Anomalous density of states of a Luttinger liquid in contact with a sup erconductor 1 2 3 1;4 C. Winkelholz , Rosario Fazio , F.W.J. Hekking , and Gerd Schon 1 Institut fur Theoretische Festkorperphysik, Universtitat Karlsruhe, 76128 Karlsruhe, Germany 2 Istituto di Fisica, Universit a di Catania, viale A. Doria 6, 95129 Catania, Italy 3 Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK 4 Department of Technical Physics, Helsinki University of Technology, 02150 Espoo, Finland Abstract We study the frequency and space dep endence of the lo cal tunneling density of states of a Luttinger liquid LL which is connected to a sup erconduc- tor. This coupling strongly mo di es the single-particle prop erties of the LL. It signi cantly enhances the density of states near the Fermi level, whereas this quantityvanishes as a p ower law for an isolated LL. The enhancement is due to the interplaybetween electron-electron interactions and multiple back-scattering pro cesses of low-energy electrons at the interface b etween the LL and the sup erconductor. This anomalous b ehavior extends over large dis- tances from the interface and may b e detected by coupling normal prob es to the system. Typ eset using REVT X E 1 Transp ort in low-dimensional structures is strongly in uenced by electron-electron in- teractions. A paradigm mo del to describ e interactions in one-dimensional systems is the Luttinger liquid LL. The low-lying excitations of the electron system consist of charge and spin waves, rather than quasiparticles [1,2]. As a consequence, the presence of a barrier in the liquid leads to p erfectly re ecting for repulsiveinteractions or transmitting for attractiveinteractions b ehavior at low energies [3]. One of the most striking characteristic prop erties of Luttinger liquids is the b ehavior of the density of states DOS close to the Fermi energy. Contrary to Fermi liquids, whose quasiparticle residue is nite, LLs have a DOS whichvanishes near the Fermi energy as a power law, g +4=g 4=8 N ! ! : 1 The exp onent g dep ends on the strength of the electron-electron interaction: it is smaller larger than two for repulsive attractive interactions. In the non-interacting case g =2, and the DOS is constantasinFermi liquids [4]. Recently it b ecame p ossible to fabricate interfaces b etween a sup erconductor S and a two-dimensional electron gas 2DEG [5]. An excess low-voltage conductance due to Andreev scattering has b een observed in Nb-InGaAs junctions [6], as well as a sup ercurrent through a 2DEG in an InGaAs/InAlGaAs heterostructure with Nb contacts [7,8]. If the 2DEG is gated to form a quantum wire, it should b e p ossible to study transp ort through Sup erconductor - Luttinger Liquid S-LL interfaces. The Josephson current through a S-LL-S system has b een calculated [9,10], as well as the I -V characteristics of a tunnel junction b etween a sup erconductor and a chiral LL [11]. The latter can b e realized, e.g., in the fractional quantum Hall regime. The proximity e ect [12] mo di es the prop erties of a normal metal N in contact with a sup erconductor. The leakage of Co op er pairs induces a non-vanishing pair amplitude in N, de ned as F ~r =h ~r ~ri, where ~r is the annihilation op erator for an electron " s with spin s. The pair amplitude is a two-particle prop erty, related to the probability to nd two time-reversed electrons at a p osition ~r . In a clean normal metal at zero temp erature, F ~r decays as 1=r away from the N-S interface. In a LL with repulsiveinteraction in con- tact with a sup erconductor, F ~r decays as 1=r , where >1 dep ends on the strength of interaction [10]. These results hold as long as ~r lies within the temp erature-dep endent coher- ence length =hv =k T from the interface to the sup erconductor. At larger distances, N F B F ~r decays exp onentially on the length scale . The reason for the decay of the pair N amplitude is that the two electrons lo ose their relative phase coherence over this distance. Single electrons, however, lo ose phase coherence only at a much larger distance, namely the phase-breaking length