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AIP conference proceedings

Kondo Effect in a Triple Quantum-Dot Array

Akira Oguri, Yunori Nisikawa, Alex C. Hewson

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We study the ground‐state properties of a triple quantum dot based on a three‐site Hubbard model connected to two non‐interacting leads. Using the numerical renormalization group (NRG), the many‐body phase shifts and dc conductance are calculated away from half‐filling as a function of the onsite energy εd, which corresponds to the gate voltage. The results for conductance g show the typical Kondo plateaus of the Unitary limit g ≈ 2e2/h at some finite ranges of the gate‐voltage corresponding to odd Nel (≈ 1, 3 and 5), where Nel is the total number of electrons in the triple dot. The conductance shows broad minima for even Nel(≈ 2 and 4). The local charge Nel shows a staircase behavior as a function of εd.

DOI
10.1063/1.2355217

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We study the ground‐state properties of a triple quantum dot based on a three‐site Hubbard model connected to two non‐interacting leads. Using the numerical renormalization group (NRG), the many‐body phase shifts and dc conductance are calculated away from half‐filling as a function of the onsite energy εd, which corresponds to the gate voltage. The results for conductance g show the typical Kondo plateaus of the Unitary limit g ≈ 2e2/h at some finite ranges of the gate‐voltage corresponding to odd Nel (≈ 1, 3 and 5), where Nel is the total number of electrons in the triple dot. The conductance shows broad minima for even Nel(≈ 2 and 4). The local charge Nel shows a staircase behavior as a function of εd.
We describe a formulation to deduce the phase shifts, which determine the ground-state properties of interacting quantum-dot systems with the inversion symmetry, from the fixed-point eigenvalues of the numerical renormalization group (NRG). Our approach does not assume the specific form of the Hamiltonian nor the electron-hole symmetry, and it is applicable to a wide class of quantum impurities connected to noninteracting leads. We apply the method to a triple dot which is described by a three-site Hubbard chain connected to two noninteracting leads, and calculate the dc conductance away from half-filling. The conductance shows the typical Kondo plateaus of Unitary limit in some regions of the gate voltages, at which the total number of electrons N_el in the three dots is odd, i.e., N_el =1, 3 and 5. In contrast, the conductance shows a wide minimum in the gate voltages corresponding to even number of electrons, N_el = 2 and 4. We also discuss the parallel conductance of the triple dot connected transversely to four leads, and show that it can be deduced from the two phase shifts defined in the two-lead case.
The ground-state properties of quartet quantum-dot arrays are studied using the numerical renormalization group (NRG) method with a four-site Hubbard model connected to two noninteracting leads. Specifically, we calculate the conductance and local charge in the dots from the many-body phase shifts, which can be deduced from the fixed-point eigenvalues of NRG. As a function of the on-site energy ϵd{ϵ}_{d}ϵd​ which corresponds to the gate voltage, the conductance shows alternatively wide peaks and valleys. Simultaneously, the total number of electrons Nel{N}_{\mathrm{el}}Nel​ in the four dots shows a quantized staircase behavior due to a large Coulomb interaction UUU. The conductance plateaus of the unitary limit emerging for odd Nel{N}_{\mathrm{el}}Nel​ are caused by the Kondo effect. The valleys of the conductance emerge for even Nel{N}_{\mathrm{el}}Nel​, and their width becomes substantially large at half-filling. It can be regarded as a kind of the Mott-Hubbard insulating behavior manifesting in a small system. These structures of the plateaus and valleys become weak for large values of the hybridization strength $\ensuremath{\Gamma}$ between the chain and leads. We also discuss the parallel conductance for the array connected to four leads.
Using Wilson's numerical renormalization group (NRG) technique we compute zero-bias conductance and various correlation functions of a double quantum dot (DQD) system. We present different regimes within a phase diagram of the DQD system. By introducing a negative Hubbard U on one of the quantum dots, we simulate the effect of electron-phonon coupling and explore the properties of the coexisting spin and charge Kondo state. In a triple quantum dot (TQD) system a multi-stage Kondo effect appears where localized moments on quantum dots are screened successively at exponentially distinct Kondo temperatures.
The transport property and phase transition for a parallel triple dot device are studied by adopting Wilson's numerical renormalization group technique, focusing on the effects of level spacings between neighboring dot sites. By keeping dot 2 at the half-filled level and tuning the level differences, it is demonstrated that the system transits from local spin quadruplet to triplet and doublet sequently, and three kinds of Kondo peaks at the Fermi surface could be found, which are separated by two Kosterlitz–Thouless type quantum phase transitions and correspond to spin–3/2, spin–1, and spin–1/2 Kondo effect, respectively. To obtain a detailed understanding of these problems, the charge occupation, the spin—spin correlation, the transmission coefficient, and the temperature-dependent magnetic moment are shown, and necessary physical arguments are given.
Using exact diagonalization of small clusters and Dyson equation embedding techniques, the conductance GGG of linear arrays of quantum dots is investigated. The Hubbard interaction induces Kondo peaks at low temperatures for an odd number of dots NNN and a filling of one particle per dot. Remarkably, for N=3,5,…N=3,5,\dots{}N=3,5,… the Kondo peak is split in half by a deep minimum, and the conductance vanishes at the gate voltage that induces particle-hole symmetry in the system. Tentative explanations for this unusual effect are proposed, including an interference process between two channels contributing to GGG, with one more and one less particle than the exactly solved cluster ground state. The Hubbard interaction and fermionic statistics of electrons also appear to be important to understand this phenomenon. Although most of the calculations used a particle-hole symmetric Hamiltonian and formalism, results also presented here show that the conductance dip exists even when this symmetry is broken. The conductance cancellation effect obtained using numerical techniques is potentially interesting, and other many-body techniques should be used to confirm its existence.
We investigate electronic transport through a parallel double quantum dot (DQD) system with strong on-site Coulomb interaction and capacitive interdot coupling. By applying the numerical renormalization group (NRG) method, the ground state of the system and the transmission probability at zero temperature have been obtained. For a system of quantum dots with degenerate energy levels and small interdot tunnel coupling, the spin correlations between the DQDs is ferromagnetic and the ground state of the system is a spin-1 triplet state. The linear conductance will reach the unitary limit (2e(2)/h) due to the underscreened Kondo effect at low temperature. As the interdot tunnel coupling increases, there is a quantum phase transition from ferromagnetic to antiferromagnetic spin correlation in DQDs and the linear conductance is strongly suppressed.