍ(3/8)(2Ϫg)vF /d the Josephson cur- ͑a͒ v 2 Ic ͑0͒ F ͱcosh ͑2Td/vF͒Ϫ1 rent decreases due to the shortening of the phase coherence 6658 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53
Note that we estimated the Josephson current assuming fixed Josephson phase difference between the superconduct- ors. Thermal fluctuations of the Josephson phase would (a) smear the critical current at temperatures T*ϳEJϵIc /2e, provided that the superconductors are coupled by the LL only. Using Eq. ͑17͒ for the noninteracting case ͓with (a) Fw (0)ϭ1͔, one obtains that the temperature T* is by a 2 2 factor 2G1G2 /(4e ) Ӷ1 smaller than the characteristic temperature scale vF /d for the LL. Hence, in order to ob- serve nontrivial temperature dependence of the critical cur- rent, one has to fix the phase difference between the super- conductors, e.g., by means of an additional Josephson junction.
B. The case vF /dӷ⌬ In this limit, the electrons propagate fast and indepen- FIG. 3. The critical current of the wire as a function of the dently through the LL on a time scale 1/⌬. A typical contri- temperature (tϭTd/2vF) for various values of the interaction bution is depicted in Fig. 2͑b͒. The Cooperon ͑15͒ can be strength gϭ1.0,1.25,1.75,2.0. approximated as
͑b͒ length. Although the maximum is not very pronounced, the ⌸L ͑0,d;1 ,...,4͒ crossover temperature shifts to higher values as the interac- ˆ ˆ † ˆ ˆ † tion strength increases ͑see Fig. 3͒. This results in a wider Ϸ͗⌿L,ϩ͑0,1͒⌿L,ϩ͑d,2͒͗͘⌿L,Ϫ͑0,3͒⌿L,Ϫ͑d,4͒͘, temperature range in which the critical stays almost constant. where we substitute It is evident from Eq. ͑21͒ that the coefficient responsible for the anomalous temperature dependence vanishes in the ab- ˆ ˆ † ͗⌿L,ϩ͑0,i͒⌿L,ϩ͑d, j͒͘ϷC␦͑iϪ j͒. sence of interaction, thus restoring the T2 suppression of the The constant C is determined by integration of the time- critical current ͑20͒. For high temperatures Tӷv/2d the suppression becomes exponential, ordered single-particle correlator of the LL, I͑a͒͑T͒ϰT2/gexp͑Ϫ2Td/v ͒. ͑22͒ /2 c F ˆ ˆ † Cϭ d͗T⌿L,ϩ͑0,͒⌿L,ϩ͑d,0͒͘. ͵Ϫ/2 The full temperature dependence of the critical current, calculated by numerical integration of Eqs. ͑17͒, ͑18͒ is The temperature-dependent Josephson current is found to be shown in Fig. 3. We see that for moderate strength of the ͑b͒ ͑b͒ interaction gϳ1 the Josephson current will maintain an ap- IJ ϭIc ͑T͒sin, preciable value up to a temperature TϳvF /d, which is of the where the critical current is given by order of 0.7 K for typical experimental parameters vFϭ 3.0 5 31 10 m/s and dϭ3 m. Moreover, the value of the critical G G (a) ͑b͒ 1 2 ͑b͒ current Ic (Tϭ0)Ϸ 22 nA ͓estimated for the parameters Ic ͑T͒ϭe⌬ 2 2 Fw , 2 ͑4e /͒ given above and Gi /(4e )ϭ0.3͔, is large enough to be mea- sured experimentally. with
g /4ϩ1/g Ϫ1 g /4ϩ1/g g /4ϩ1/g ϩ1 vF/2d ϩ ͑b͒ 1 g d 2 Fw ϭ dx sin ͫ kFdͬ ͫ 2 ͬ ͫ vFͬ ͭ ͵0 ͩ 2 ͪ 2 gj/16ϩ1/4g j 2 ϫ ͟ . ͑23͒ jϭ, ͫ cosh͑2d/v j͒Ϫcos2dx/vFͬ ͮ
2 The phase factor j is given by 2 1 F͑b͒ϭ arctan . w sinh d/v dx d ͫ ͩ ͑ F͒ͪͬ ϭarctan cot tanh . j ͫ ͩ v ͪ ͩ v ͪͬ F j (b) At zero temperature we find Fw ϭ1. The resulting Joseph- (b) In the noninteracting case gϭgϭ2, Fw can be calcu- son current thus is independent of the distance d between the lated explicitly: contacts, analogous to the result obtained in Ref. 6. At finite 53 dc AND ac JOSEPHSON EFFECT IN A . . . 6659