L . Indeed, recent exp eriments [8,13] have shown that interference e ects due to single quasiparticles in N-S systems p ersist over distances much larger than .We, therefore, exp ect quite generally a considerable in uence of sup erconductivityon N single particle prop erties over distances where the pair amplitude has already decayed. In order to investigate these prop erties in a strongly interacting system we study the lo cal single-particle density of states DOS of a LL in contact with a sup erconductor. We nd that the lo cal DOS is substantially enhanced near the Fermi energy as compared to the p ower law decay of an isolated LL cf. Eq.1. This result should b e contrasted with the b ehavior of the pair amplitude a two-particle prop erty whichis suppressed in the interacting system. 2 The anomalous enhancement is a result of the interplaybetween the scattering of low-energy electrons at the S-LL interface and the electron-electron interactions in the LL [14]. As for the space dep endence, the DOS do es not decay in the same fashion as the pair amplitude away from the S-LL interface. It remains enhanced up to distances of the order of the mean free path, whichmaybemuch larger than for a clean quantum wire. Hence, the e ect N cannot simply b e explained in terms of a nite density of Co op er pairs in the LL. Both the frequency and the space dep endence of the DOS can b e detected exp erimentally by coupling normal metal tunneling prob es to the LL at some distance from the sup erconductor. In the inset of Fig. 1 weschematically draw a 'clean' S-LL interface, consisting of a LL in go o d contact with S. The shaded area indicates a tunnel junction b etween the LL and a normal metal prob e. We will also study a LL of nite length connected to two sup erconductors S-LL-S system. In such a system Andreev b ound states exist b elow the gap [15]. Finally,we will comment on the case of a LL connected to S by means of a tunnel barrier. The anomalous enhancement of the DOS is found in this case as well. The Hamiltonian of a LL can b e written in b osonized form as h =1 " Z X 1 g 2 j 2 2 ^ H = dx r ; 2 v r + L j j j 2 2 g j j where j = ; , and v =2=g v are the renormalized interaction-dep endentFermi velo c- j j F ities. We restrict ourselves to repulsive, spin-indep endentinteractions; hence g < 2 and g = 2. The Fermi eld op erators are decomp osed in terms of right- and left-moving Fermion ik x ik x F F op erators and , resp ectively, = e + e , where k is the Fermi +;s ;s s +;s ;s F wavevector. The elds in turn can b e expressed through Boson op erators ;s p p y i [ x+ x] s s = e ; 3 0 ;s 1 1 p p where = + s and = + s . The density of electrons p er spin in the LL s s 2 2 is = k =2 . Maslovetal. [10] recently develop ed a b osonization scheme to treat clean 0 F S-LL interfaces. For a LL coupled to two sup erconductors at a distance L, they obtained the following normal mo de expansion for the the elds r r X x i g y ^ ^ x= J + + sinqxb b ; 4 q ;q ;q 2 2L 2 2 q>0 r X i 1 g 0 y ^ ^ p + sinqxb + b ; 5 x= q ;q ;q 2 2 q>0 s r X 2 x i y ^ ^ x= b ; 6 M + sinqxb ;q q ;q 2 2L 2 g q>0 s X 1 i 2 0 y ^ ^ p x= + sinqxb + b : 7 q ;q ;q 2 g q>0 y ^ Here, b are Bose op erators and = expfq =2g where is a short range cut-o . The q j;q phase di erence b etween the two sup erconductors is ; J and M describ e the top ological 0 0 excitations satisfying the constraint J + M = o dd. Finally, and are canonically conjugate to J;M resp ectively. The lo cal density of states p er spin of the LL measured at 3 a distance x from the sup erconducting contact is obtained from the retarded one-electron 0 y 0 Green's function of the LL, G x; x ; t ihf x; t; x ;0gi t, R s s Z 1 1 i! t N x; ! = Im dte G x; x; t : 8 R 1 We rst discuss the space and frequency dep endence of the DOS of a LL contacted at x = 0 with a sup erconductor, which corresp onds to the limit L !1 in the mo de expansion given by Eqs. 4 { 7. In this case only the non-zero mo des q>0 contribute to the y lo cal DOS. The correlation function h x; t x; 0i can b e evaluated using the b oson s s representation Eq.