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Physical review. B, Condensed matter

Resonant tunneling and low-energy impurity behavior in a resonant-level model

Vadim V. Ponomarenko

Published Aug 15, 1993• 30 citations
Publisher

Abstract

We study the effect of screening of an impurity charge by conduction electrons on resonant tunneling at arbitrary energy of the impurity level. We examine the exactly solvable model of the resonant level. We find the ground-state solutions of the model at all possible Coulomb interactions. The principle of equivalence between its universal behavior and the Kondo model is proved. However, the dependence of its scale invariant variable on the bare variables has different forms in two different regions of the parameter variations. This means that besides the well-known Kondo-like fixed point of the renormalization at weak Coulomb interaction a new stable point exists. The latter determines a singular low-energy behavior of the impurity. Based on the solution, the tunneling amplitude is calculated. It is shown that for a narrow gap or quasi-one-dimensional junction a severe renormalization of the one-electron half-width ${\mathrm{\ensuremath{\Gamma}}}_{0}$ occurs ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{R}}$\ensuremath{\propto}${\mathrm{\ensuremath{\Gamma}}}_{0}$(${\mathrm{\ensuremath{\Gamma}}}_{0}$/D${)}^{\mathrm{\ensuremath{\alpha}}/(1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}})}$. The comparison with perturbation results is discussed. Finally the conductance of the junction with randomly located impurities is calculated to clarify the crucial role of the non-Fermi-liquid behavior on the resonant tunneling under certain conditions.

DOI
10.1103/physrevb.48.5265

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We study the effect of screening of an impurity charge by conduction electrons on resonant tunneling at arbitrary energy of the impurity level. We examine the exactly solvable model of the resonant level. We find the ground-state solutions of the model at all possible Coulomb interactions. The principle of equivalence between its universal behavior and the Kondo model is proved. However, the dependence of its scale invariant variable on the bare variables has different forms in two different regions of the parameter variations. This means that besides the well-known Kondo-like fixed point of the renormalization at weak Coulomb interaction a new stable point exists. The latter determines a singular low-energy behavior of the impurity. Based on the solution, the tunneling amplitude is calculated. It is shown that for a narrow gap or quasi-one-dimensional junction a severe renormalization of the one-electron half-width \ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​ occurs \ensuremathΓR{\mathrm{\ensuremath{\Gamma}}}_{\mathit{R}}\ensuremathΓR​\ensuremath{\propto}\ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​(\ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​/D)\ensuremathα/(1\ensuremath−\ensuremathα){)}^{\mathrm{\ensuremath{\alpha}}/(1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}})})\ensuremathα/(1\ensuremath−\ensuremathα). The comparison with perturbation results is discussed. Finally the conductance of the junction with randomly located impurities is calculated to clarify the crucial role of the non-Fermi-liquid behavior on the resonant tunneling under certain conditions.
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In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
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We generalize the fermionic renormalization group method to describe analytically transport through a double barrier structure in a one-dimensional system. Focusing on the case of weakly interacting electrons, we investigate thoroughly the dependence of the conductance on the strength and the shape of the double barrier for arbitrary temperature T. Our approach allows us to systematically analyze the contributions to renormalized scattering amplitudes from different characteristic scales absent in the case of a single impurity, without restricting the consideration to the model of a single resonant level. Both a sequential resonant tunneling for high T and a resonant transmission for T smaller than the resonance width are studied within the unified treatment of transport through strong barriers. For weak barriers, we show that two different regimes are possible. Moderately weak impurities may get strong due to a renormalization by interacting electrons, so that transport is described in terms of theory for initially strong barriers. The renormalization of very weak impurities does not yield any peak in the transmission probability; however, remarkably, the interaction gives rise to a sharp peak in the conductance, provided asymmetry is not too high.
Tunneling impurities in metals have been known for some time to have the potential for exhibiting Kondo-like physics. However, previous models based on an impurity hopping between two equivalent positions have run into trouble due to the existence of relevant operators that drive the system away from the non-Fermi-liquid Kondo fixed point. In the case of an impurity hopping among positions with higher symmetry, such as triangular symmetry, it is shown here that the non-Fermi-liquid behavior at low temperatures can be generic. Using various bosonization techniques, the fixed point is shown to be stable. However, unlike the conventional two-channel Kondo (2CK) model, it has four leading irrelevant operators, implying that while the form of the singular temperature dependence of physical quantities is similar to the 2CK model, there will not be simple universal amplitude ratios. The phase diagram of this system is analyzed and a critical manifold is found to separate the non-Fermi-liquid from a conventional Fermi liquid fixed point. Generalization to higher symmetries, such as cubic, and the possibility of physical realizations with dynamic Jahn-Teller impurities are discussed.
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