2 temperatures the Josephson current is suppressed. If where we dropped terms of the order of O͓(2Ϫg) 2 TӶvF /d, the suppression is linear in T: ϩ(2Ϫg)(Td/vF) ]. We find again an anomalous depen- dence on temperature, like the one we discussed in the case I͑b͒͑T͒ 4Td c vF /dӶ⌬. ͑b͒ Ӎ1Ϫ . Ic ͑0͒ vF In the interacting case, at Tϭ0, the critical current is sup- pressed, V. dc-JOSEPHSON CURRENT THROUGH A RING
͑b͒ g/4ϩ1/gϪ1 Ic ͑0͒ϰ1/͑kFd͒ . In case of the ring with circumference L ͓Fig. 1͑b͔͒,we At finite temperatures, and for weak interactions we obtain should take into account the contribution to the Josephson current due to the topological part, see Eqs. ͑4a͒, ͑4b͒, and ͑b͒ 1/2ϩg/8ϩ1/2g Ic ͑T͒ 3 Td 2Ϫg Td Ϫ1ϳϪ ϩ , ͑8͒. The Cooperon for the ring in case of a symmetric setup I͑b͒͑0͒ 2 ͫ v ͬ 2 v dϭL/2ӷv /⌬ is evaluated along the same lines as before. It c F F F ͑24͒ is then straightforward to get the Josephson current
2evF G1G2 IJϭ 2 2 ͗Fr͑g ,L,⑀,M ,J͒sin͑ϩ⑀M /2ϩJ/2͒͘J,M , ͑25͒ L ͑4e ͒ ⑀ϭϮ͚ 1 where
2/g Ϫ1 2/g 2 1 vF/2L 1 1 2 Frϭ dx cosh ͑M Ϫ4 f ͒xϩ⑀Jx 2 ͫk Lͬ ͵Ϫv /2L ͫcosh͑2x/g ͒ͬ cosh͑x͒ ͫͩ g ͪ ͬ F F
2 1/g j 1ϩ͚nexp͑2v jn /L͒ ϫ ͟ 2 , ͑26͒ jϭ, ͫ1ϩ͚nexp͑2v jn /L͒cos„2n j͑x͒…ͬ
with tion of J j and M j is found to be (J ,J ,M ,M )ϭ ͑0,0,0,0͒;iffϩf⌽Ͼ1/2 the configuration is ͑2,0,2,0͒. 1 2v jx Hence, the ground state of the system can be changed by ͑x͒ϭ arcos cosh . j 2 ͫ ͩ v ͪͬ varying either the flux or the gate voltage. As a result, the F Josephson current changes, see Eqs. ͑25͒, ͑26͒. This is illus- trated in Fig. 4, where the critical current I ͑we write The brackets, ͗•••͘J,M , should be considered as the thermal c average over the topological excitations weighted by the ap- IJϭIcsin where Ic can be positive or negative͒ is plotted as propriate Boltzmann factor and subject to the topological a function of f and T at fixed f ⌽ϭ0.2. For Tϭ0, the criti- constraints. cal current shows a maximum and a sharp jump at f ϭ0.3 For zero temperature this calculation involves only the ground state ͑see Ref. 25 for details͒. The dependence of the critical current on f ⌽ and f has been found to show very rich behavior. In the present study, we will focus on the effect of finite temperatures. In particular, we will investigate how robust the structure, found in Ref. 25 is against thermal fluctuations. Two remarks are in order at this point. First, throughout this section, we assume that the linearization point of the original electron spectrum corresponds to odd values of N0 ͑for even N0 the picture is the same, apart from a relative shift of f ⌽ and f ). Second, the Josephson current depends periodically on both f ⌽ and f with period 1. However, since the original problem has additional symmetries f ⌽ Ϫ f ⌽ and f Ϫ f ͑together with a change of sign of the→ corre- → sponding topological numbers, J j and M j), it is enough to consider f ⌽ and f in the intervals 0Ͻ f ⌽Ͻ1/2 and FIG. 4. The critical current through the ring ͕normalized to 2 2 0Ͻ f Ͻ1/2. (2evF /L)͓G1G2 /(4e /ប) ͔͖ at a fixed value of the flux It is instructive to discuss first the noninteracting case f ⌽ϭ0.2 as a function of the gate voltage and the temperature gϭ2, for Tϭ0. If f ϩ f ⌽Ͻ1/2 the ground-state configura- (tϭTL/vF) in the noninteracting case. 6660 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53 where the states ͑0,0,0,0͒ and ͑2,0,2,0͒ are degenerate. At this value of the gate voltage, the number of electrons on the ring (M ) increases