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arXiv:1010.1580 · Physical Review B

Transport in carbon nanotubes: Two-level SU(2) regime reveals subtle competition between Kondo and intermediate valence states

C. A. Büsser, E. Vernek, P. A. Orellana, Gustavo A. Lara, E. H. Kim, Adrian E. Feiguin, E. V. Anda, George Balster Martins

Published Mar 14, 2011• 29 citations
arXivHTMLPDF

Abstract

In this work, we use three different numerical techniques to study the charge transport properties of a system in the two-level SU(2) (2LSU2) regime, obtained from an SU(4) model Hamiltonian by introducing orbital mixing of the degenerate orbitals via coupling to the leads. SU(4) Kondo physics has been experimentally observed, and studied in detail, in carbon nanotube quantum dots. Adopting a two-molecular-orbital basis, the Hamiltonian is recast into a form where one of the molecular orbitals decouples from the charge reservoir, although still interacting capacitively with the other molecular orbital. This basis transformation explains in a clear way how the charge transport in this system turns from double- to single-channel when it transitions from the SU(4) to the 2LSU2 regime. The charge occupancy of these molecular orbitals displays gate-potential-dependent occupancy oscillations that arise from a competition between the Kondo and the intermediate valence states. The determination of whether the Kondo or the intermediate valence state is more favorable, for a specific value of gate potential, is assessed by the definition of an energy scale ${T}_{0}$, which is calculated through density matrix renormalization group (DMRG). We speculate that the calculation of ${T}_{0}$ may provide experimentalists with a useful tool to analyze correlated charge transport in many other systems. For that, a current work is under way to improve the numerical accuracy of its DRMG calculation and explore different definitions.

