Tunneling in one-dimensional non-Luttinger electron liquid.

The conductance of a weakly interacting electron gas in the presense of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction, a simple renormalization group approach can be used instead of the standard bosonization procedure. Our technique allows to take into account the backscattering of electrons that leads to a non-Luttinger-liquid behavior of the low-temperature conductance. In the presence of magnetic field, the backscattering may give rise to a peak in differential conductance at bias equal to the Zeeman energy. PACS numbers: 73.20.Dx, 73.40.Gk Typeset using REVTEX 1 Recent progress in semiconductor technology renewed the interest in transport properties of one-dimensional (1D) electron systems. It is well known that in a clean short channel the electron transport is ballistic, and the conductance is quantized [1]. The ballistic conductance is destroyed, however, by scattering on impurities in longer channels [2]. It is reasonable to expect that in a sufficiently clean system it is possible to form a long 1D channel with a single impurity. Transport properties of such a system are determined by scattering of electrons on the impurity. In the simplest case of non-interacting electrons the conductance is related to the transmission coefficient by Landauer formula [3]. Electron-electron interaction in a 1D electron gas renormalizes significantly the scattering caused by an impurity potential [4]. The low-energy properties of a 1D interacting electron system are usually described by Luttinger model (see, e.g., [5]). This approach allows to treat the renormalizations of the impurity potential at any strength of interaction between electrons [6]. The resulting conductance G demonstrates the power-law dependence on temperature at T → 0. However, this method does not allow to calculate the conductance at higher temperatures where the power-law asymptotics fails. Besides, for the system of spin2 electrons, the Luttinger model neglects backscattering in electron-electron collisions. The backscattering processes lead to additional renormalizations that affect the low-temperature conductance. Below we present an alternative renormalization group (RG) approach that is valid only in the case of weakly interacting electrons. The advantage of our approach is in its ability to describe the conductance at any temperature. For a model with spinless electrons it allows us to show explicitly the crossover from the Fermi-gas to the low-temperature Luttinger liquid behavior. For the realistic case of spin2 electrons, the backscattering processes can be incorporated into the renormalization procedure. Striking differences from Luttinger-liquid behavior occur in the case of a short-range interaction between electrons. The temperature dependence of conductance becomes non-monotonic, and application of a magnetic field creates a peak in the differential conductance at bias equal to the energy of the Zeeman splitting. We start with a non-interacting 1D electron gas in the presense of a single scatterer. In the simplest case of a small-size scatterer we model its potential by δ-function. The electron scattering is characterized by the transmission and reflection amplitudes t0 and r0. In terms of these amplitudes the scattering wave functions have the form: ψk(x) = 1 √ 2 { e + r0e , x < 0, t0e , x > 0, (1) ψ−k(x) = ψk(−x). (2) Here ψk and ψ−k describe the scattering of electrons incoming from the left and right correspondingly, k > 0. Scattering disturbs the electron density around the barrier. In the presense of electron-electron interaction this modulation of density leads to an additional scattering of electrons. If the interaction potential V (y − z) is weak, the correction to the wave functions (1) may be found in the Born approximation, δψk(x) = ∫ ∞ −∞ Gk(x, y)VH(y)ψk(y)dy − ∫∫ ∞ −∞ Gk(x, y)Vex(y, z)ψk(z)dzdy. (3)