by two. Since electronic states are dou- bly degenerate in spin and nonzero flux is applied, the two electrons will occupy the same ͑clockwise or counterclock- wise moving͒ single-particle state. Therefore, the net current evFJ /L increases by 2 quanta evF /L, while the topological numbers M and J related to spin remain unchanged. At the jump, Ic changes sign. This reflects the fact that the ring acts as a junction (IcϽ0) in the state ͑2,0,2,0͒, as can be seen from Eq. ͑25͒. Therefore, for noninteracting electrons, the critical current shows two jumps per period of the gate volt- age dependence. The same is true for the dependence of Ic on the magnetic flux. This picture is correct for any generic point on the line f ϩ f ⌽ϭ1/2. At the end points ( f , f ⌽)ϭ(0,0.5) and FIG. 5. The same as in the previous figure for the interacting (0.5,0), no jumps of the critical current occur one can say ͑ case; for this plot we choose gϭ1.75 but the result is rather ge- that two jumps in opposite directions merge together͒. In- neric for repulsive interaction ͕the critical current is normalized to 2 2 stead, the critical current shows a resonance. The resonance (2evF /L)͓G1G2 /(4e /ប)͔ ͖. occurs due to alignment of two spin-degenerate energy levels ͑for clockwise and counterclockwise moving electrons͒ with function of f for Tϭ0 ͓this can be seen from Eqs. ͑25͒, the chemical potential of the superconductors.25 ͑26͒; the flux f enters to these equations only implicitly, via At a finite temperature, both the nonzero modes and the ⌽ topological numbers͔. Depending on the gate voltage, the topological excitations are thermally activated. Thermal ac- critical current can show zero, two, or four jumps per period tivation of the nonzero modes leads to an overall suppression of the flux dependence.25 of the critical current, as it has been discussed for the wire in The state ͑1,1,1,1͒ is the ground state in a strip of width Sec. IV. Thermal activation of the states ͑0,0,0,0͒ and ␦ f ϭ͓1Ϫ(g /2)2͔/4. This determines the energy ͑2,0,2,0͒ will lead to a smearing of the jump. Moreover, at ␦EӍ͓1Ϫ(g /2)2͔v /(g L) needed to create topological finite temperature there will be a nonvanishing probability to excitations. The features related to the state ͑1,1,1,1͒ will be activate other topological excitations which can contribute to smeared at temperatures Tϳ␦E. Therefore, for weak inter- the Josephson current. In the plotted temperature range, only action 2Ϫg Ӷ1, the interaction effects will disappear at one additional state ͑1,1,1,1͒ with one extra electron on the much lower temperatures TӶv /L than the parity effects. ring can be activated. As a result, the negative critical current F For example, the features related to the configuration of the state ͑2,0,2,0͒͑at f տ0.3) will be partially compen- ͑1,1,1,1͒ in Figs. 5 and 6 are seen only in the temperature sated by the positive critical current of the state ͑1,1,1,1͒, the range TϽ␦Eϳ0.1 K whereas the overall dependence is ro- occupancy of which increases with temperature. Note that bust up to the temperatures Tϳ1 K. However, it is worth- the jump at f ϭ0.3 remains visible up to temperatures of the while to stress that the ͑1,1,1,1͒ state survives at much higher order of Tϳv /Lϳ1K͑for the parameters mentioned F temperatures when the interaction strength is increased. above and Lϭ2 m͒. Hence, the parity effect causing the The behavior we