Effect of electron-phonon coupling on transmission through Luttinger liquid hybridized with resonant level Alexey Galda, Igor V. Yurkevich and Igor V. Lerner School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom (Dated: January 28, 2013) We show that electron-phonon coupling strongly affects transport properties of the Luttinger liquid hybridized with a resonant level. Namely, this coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant or antiresonant transmission in the Fermi gas. This leads to a rich phase diagram for a metal-insulator transition induced by the hybridization with the resonant level. PACS numbers: 73.20.Mf Low-temperature electronic properties of one- dimensional (1D) systems (like quantum wires or nanotubes) are strongly affected by electron-electron interactions. Electrons in such systems form a Luttinger Liquid (LL) [1]. An arbitrarily weak repulsion in a clean LL leads to power-law decay of various correlation functions with exponents depending on the interaction strength. Such a decay which is a characteristic property of the LL has been been experimentally observed in carbon nanotubes [2] and various quantum wires [3] (see Ref. [4] for a recent review). Inserting a potential impurity or a weak link (e.g., a tunnel barrier) into the LL results in a power-law sup- FIG. 1. The geometries considered: resonant-barrier geome- pression of a local density of states (LDoS) at the im- try (top) and side-attached geometry (bottom), also referred purity site and thus a suppression of the conductance at to in the text as RBG and SAG. low temperatures T [5, 6],changes in characteristics of the Fermi-edge singularity [7] and Friedel oscillations [8], etc. If the barrier interrupting the LL carries a discrete with phonons. localized state resonant with the electron Fermi energy, It is known that the electron-phonon (el-ph) coupling its hybridization with the electronic states in the leads re- in combination with the electron-electron (el-el) repul- sults in a sharp resonant transmission [9]. Similar to the sion results in the formation of two polaron branches with Fermi liquid, it is described by the Breit{Wigner formula different propagation velocities in the clean LL (see, for example, [12, 13]). The formation of polarons modifies but with the resonance width Γ0 replaced by an energy- dependent effective width Γ(") vanishing at the Fermi the values of exponents in power laws characteristic for level. Such a resonant transmission can be realized, e.g., the LL. The exponents can change signs as functions of by inserting into a 1D quantum wire a double barrier with the relative strength of the el-el and el-ph coupling and a resonant level or a weakly coupled quantum dot (QD) of the ratio of the Fermi to sound velocities. The ef- fect of this is especially pronounced for the LL with an with sufficiently large level spacing δ (δ T; Γ0) and one level in resonance. We would refer to this geometry as embedded potential scatterer. While a single scatterer resonant-barrier. embedded into the phononless LL makes it going from In a dual geometry, when a QD with such a reso- an ideal metal to an ideal insulator (at T = 0) [5], in nant level is side-attached to the LL, transmission be- the presence of the el-ph coupling such a transition can comes antiresonant: it is reflectance rather than trans- be reversed for a weak scatterer [14] or, in general, can mittance which is described by the Breit-Wigner formula become dependent on the scatterer strength [15]. but with the width Γ(") being power-law divergent at the In this Letter we show that the el-ph coupling results Fermi level [10] (with the divergence cut by a tempera- also in a drastic change of electronic transport trough the ture T ). Both geometries are realistic: transmission and LL hybridized with a resonant level both in the resonant- tunnelling measurements in the presence of controlled de- barrier and in the side-attached geometry (see fig. 1). fects have already been performed in both quantum wires We will proceed as follows. First we introduce the [3] and carbon nanotubes for various types of defects [11]. action for the LL with the Coulomb repulsion, el-ph cou- Naturally in any realistic geometry electrons in the LL pling and hybridization with a resonant level. By inte- are inevitably interact not only with each other but also grating out the phonon fields and the fields describing 2 the resonant-level electron, we obtain an effective action Finally, the tunnelling action for both geometries (as- in terms of only the fields describing the conduction elec- suming that the impurity or QD carrying the resonant trons. Then we employ the functional bosonization in level is inserted in the LL or side-attached to it at x = 0) form developed in [16] to describe the polaron formation has the form: in the presence of the resonant level in terms of the mixed Z n ¯ X ¯ o fermion-bosonic action. Finally we use the renormaliza- ST = it id(t)(@t + "0) d(t) + t0d(t) µ(t) + h:c: ; tion group (RG) analysis in form similar to that for the µ phononless LL [5, 10] to calculate transmission through (4) such a polaronic liquid as well as the effect of the el-ph coupling and hybridization on the electronic LDoS in the where µ(t) ≡ µ(x = 0; t), d(t) is the field correspond- vicinity of the QD. ing to the electron localized at the resonant level with the energy "0 counted from the Fermi level. In the side- attached geometry (SAG) the index µ simply labels right- EFFECTIVE ACTION and left-movers while in the resonant-barriers geometry (RBG) it refers to the left and right electron subsystems The action consists of three terms, S = SLL + Sel−ph + separated by the barrier. In this case the electron can ST. We assume the usual LL decoupling of the (spin- leave the left subsystem to the QD and enter it from less) electron field into the sum of right- and left-moving the QD only as a right- and left-mover, respectively { electrons, and conversely for the right subsystem so that the labels µ = ±1 mean ip x −ip x (ξ) = R(ξ)e F + L(ξ)e F ; ξ ≡ (x; t); ¯ ¯ ¯ ¯ − = `;L ; − = `;R ; + = r;R ; + = r;L (5) with R;L being labelled by η = ±1 below, and unit with ~ = 1 are used. Then the LL part of the action, which in- where L; R refer to the left- and right movers, as be- cludes the electron kinetic energy and the density-density fore, and `; r to the left and right subsystems. We have interaction, has the following form: assumed the tunnelling amplitude t0 to be the same in X Z 1 Z all channels. This is always the case for the SAG, but S = iξ ¯ (ξ) i@ (ξ) − iξ V n2(ξ) : (1) LL η η η 2 0 not necessarily for the RBG if it is implemented as an η=±1 asymmetric double-barrier. In the latter case the reso- nant properties could be different for t 6= t but we leave Here n ≡ ( ¯ + ¯ ) is the electron density, V 1 2 R R L L 0 it aside here. is the screened Coulomb interaction, iξ stands for ix it The action defined by Eqs. (1), (3) and (4) is quadratic and @η ≡ @t + ηvF @x. The free conduction electron 0 0 in fields d and φ which can thus be integrated out. The Green function, gη(ξ − ξ ), is defined by @ηgη(ξ − ξ ) = δ(ξ − ξ0) so that its Fourier transform is given by integration over the phonon fields results only in substi- tuting V0 in the action (1) by the dynamical coupling, −1 gη("; q) = [" + iδ sgn " − ηvF q] : (2) 2 V (ξ) = V0 + g D0(ξ) : (6) We use here the zero-temperature formalism: the only role played by temperature is providing an alternative Performing the integration over the field d(t) results in low-energy cutoff in the RG equations below. transforming the tunnelling term (4) in the action to the It is not particularly important for what follows how following one: we model the phonon action. We choose a model of 1D Z ~ X 0 ¯ 0 0 acoustic phonons linearly coupled to the electron density, ST = − it it µ(t)Σµ,ν (t − t ) ν (t ) ; assuming the phonon spectrum to be linear with a cutoff µ,ν (7) at the Debye frequency, !D = cqD . Then the phonon ^ vF Γ0 action (neglecting possible electron backscattering which Σ(^ ") = ; Γ^0 ≡ Γ0(1^ +σ ^x) ; " − "0 + iδ sgn " is justified for T !D) has the following form: Z where Σ(t − t0) is the Fourier transform of Σ("). Γ^ is the h 1 −1 i 0 Sel−ph = iξ − φ(ξ) D φ(ξ) + g φ(ξ) n(ξ) : (3) 2 0 matrix with all elements equal to the tunnelling rate Γ0 ≡ 2 −1 πν0jt0j , where ν0 = (πvF ) is the one-particle DoS of Here φ(ξ) is the phonon field, g the el-ph coupling con- the conduction electrons in the absence of interactions. stant and D0(ξ) the free phonon propagator with the The action given by Eqs. (1) (with V0 = 0) and (7) Fourier transform given by describes the resonant transmission through the Fermi 2 gas hybridized with the resonant level. The hybridiza- !q D0(!; q) = ;!q = cq : tion makes the electron Green function to acquire an !2 − !2 + iδ 0 0 q off-diagonal part, gµ(ξ − ξ ) ! Gµν (ξ; ξ ), describing 3 the resonance-induced backscattering in the SAG or left- transformation which introduces the new bosonic field to-right connection in the RBG.