DOI
10.1103/physrevb.83.125404

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•Jan 1, 2011•0 citations
In this work, we use three different numerical techniques to study the charge transport properties of a system in the two-level SU(2) (2LSU2) regime, obtained from an SU(4) model Hamiltonian by introducing orbital mixing of the degenerate orbitals via coupling to the leads. SU(4) Kondo physics has been experimentally observed, and studied in detail, in carbon nanotube quantum dots. Adopting a two-molecular-orbital basis, the Hamiltonian is recast into a form where one of the molecular orbitals decouples from the charge reservoir, although still interacting capacitively with the other molecular orbital. This basis transformation explains in a clear way how the charge transport in this system turns from double- to single-channel when it transitions from the SU(4) to the 2LSU2 regime. The charge occupancy of these molecular orbitals displays gate-potential-dependent occupancy oscillations that arise from a competition between the Kondo and the intermediate valence states. The determination of whether the Kondo or the intermediate valence state is more favorable, for a specific value of gate potential, is assessed by the definition of an energy scale T0{T}_{0}T0​, which is calculated through density matrix renormalization group (DMRG). We speculate that the calculation of T0{T}_{0}T0​ may provide experimentalists with a useful tool to analyze correlated charge transport in many other systems. For that, a current work is under way to improve the numerical accuracy of its DRMG calculation and explore different definitions.
We study finite-temperature properties of the Kondo effect in a carbon nanotube (CNT) quantum dot using the Wilson numerical renormalization group (NRG). In the absence of magnetic fields, four degenerate energy levels of the CNT consisting of spin and orbital degrees of freedom give rise to the SU(4) Kondo effect. We revisit the universal scaling behavior of the SU(4) conductance for quarter- and half-filling in a wide temperature range. We find that the filling dependence of the universal scaling behavior at low temperatures TTT can be explained clearly with an extended Fermi-liquid theory. This theory clarifies that a T2{T}^{2}T2 coefficient of conductance becomes zero at quarter-filling, whereas the coefficient at half-filling is finite. We also study a field-induced crossover from the SU(4) to SU(2) Kondo state observed at the half-filled CNT dot. The crossover is caused by the matching of the spin and orbital Zeeman splittings, which lock two levels among the four at the Fermi level even in magnetic fields BBB. We find that the conductance shows the SU(4) scaling behavior at ${\ensuremath{\mu}}_{B}B<{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$ and it exhibits the SU(2) universality at ${\ensuremath{\mu}}_{B}B\ensuremath{\gg}{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$, where TKSU(4){T}_{K}^{\mathrm{SU}(4)}TKSU(4)​ is the SU(4) Kondo temperature. To clarify how the excited states evolve along the SU(4) to SU(2) crossover, we also calculate the spectral function. The results show that the Kondo resonance width of the two states locked at the Fermi level becomes sharper with increasing fields. The spectral peaks of the other two levels moving away from the Fermi level merge with atomic limit peaks for ${\ensuremath{\mu}}_{B}B\ensuremath{\gtrsim}{k}_{B}{T}_{K}^{\mathrm{SU}(4)}$.
We investigate theoretically the linear and nonlinear conductance through a nanostructure with two-fold degenerate single levels, corresponding to the transport through nanostructures such as a carbon nanotube, or double dot systems with capacitive interaction. It is shown that the presence of the interaction asymmetry between orbits/dots affects significantly the profile of the linear conductance at finite temperature, and, of the nonlinear conductance, particularly around half-filling, where the two-particle Kondo effect occurs. Within the range of experimentally feasible parameters, the SU(4) universal behavior is suggested, and comparison with relevant experiments is made.
We analyze, from a quantum information theory perspective, the possibility of realizing an SU(4) entangled Kondo regime in semiconductor double quantum dot devices. We focus our analysis on the ground-state properties and consider the general experimental situation where the coupling parameters of the two quantum dots differ. We model each quantum dot with an Anderson-type Hamiltonian including an interdot Coulomb repulsion and tunnel couplings for each quantum dot to independent fermionic baths. We find that the spin and pseudospin entanglements can be made equal, and the SU(4) symmetry recovered, if the gate voltages are chosen in such a way that the average charge occupancies of the two quantum dots are equal, and the double occupancy on the double quantum dot is suppressed. We present density matrix renormalization group numerical results for the spin and pseudospin entanglement entropies, and analytical results for a simplified model that captures the main physics of the problem.
In this thesis we deal with the description of quantum impurity systems. In order to achieve such an aim, we exploit in the first part of this work the Reduced Density Matrix (RDM) formalism, in the second one the Density Matrix-Numerical Renormalization Group (DM-NRG) method. The first half of the thesis deals with the derivation and the application of out-of-equilibrium and non-perturbative approximations for the dynamics of the Single Impurity Anderson Model (SIAM) in the perturbative, intermediate coupling and Kondo cross-over regimes. In this part of the thesis, the main aim is to develop new technical tools useful to extend the RDM formalism from the well established perturbative regime to the intermediate and, in the future, to the strong coupling regimes. In this part, we analyse the advantages and drawbacks of the so-called Dressed Second Order (DSO), Dressed DSO (DDSO) and the Resonant Tunnelling Approximation (RTA). In the DSO we include processes capable to describe the broadening of the Coulomb blockade peaks but the DSO itself, as the RTA, is not capable to describe the Kondo resonance width properly. In the DDSO we propose a prescription to solve this issue and to describe properly the exponential dependence of the Kondo resonance width. However, in the DDSO the vertex corrections are not included and this yields a spurious