described here is rather generic for all jump is quite robust against thermal fluctuations. values of f , f and for various values of the interaction. An important feature of the noninteracting case at Tϭ0is ⌽ that the various possible ground-state configurations may dif- fer by an even number of electrons only. The situation changes drastically when repulsive interactions are switched on. In addition to the states ͑0,0,0,0͒ and ͑2,0,2,0͒, the state ͑1,1,1,1͒ can act as a ground-state configuration.25 This hap- pens for parameters f , f ⌽ within the range 2 2 2 1ϩ3(g/2) Ͻ8͓ f ϩ(g/2) f ⌽͔Ͻ3ϩ(g/2) . Within this ‘‘strip,’’ it is energetically more favorable to add a single electron, rather than a pair of electrons to the ring, due to repulsive electron-electron interactions. The Josephson cur- rent in the state ͑1,1,1,1͒ differs from the current in the states ͑0,0,0,0͒ and ͑2,0,2,0͒, see Eqs. ͑25͒, ͑26͒. For example, for gϭ1.75 and f ⌽ϭ0.2 ͑Fig. 5͒ the state ͑1,1,1,1͒ occurs in the range 0.259Ͻ f Ͻ0.318. Indeed, one sees two pro- nounced jumps of Ic at the borders of this interval in Fig. 5 ͑at low temperatures͒. Generally, for interacting electrons the critical current shows four jumps per period of the gate volt- FIG. 6. The critical current through the ring at a fixed value of 25 age dependence. the gate voltage f ϭ0.2 as a function of the flux and the tempera- Similar jumps are seen also at the dependence of the criti- ture (tϭTL/vF) in the interacting case gϭ1.75 ͕the critical cur- 2 2 cal current on the flux, Fig. 6. Moreover, Ic is a stepwise rent is normalized to (2evF /L)͓G1G2 /(4e /ប) ͔͖. 53 dc AND ac JOSEPHSON EFFECT IN A . . . 6661
expression which has the same structure as Eq. ͑12͒. The integrals are now taken over real times. It is convenient to depict the times t,t1 ,t2 ,t3 of tunneling events on the Keldysh contour33 ͑the Josephson current is calculated at a time t). Again, we will consider two cases of long (vF /dӶ⌬) and short (vF /dӷ⌬) distance between the con- tacts. The relevant diagrams are shown in Fig. 7 for both cases. We restrict our consideration to the case of a quantum wire at zero temperature.
A. The case vF /dӶ⌬ For a large distance between the contacts the tunneling of two electrons to/from a superconductor is a fast process on the time scale of their propagation through LL. The Joseph- son current is described by diagrams of the type shown in FIG. 7. Relevant diagrams for ac-Josephson current: ͑a͒ Fig. 7͑a͒. The Josephson current is then given by vF /dӷ⌬ and ͑b͒ vF /dӶ⌬. The shaded area indicates the electron- electron interaction. G1G2 I ͑t͒ϭ42ev2 J F ͑4e2͒2 What is specific is the configuration of the two superconduct- ors: they are connected symmetrically to the ring. If the ϱ ϫRe ͚ ϮeϮ2ieVt dtЈeϯieVtЈ⌸͑tЈ͒ , ͑28͒ points on the ring at which the electrodes are attached would ͫ Ϯ ͵0 ͬ form a generic angle, a more complicated interference pat- (a) tern would arise. In the symmetric setup, the maximum Jo- where ⌸(t)ϭ⌸L (0,d;it,it,0,0) is the Cooperon propagator sephson current occurs either at ϭ0oratϭ. In the ͑14͒ in real time taken at coinciding time arguments. The nonsymmetric setup the maximum Josephson current would leading contribution stems from the terms in Eq. ͑2͒ with nϭϮ1, occur at a value (J ,J ,M ,M ) which depends on the values of topological numbers. The critical current should then be found by maximizing the resulting