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Physical review. B, Condensed matter

Two-channel versus infinite-channel Kondo models for the single-electron transistor

Gergely Zaránd, Gergely T. Zimányi, Frank K. Wilhelm

Published Sep 15, 2000• 28 citations
Publisher

Abstract

We investigate the low-temperature dynamics of single electron boxes and transistors close to their degeneracy point using renormalization-group methods. We show that intermode scattering is a relevant perturbation and always drives the system to the two-channel Kondo fixed point, where the two channels correspond to the real spins of the conduction electrons. However, the crossover temperature ${T}^{*},$ below which Matveev's two-channel Kondo scenario [K.A. Matveev, Phys. Rev. B 51, 1743 (1995)] develops decreases exponentially with the number of conduction modes in the tunneling junctions and is extremely small in most cases. Above ${T}^{*}$ the ``infinite channel model'' of G. Falci, G. Sch\"on, and G.T. Zimany, Phys. Rev. Lett. 74, 3257 (1995) turns out to be a rather good approximation. We discuss the experimental limitations and suggest an experimental setup to observe the multichannel Kondo behavior.

DOI
10.1103/physrevb.62.8137

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We investigate the low-temperature dynamics of single electron boxes and transistors close to their degeneracy point using renormalization group methods. We show that intermode scattering is a relevant perturbation and always drives the system to the two-channel Kondo fixed point, where the two channels correspond to the real spins of the conduction electrons. However, the crossover temperature T^*, below which Matveev's two-channel Kondo scenario [K.A. Matveev, Phys. Rev. B 51, 1743 (1995)] develops decreases exponentially with the number of conduction modes in the tunneling junctions and is extremely small in most cases. Above T^* the 'infinite channel model' of Ref.{onlinecite{Schoen}} turns out to be a rather good approximation. We discuss the experimental limitations and suggest a new experimental setup to observe the multichannel Kondo behavior.

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We perform numerical renormalization group (NRG) as well as analytical calculations for the two-channel Kondo (2CK) model to obtain the dependence of the Kondo temperature ${T}_{K}$ on the dimensionless (bare) spin exchange coupling $g$ over the complete parameter range from $g⪡1$ to $g⪢1$. We show that there exists a duality between the regimes of small and large coupling. It is unique for the two-channel model and enables a mapping between the strong and the weak coupling cases via the identification $g\ensuremath{\leftrightarrow}3∕(2g)$, implying an exponential dependence of ${T}_{K}$ on $1∕g$ and $g$, respectively, in the two regimes. This agrees quantitatively with our NRG calculations where we extract ${T}_{K}(g)$ over the complete parameter range and obtain a nonmonotonic ${T}_{K}(g)$ dependence, strongly peaked at the 2CK fixed point coupling ${g}^{*}$. These results may be relevant for resolving the long-standing puzzle within the 2CK interpretation of certain random defect systems as to why no broad distribution of ${T}_{K}$ is observed in those systems.

