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Europhysics Letters (EPL)

Tunnel density of states in a Luttinger constriction

Yuli V. Nazarov, A. A. Odintsov, D. V. Averin

Published Jan 20, 1997• 13 citations
Publisher

Abstract

We study Luttinger constriction that is a finite region of interacting electrons adiabatically connected to the reservoirs of non-interacting electrons. We have shown that, in contrast to an infinite Luttinger liquid, the density of states (DOS) remains finite at the Fermi surface although it is strongly suppressed in comparison with the non-interacting value. The finite-size effects give rise to an additional staircase structure in the energy dependence of the DOS at higher energies.

DOI
10.1209/epl/i1997-00133-6

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We study Luttinger constriction that is a finite region of interacting electrons adiabatically connected to the reservoirs of non-interacting electrons. We have shown that, in contrast to an infinite Luttinger liquid, the density of states (DOS) remains finite at the Fermi surface although it is strongly suppressed in comparison with the non-interacting value. The finite-size effects give rise to an additional staircase structure in the energy dependence of the DOS at higher energies.
We study the frequency and space dependence of the local tunneling density of states (DOS) of a Luttinger liquid (LL) connected to a superconductor. This coupling strongly modifies the single-particle properties of the LL. It significantly enhances the DOS near the Fermi level, whereas this quantity vanishes as a power law for an isolated LL. The enhancement is due to the interplay between electron-electron interactions and multiple backscattering processes of low-energy electrons at the interface between the LL and the superconductor. This anomalous behavior extends over large distances from the interface and may be detected by coupling normal probes to the system.
We develop a theory of tunneling spectroscopy of interacting electrons in a nonequilibrium quantum wire coupled to reservoirs. The problem is modeled as an out-of-equilibrium Luttinger liquid with spatially dependent interaction. The interaction leads to the renormalization of the tunneling density of states, as well as to the redistribution of electrons over energies. Energy relaxation is controlled by plasmon scattering at the boundaries between regions with different interaction strengths and affects the distribution function of electrons in the wire as well as that of electrons emitted from the interacting regions into noninteracting electrodes.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
In a recent Physical Review Letter, Oreg and Finkel'stein (OF) have calculated the electron density of states (DOS) for tunneling into a repulsive Luttinger liquid close to the location of an impurity. The result of their calculation is a DOS which is enhanced with respect to the pure system, and moreover diverging for not too strong repulsion. In this Comment we intend to show that OF's calculation suffers from a subtle flaw which, being corrected, results into a DOS not only vanishing at zero frequency but in fact suppressed in comparison with the DOS of a pure Luttinger liquid.
Tunneling density of states (DOS) in Luttinger liquid has a dip at zero energy, commonly known as the zero-bias anomaly. In the presence of a magnetic field, in addition to the zero-bias anomaly, the DOS develops two peaks separated from the origin by the Zeeman energy. We show that these finite-bias anomalies are characterized by a power-law behavior of the DOS and the differential conductance, and find the corresponding exponents at arbitrary strength of the electron-electron interaction. The developed theory is applicable to various kinds of quantum wires, including carbon nanotubes.
We examine the local density of states (DOS) at low energies numerically and analytically for the Hubbard model in one dimension. The eigenstates represent separate spin and charge excitations with a remarkably rich structure of the local DOS in space and energy. The results predict signatures of strongly correlated excitations in the tunneling probability along finite quantum wires, such as carbon nanotubes, atomic chains or semiconductor wires in scanning tunneling spectroscopy (STS) experiments. However, the detailed signatures can only be partly explained by standard Luttinger liquid theory. In particular, we find that the effective boundary exponent can be negative in finite wires, which leads to an increase of the local DOS near the edges in contrast to the established behavior in the thermodynamic limit.