function of the 2 Ϫ1/g j ⌸͑t͒ϭ20 ͟ ͕͓1ϩikF͑vjtϩd͔͓͒1ϩikF͑vjtϪd͔͖͒ . phase difference. jϭ, ͑29͒ VI. ac-JOSEPHSON EFFECT In particular, for noninteracting electrons (gϭ2) we obtain The effect of a finite dc bias voltage eVӶ2⌬ applied 2evF G1G2 eVd between the superconductors S1 and S2, will be twofold. IJ͑t͒ϭ 2 2 sin 2eVtϪ . ͑30͒ First of all, the phase difference between S1 and S2 will d ͑4e ͒ ͩ vF ͪ acquire a time dependence, according to the Josephson rela- This result means that the Josephson current acquires an ad- tion ˙ ϭJϭ2eV. As a result, the Josephson current will ditional phase shift due to the propagation of electrons be- oscillate as a function of time at a frequency J ͑ac- tween the contacts. For interacting electrons we computed Josephson effect, see Ref. 32͒. Secondly, a dc subgap current the Josephson current numerically. We split IJ into sinusoidal will be induced, due to Andreev reflection at both junctions. and cosinusoidal components, This current is dissipative, energy will be dissipated in the 2/gϪ1 LL. In a typical experiment one thus will find a current with 2evF G1G2 1 eV I ͑t͒ϭ J ,g sin͑2eVt͒ both a dc and an ac component. In this section, we will J d ͑4e2͒2 ͩ k dͪ sͩ v /d ͪ F ͭ F mainly concentrate on the ac-Josephson current, and estimate the dc component at the end. eV ϩJ ,g cos͑2eVt͒ . ͑31͒ In the presence of a bias voltage V between the supercon- cͩv /d ͪ F ͮ ductors, the imaginary time formalism cannot be applied and Josephson current is found by calculating the average of the The amplitudes Jc(s) of the two components and the phase corresponding Heisenberg operator. Using the interaction ϭϪarctan(Jc /Js) of the Josephson current are shown in ˆ Fig. 8 as functions of the voltage for two values of the inter- representation with the unperturbed Hamiltonian H0 , see Eq. ͑9͒, one obtains action parameter, gϭ1.75 and 1. One sees that the deviation from the simple result ͑30͒ for noninteracting electrons † I͑t͒ϭ͗Uˆ ͑t͒Iˆ͑t͒Uˆ ͑t͒͘, ͓which corresponds to Jsϭcos(eVd/vF), JcϭϪsin(eVd/vF), and ϭeVd/vF͔ increases with the increase of the interac- t tion. This deviation becomes striking in the dependence of ˆ ˆ U͑t͒ϭT exp Ϫi HT͑tЈ͒dtЈ . ͑27͒ the ac current amplitude J ϭͱJ2ϩJ2 on the voltage, Fig. 9. ͫ ͵Ϫϱ ͬ a s c Apart from the noninteracting case ͓Ja(V)ϭconst͔, one sees ˆ We expand ͑27͒ to the fourth order in HT and keep the Jo- pronounced oscillations of the current amplitude. These os- sephson terms in the current. These are proportional to cillations are due to the difference in the velocities of the exp(Ϯ2ieVt). As a result the Josephson current is given by an charge (v) and spin (v) excitations. The period ␦V of the 6662 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53
tions of creation and annihilation operators͒ over the range tϪt1ϳt2Ϫt3ϳd/vF and tϪt2ϳ1/⌬͓see Fig. 7͑b͔͒. For this (b) reason, we can present ⌸L as a product of two single- particle propagators and integrate the latter over the ‘‘fast’’ variables tϪt1 and t2Ϫt3 ͑from 0 to ϱ) as we did for the dc case. The last integration over the ‘‘slow’’ variable sϭtϪt2 involves the product of two anomalous Green functions with an exponent containing the time-dependent Josephson phase:
ϱ ϩ dsFS1͑0,is͒FS2͑0,is͒exp͑ieVs͒. ͵0 Hence, for short distances between the contacts the presence of LL does not influence the voltage dependence of ac- Josephson current. The latter is still given by the simple for- mula