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We investigate the low-temperature dynamics of single electron boxes and transistors close to their degeneracy point using renormalization-group methods. We show that intermode scattering is a relevant perturbation and always drives the system to the two-channel Kondo fixed point, where the two channels correspond to the real spins of the conduction electrons. However, the crossover temperature T∗,{T}^{*},T∗, below which Matveev's two-channel Kondo scenario [K.A. Matveev, Phys. Rev. B 51, 1743 (1995)] develops decreases exponentially with the number of conduction modes in the tunneling junctions and is extremely small in most cases. Above T∗{T}^{*}T∗ the ``infinite channel model'' of G. Falci, G. Sch"on, and G.T. Zimany, Phys. Rev. Lett. 74, 3257 (1995) turns out to be a rather good approximation. We discuss the experimental limitations and suggest an experimental setup to observe the multichannel Kondo behavior.
We investigate the low-temperature dynamics of single electron boxes and transistors close to their degeneracy point using renormalization group methods. We show that intermode scattering is a relevant perturbation and always drives the system to the two-channel Kondo fixed point, where the two channels correspond to the real spins of the conduction electrons. However, the crossover temperature T^*, below which Matveev's two-channel Kondo scenario [K.A. Matveev, Phys. Rev. B 51, 1743 (1995)] develops decreases exponentially with the number of conduction modes in the tunneling junctions and is extremely small in most cases. Above T^* the 'infinite channel model' of Ref.{onlinecite{Schoen}} turns out to be a rather good approximation. We discuss the experimental limitations and suggest a new experimental setup to observe the multichannel Kondo behavior.
A detailed and comprehensive study of the one-impurity multichannel Kondo model is presented. In the limit of a large number of conduction electron channels k>>1, the low-energy fixed point is accessible to a renormalization group improved perturbative expansion in 1/k. This straightforward approach enables us to examine the scaling, thermodynamics and dynamical response functions in great detail and make clear the following features: (i) the criticality of the fixed point, (ii) the universal non-integer degeneracy, and (iii) that the compensating spin cloud has a spatial extent of the order of one lattice spacing.
We consider an atom in a double-well potential [parametrized by a two-level system (TLS)] interacting with screened conduction electrons. The tunneling of the TLS is assisted by the Fermi gas. Close to the strong-coupling fixed point the renormalization-group equations for the noncommutative model in the case of spinless fermions, but an arbitrary number of relevant orbital channels, lead to the n-channel Kondo problem with S=1/2. By solving the thermodynamic Bethe-ansatz equations for the n-channel Kondo problem numerically in the presence of a magnetic field, we discuss the low-temperature properties of the TLS close to the fixed point. The Zeeman splitting corresponds to the energy difference between the two positions of the tunneling atom. The atom is not localized at one of the potential minima. The susceptibility diverges as H\ensuremath{\rightarrow}0 and T\ensuremath{\rightarrow}0, indicating that the symmetric TLS is unstable against a local lattice deformation via coupling to phonons. The lattice distortion disappears above a critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​. The ground-state equilibrium situation of the TLS corresponds to H\ensuremath{\ne}0 and a Fermi-liquid picture applies. The values for the specific heat become very large as H\ensuremath{\rightarrow}0. For small fields the specific heat shows a double-peak structure, which is particularly pronounced for n=2.
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable ``zero-temperature'' fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.
The two-channel Kondo (2CK) effect with its exotic ground state properties has remained difficult to realize in physical systems. At low energies, a quantum impurity with orbital degree of freedom, like a proton bound in an interstitial lattice space, comprises a 3-level system with a unique ground state and (at least) doubly degenerate rotational excitations with excitation energy Delta0Delta_0Delta0​. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings JJJ), while the electron spin is conserved. We show by extensive numerical renormalization group (NRG) calculations that without fine-tuning of parameters this system exhibits a 2CK fixed point, due to Kondo correlations in the excited-state doublet whose degeneracy is stabilized by the host lattice parity, while the channel symmetry (electron spin) is guaranteed by time reversal symmetry. We find a pronounced plateau in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between the high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the 2CK ground state value, S(0)=kBlnsqrt2S(0) = k_B lnsqrt{2}S(0)=kB​lnsqrt2. This indicates a downward renormalization of the doublet below the non-interacting ground state, thus realizing the 2CK fixed point, in agreement with earlier conjectures. We mapped out the phase diagram of the model in the J−Delta0J-Delta_0J−Delta0​ plane. The Kondo temperature TKT_KTK​ shows non-monotonic JJJ-dependence, characteristic for 2CK systems.newline indent Beside the two-channel Kondo effect of the model, we also study the single-channel version, which is realized by applying a strong magnetic field to the conduction band electrons so that their degeneracy is lifted and consequently having only one kind of electrons scattering off the impurity. This single-channel case is easier to analyze since the Hilbert space is not as large as that of the 2CK. We equally find a downward renormalization of the excited state energy by the Kondo correlations in the SU(2) doublet. In a wide range of parameter values this stabilizes the single-channel Kondo fixed point and a phase diagram is also mapped out for the model. In the single-channel version a plateau is found in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the single-channel Kondo ground state value, S(0)=kBln1S(0) = k_B ln 1S(0)=kB​ln1.
We perform numerical renormalization group (NRG) as well as analytical calculations for the two-channel Kondo (2CK) model to obtain the dependence of the Kondo temperature TK{T}_{K}TK​ on the dimensionless (bare) spin exchange coupling ggg over the complete parameter range from g⪡1g⪡1g⪡1 to g⪢1g⪢1g⪢1. We show that there exists a duality between the regimes of small and large coupling. It is unique for the two-channel model and enables a mapping between the strong and the weak coupling cases via the identification $g\ensuremath{\leftrightarrow}3∕(2g)$, implying an exponential dependence of TK{T}_{K}TK​ on 1∕g1∕g1∕g and ggg, respectively, in the two regimes. This agrees quantitatively with our NRG calculations where we extract TK(g){T}_{K}(g)TK​(g) over the complete parameter range and obtain a nonmonotonic TK(g){T}_{K}(g)TK​(g) dependence, strongly peaked at the 2CK fixed point coupling g∗{g}^{*}g∗. These results may be relevant for resolving the long-standing puzzle within the 2CK interpretation of certain random defect systems as to why no broad distribution of TK{T}_{K}TK​ is observed in those systems.