conductance peak in the empty orbital regime (Coulomb valley with zero electrons). For this reason we turn our attention to the inclusion of the vertex corrections within the RTA framework. In the second part, we describe transport and thermodynamic properties of carbon-nanotubes (CNTs) based quantum dots using the DM-NRG method. We present results on the spectral function and the linear conductance in cooperation with Catalin Pascu Moca and Gergely Zarand from the Universities of Oradea and Budapest, respectively. In order to model the CNTs we consider an extended Anderson Hamiltonian. Exploiting such an Hamiltonian we study the crossover from the SU(4) to the SU(2) Kondo effect. We demonstrate numerically and analytically that the Kondo temperature scales inversely with respect to the SU(4)-symmetry breaking parameter Δ\DeltaΔ, tending to the Kondo temperature of the SU(2) Kondo effect in the large Δ\DeltaΔ limit. Furthermore we study the CNT quantum dot in the presence of a parallel and perpendicular (with respect to the CNT axis) magnetic field. We find necessary conditions to observe the so-called ``Kondo revivals'' for specific values of the magnetic field. In the final part of this thesis, we model a real world device realised by Jean-Pierre Cleuziou, Ngoc-Viet Nguyen and Wolfgang Wernsdorfer in Grenoble by means of the previously studied Hamiltonian. For this system we search for the best set of parameters to reproduce the experimentally measured linear conductance and we study the underlying Kondo state evaluating the CNT added entropy and specific heat.
Carbon nanotube quantum dot has four-fold degenerate one-particle levels, which bring a variety to the Kondo effects taking place in a wide tunable-parameter space. We theoretically study an emergent SU(2) symmetry that is suggested by recent magneto-transport measurements, carried out near two electrons filling. It does not couple with the magnetic field, and emerges in the case where the spin and orbital Zeeman splittings cancel each other out in two of the one-particle levels among four. This situation seems to be realized in the recent experiment. Using the Wilson numerical renormalization group, we show that a crossover from the SU(4) to SU(2) Fermi-liquid behavior occurs as magnetic field increases at two impurity-electrons filling. We also find that the quasiparticles are significantly renormalized as the remaining two one-particle levels move away from the Fermi level and are frozen at high magnetic fields. Furthermore, we consider how the singlet ground state evolves during such a crossover. Specifically, we reexamine the SU(N) Kondo singlet for M impurity-electrons filling in the limit of strong exchange interactions. We find that the nondegenerate Fermi-liquid fixed point of Nozières and Blandin can be described as abosonic Perron–Frobenius vector for M composite pairs, each of which consists of one impurity-electron and one conduction-hole. This interpretation in terms of the Perron–Frobenius theorem can also be extended to the Fermi-liquid fixed-point without the SU(N) symmetry.
The first topic of this thesis is the study of many-body effects in an one-dimensional strongly correlated electronic system - the Kondo lattice model. This system is tackled numerically by means of the density matrix renormalization group, since analytic method, i.e., perturbation the- ory fail due to competing coupling constants. The Kondo lattice model consists of a conduction band of electrons which couple via a spin exchange coupling to a localized spin lattice. We study the spectral properties of the one-dimensional Kondo lattice model as a function of the exchange coupling, the band filling, and the quasimomentum in the ferromagnetic and paramagnetic phases. We compute the dispersion relation of the quasiparticles, their lifetimes, and the Z factor. The exact ground state and the quasiparticle-dispersion relation of the Kondo lattice model with one conduction electron are well known. The quasiparticle could be identified as the spin polaron. Our calculations of the dispersion relation for partial band fillings give a result similar to the one-electron case, which suggests that the quasiparticle in both cases is the spin polaron. We find that the quasiparticle lifetime differs by orders of magnitude between the ferromagnetic and paramagnetic phases and depends strongly on the quasimomentum. Further- more, we study the effects of the Coulomb interaction on the phase diagram, the static magnetic susceptibility and electron spin relaxation. We show that onsite Coulomb interaction supports ferromagnetic order and nearest neighbor Coulomb interaction drives, depending on the elec- tron filling, either a paramagnetic or ferromagnetic order. Furthermore, we calculate electron quasiparticle life times, which can be related to electron spin relaxation and decoherence times, and explain their dependence on the strength of interactions and the electron filling in order to find the sweet spot of parameters where the relaxation time is maximized. We find that effective exchange processes between the electrons dominate the spin relaxation and decoherence rate. In the second topic of this thesis, we numerically calculate the electron transport through carbon nanotube based quantum dot devices. We use a master equation’s approach in first order of the tunneling rate to the leads and an extended constant interaction model to model the carbon nanotube system. This work has been done in collaboration with two experimental groups and we compare their respective experimentally obtained data to our numerical calculations. In both collaborations striking similarity between the numerical data and the experimental data is found. In the first collaboration transport through a carbon nanotube peapod, i.e, a carbon nanotube filled with fullerenes, has been measured. We identify a small hybridization between a fullerene molecule and the surrounding carbon nanotube to be of crucial importance for the understanding of the transport data. In the second collaboration, electron transport through a carbon nanotube rope, i.e., a bundle of carbon nanotubes has been measured. Also here, hybridization between the different nanotubes plays a crucial role. Furthermore, an external magnetic field is applied, which enables the identification of specific spin states of the compound quantum dot system. This might be important for future applications of such devices in spin-dependent electronics.