͑b͒ IJ͑t͒ϭ͑2/͒K͑eV/2⌬͒Ic ͑0͒sin͑2eVt͒, (b) where K(x) is an elliptic integral and Ic (0) is the critical current in the dc case ͓cf. Eq. ͑23͒ in the limit of zero tem- perature͔. The effect of the interaction is only to reduce the value of the critical current, while its voltage dependence is analogous to that of the critical current of a conventional FIG. 8. The voltage dependence of sinusoidal ͑solid line͒ and Josephson junction.34 cosinusoidal ͑dashed line͒ components and the phase ͑dotted line͒ We conclude this section with an estimate of the dissipa- of ac-Josephson current; ͑a͒ gϭ1.75 and ͑b͒ gϭ1. tive dc current due to Andreev reflection at both junctions. For a single junction between a superconductor and a LL oscillations corresponds to 2 difference between the phases with repulsive interactions, the subgap current Is(V)asa 26,27,35 of charge (eVd/v) and spin (eVd/vF) excitations. Using function of the applied voltage V is given by 2/g Ϫ1 the relation vϭ2vF /g we obtain e␦V/(vF /d) I (V)ϳV͉V͉ . For the system of Fig. 1͑a͒, which con- Ϫ1 s ϭ2(1Ϫg/2) Ӎ50.4,12.6 for gϭ1.75 and 1, respec- sists of two junctions in series, the lowest-order contribution tively. This is in very good argeement with the period of to the subgap current stems from sequential tunneling. Em- oscillations in Fig. 9. Therefore, the ac-Josephson effect can ploying a rate equation approach, we find for this contribu- be used as a tool for the observation of spin-charge separa- tion tion in the LL. 2/gϪ1 2/g G1G2 2eV ͑2/g͒ Is͑V͒ϭ2e 2 2 2eV B. The case vF /dӷ⌬ ͑4e ͒ ͫvFkFͬ ⌫͑1ϩ2/g͒
At short distances between the contacts, the two electrons g/2 2/g 2͑G1G2͒ propagate fast through the LL on the time scale 1/⌬. The ϫ . ͑32͒ 2 g/2 2 g/2 relevant diagrams are similar to the graph shown in Fig. 7͑b͒. ͫ͑G1͒ ϩ͑G2͒ ͬ The main contribution to the Josephson current comes from Comparing this result with the critical current we see that the the integration of the two-particle propagators of the type (b) dissipative component is much smaller at low voltages. In ⌸L (0,d;it,it1 ,it2 ,it3), Eq. ͑15͒͑with possible permuta- order to get a complete description at finite voltages, one has to solve the corresponding equation for nonlinear resistively shunted-junction model.9 This will be discussed in a forth- coming publication.36
VII. DISCUSSION In this paper, we studied the ac- and dc-Josephson effect in a single-mode quantum wire and quantum ring connected to two superconductors by tunnel junctions. Repulsive inter- actions were treated in the framework of the Luttinger model. Interactions were found to have a drastic influence on both dc- and ac-Josephson effect. The critical current is suppressed by interactions at zero temperature. The results depend on the ratio between the FIG. 9. The voltage dependence of the amplitude of ac- characteristic energy បvF /d of the 1D electron system and Josephson current. Here, gϭ2,1.75,1,0.75,0.5,0.25 for the curves the superconducting energy gap ⌬. For large distances be- from top to bottom at zero voltage. tween the contacts dӷបvF /⌬ in the presence of interactions, 53 dc AND ac JOSEPHSON EFFECT IN A . . . 