Experimental Evidence for Resonant Tunneling in a Luttinger Liquid

Abstract

We have measured the low temperature conductance of a one-dimensional island embedded in a single mode quantum wire. The quantum wire is fabricated using the cleaved edge overgrowth technique and the tunneling is through a single state of the island. Our results show that while the resonance line shape fits the derivative of the Fermi function the intrinsic line width decreases in a power law fashion as the temperature is reduced. This behavior agrees quantitatively with Furusaki's model for resonant tunneling in a Luttinger-liquid.PACS numbers: 73.20. Dx, 73.23.Ad, 73.23.Ps, 73.50.Jt One-dimensional (1D) electronic systems are expected to show unique transport behavior as a consequence of the Coulomb interaction between carriers [1]. Unlike in two and three dimensions [2], where the Coulomb interaction affects the transport properties only perturbatively, in 1D it completely modifies the ground state from its well-known Fermi-liquid form and the Fermi surface is qualitatively altered even for weak interactions. Today, it is well established theoretically that the low temperature transport properties of interacting 1D-electron systems are described in terms of a Luttinger-liquid rather than a Fermi-liquid [3,4]. The difference between a Luttingerliquid and Fermi-liquid becomes dramatic already in the presence of a single impurity. According to Landauer's theory the conductance of a single channel wire with a barrier is given by G = |t| 2 · e 2 /h, where |t| 2 is the transmission probability through the barrier. This result holds even at finite temperatures, assuming the transmission probability is independent of energy, as is often the case for barriers that are sufficiently above or below the Fermi energy. In 1D, interactions play a crucial role in that they form charge density correlations. These correlations, similar in nature to charge density waves [5], are easily pinned by even the smallest barrier, resulting in zero transmission and, hence, a vanishing conductance at zero temperature. At finite temperatures the correlation length is finite and the conductance decreases as a power-law of temperature,Herewhere U is the Coulomb energy between particles and E F is the Fermi energy in the wire. Despite the vast theoretical understanding of Luttingerliquids only a handful of experiments have been interpreted using such models. For example, in clean semiconductor wires prepared by the cleaved edge overgrowth (CEO) method [6], contrary to theory, the conductance is suppressed from its universal value [7]. Although not fully understood this suppression is believed to be a result of Coulomb interactions that suppress the coupling between the reservoirs and the wire region. Other measurements done on weakly disordered wires [8] show a weak temperature dependence of the conductance that is attributed to the Coulomb forces between electrons in the wire. Finally, The strongest manifestation of interaction in the clean limit comes from tunneling experiments such as the one recently reported on single walled carbon na...

Singular or non-Fermi liquids

“…In e ect, the two-dimensional material breaks up into one-dimensional lines with weak inter-connections. 32 Related behavior is also found in numerical work [174] in the so-called, t -t-J model. 33 An important question for such models is the extent to which the interconnections between stripes are "irrelevant"-i.e., the coupled chains problem, which we brie y allude to in Section 4.10.…”

“…The end result is equivalent to Eqs. (28), (30) and (32). These methods may well turn out to be very important in ÿnding the structure of SFLs and in systematizing them.…”

“…(32). These collective modes can be characterized by the angular momentum ' of oscillation of the Fermi surface and whether the oscillation is symmetric "s" or antisymmetric "a" in spin.…”

Introduction to bosonization

“…Both models (1) and (4) can be solved exactly in one dimension (and only in one dimension) by means of the celebrated Bethe Ansatz [20,21]. However, though the Bethe Ansatz can give the spectrum of eigenvalues and eigenvectors (plus a bit more), there is still a lot of important information that it cannot give, such as correlation functions.…”

“…I follow through the mathematical details necessary to establish the bosonization technique of one-dimensional systems, which is by now a rigorous and mature method that underlies much of our understanding of these systems. It has found many applications in real quasi-one-dimensional systems such as quantum wires [1], carbon nanotubes [2] and edge states of the quantum Hall effect [3]. For the sake of motivation, I focus on two models: the Hubbard model of spin-½ ¾ fermions and the anisotropic (XXZ) Heisenberg spin-½ ¾ model.…”