6663 there is a competition between thermal suppression of coher- tion. The corresponding period depends on the ratio of the ent two-particle propagation in the wire and activation of velocities of the spin and charge excitations in the LL. tunneling at the junctions at low temperatures. As a result, A quantum wire closed to a loop ͑or quantum ring͒ shows the critical current shows maximum as a function of tem- interesting parity effects. Boundary conditions on the elec- perature. At even higher temperatures, k Tӷបv /2d, the tronic wave functions result in a discrete set of quantum B numbers, related to the number of particles and angular mo- suppression becomes exponential. mentum. We showed how these numbers can be tuned by In our model it is assumed that the superconducting elec- applying a gate voltage and a magnetic flux, and calculated trodes do not influence the uniformity of the potential along the corresponding dependence of the critical current on these the quantum wire, since they are separated from the wire by parameters. This dependence shows a rich behavior, which thick barriers. It was argued in Ref. 37 that a nonuniform can be detected in an interference experiment employing a potential in the wire will lead to an effective change of the superconducting quantum interference device. We showed boundary conditions for the electronic wave function, which that the dependence is robust to thermal fluctuations up to in turn could strongly affect our results. However, a recent experimentally measurable temperatures. calculation26 of the Josephson current through an interacting quantum wire of finite length is in agreement with our re- ACKNOWLEDGMENTS sults. This indicates that the results obtained are robust with We would like to thank I.L. Aleiner, L.I. Glazman, Yu.V. respect to the specific choice of boundary conditions. They Nazarov, G. Scho¨n, C. Winkelholz, and A.D. Zaikin for use- rather describe generic properties of the superconductor– ful discussions. The financial support of the European Com- Luttinger-liquid system. munity ͑HCM-network CHRX-CT93-0136 and HCM ERB- If a finite voltage V is applied between the junctions, the CHBI-CT94-1474͒, the Deutsche Forschungsgemeinschaft ac-Josephson effect occurs. The ac current acquires phase through SFB 195, and the Netherlands Organization for Sci- shift proportional to the distance between the tunnel junc- entific Research ͑NWO͒ is gratefully acknowledged. The au- tions. Moreover, the amplitude of ac current depends on the thors also acknowledge the kind hospitality of ISI-Torino voltage in an oscillatory fashion due to spin-charge separa- ͑Italy͒ where part of this work was done.
*Permanent address. Phys. 28, 201 ͑1979͒; D. Jerome and H.J. Schulz, ibid. 31, 299 1 J. Nitta, T. Azaki, H. Takayanagi, and K. Arai, Phys. Rev. B 46, ͑1992͒; J. Voit, Rep. Prog. Phys. 58, 977 ͑1995͒. 14 286 ͑1992͒; A. Dimoulas, J.P. Heida, B.J. van Wees, T.M. 14 C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 ͑1992͒; Klapwijk, W.v.d. Graaf, and G. Borghs, Phys. Rev. Lett. 74, 602 Phys. Rev. B 46, 15 233 ͑1992͒. ͑1995͒. 15 K.A. Matveev and L.I. Glazman, Phys. Rev. Lett. 70, 990 ͑1993͒; 2 A. Kastalsky, A.W. Kleinsasser, L.H. Greene, R. Bhat, F.P. Mil- K.A. Matveev, D. Yue, and L.I. Glazman, ibid. 71, 3351 ͑1993͒. liken, and J.P. Harbison, Phys. Rev. Lett. 67, 3026 ͑1991͒. 16 P. Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74, 3 ´ A.F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 ͑1964͓͒Sov. Phys. 3005 ͑1995͒. JETP 19, 1228 ͑1964͔͒. 17 U. Weiss, R. Egger, and M. Sassetti, Phys. Rev. B 52, 16 707 4 For a review, see, C.W.J. Beenakker, in Mesoscopic Quantum ͑1995͒. Physics, edited by E. Akkermans, G. Montambaux, and J.-L Pi- 18 D. Loss, Phys. Rev. Lett. 69, 343 ͑1992͒. chard ͑North-Holland, Amsterdam, 1995͒, and references 19 These parity effects, occurring in a normal system, are quite dif- therein. ferent from the ones discussed in Ref. 11 which are related to 5 Several papers, in Mesoscopic Superconductivity, edited by F.W.J. superconductivity. ¨ Hekking, G. Schon, and D.V. Averin, Physica B 203, 205 ͑1994͒ 20 F.D.M. Haldane, Phys. Rev. Lett. 47, 1840 ͑1981͒; J. Phys. C 14, deal with superconductor-semiconductor devices. 2585 ͑1981͒. 6 C.W.J. Beenakker and H. van Houten, Phys. Rev. Lett. 66, 3056 21 N. Byers and C.N. Yang, Phys. Rev. Lett. 7,46͑1961͒. ͑1991͒; A. Furusaki, H. Takayanagi, and M. Tsukada, ibid. 67, 22 D. Loss and P. Goldbart, Phys. Rev. B 43, 13 762 ͑1991͒;S. 132 ͑1991͒. Fujimoto and N. Kawakami, ibid. 48, 17 406 ͑1993͒. 7 B.J. van Wees, P. de Vries, P. Magnee´, and T.M. Klapwijk, Phys. 23 A.R. Gon˜i, A. Pinczuk, J.S. Weiner, J.M. Calleja, B.S. Dennis, Rev. Lett. 69, 510 ͑1992͒. 8 L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 67, 3298 ͑1991͒. F.W.J. Hekking and Yu.V. Nazarov, Phys. Rev. Lett. 71, 1625 24 ͑1993͒; Phys. Rev. B 49, 6847 ͑1994͒. S. Tarucha, T. Honda, and T. Saku, Solid State Commun. 94, 413 9 L.G. Aslamazov, A.I. Larkin, and Yu.N. Ovchinnikov, Zh. E´ ksp. ͑1995͒. 25 Teor. Fiz. 55, 323 ͑1968͓͒Sov. Phys. JETP 28, 171 ͑1969͔͒. A.A. Odintsov, R. Fazio, and F.W.J. Hekking, in Mesoscopic Su- 10 For a review, see, Single Charge Tunneling, edited by H. Grabert perconductivity ͑Ref. 5͒, p. 361; R. Fazio, F.W.J. Hekking, and and M.H. Devoret ͑Plenum, New York, 1992͒. A.A. Odintsov, Phys. Rev. Lett. 74, 1843 ͑1995͒. 26 11 R. Bauernschmitt, J. Siewert, Yu.V. Nazarov, and A.A. Odintsov, D.L. Maslov, M. Stone, P.M. Goldbart, and D. Loss ͑unpub- Phys. Rev. B 49, 4076 ͑1994͒. lished͒. 12 B.L. Altshuler, D.E. Khmelnitskii, and B.Z. Spivak, Solid State 27 M.P.A. Fisher, Phys. Rev. B 49, 14 550 ͑1994͒. Commun. 48, 841 ͑1983͒. 28 The tunnel junctions are assumed to have linear dimensions that 13 For a review, see, V.J. Emery, in Highly Conducting One- are small compared to their separation d, but large compared to
Dimensional Solids, edited by J.T. Devreese, R.P. Evrard, and the Fermi wavelength F . V.E. van Doren ͑Plenum, New York, 1979͒;J.So´lyom, Adv. 29 A.A Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of 6664 ROSARIO FAZIO, F. W. J. HEKKING, AND A. A. ODINTSOV 53
Quantum Field Theory in Statistical Physics ͑Dover, New York, bar, FL, 1980͒. 1975͒. 33 L.V. Keldysh, Zh. E´ ksp. Teor. Fiz. 47, 1515 ͑1964͓͒Sov. Phys. 30 For BCS superconductors in the clean limit, the gap ⌬ is given by JETP 20, 1018 ͑1965͔͒. 34 vF,S /, where is the correlation length. Therefore, comparing I.O. Kulik and I.K. Yanson, The Josephson Effect in Supercon- energy scales means simply comparing d and , up to the ratio ducting Tunneling Structures ͑Israel Program for Scientific
vF /vF,S of Fermi velocities in the 1D system and the supercon- Translations, Jerusalem, 1972͒. ductor. 35 C. Winkelholz, R. Fazio, F.W.J. Hekking, and G. Scho¨n ͑unpub- 31 D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 lished͒. ͑1993͒. 36 R. Fazio, F.W.J. Hekking, and A.A. Odintsov ͑unpublished͒. 32 M. Tinkham, Introduction to Superconductivity ͑Krieger, Mala- 37 M. Fabrizio and A. Gogolin, Phys. Rev. B 51, 17 827 ͑1995͒.