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arXiv:cond-mat/0511182 · Physical Review B

Numerical renormalization group approach to a quartet quantum-dot array connected to reservoirs: Gate-voltage dependence of the conductance

Yunori Nisikawa, Akira Oguri

Published Mar 15, 2006• 34 citations
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Abstract

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 ${ϵ}_{d}$ which corresponds to the gate voltage, the conductance shows alternatively wide peaks and valleys. Simultaneously, the total number of electrons ${N}_{\mathrm{el}}$ in the four dots shows a quantized staircase behavior due to a large Coulomb interaction $U$. The conductance plateaus of the unitary limit emerging for odd ${N}_{\mathrm{el}}$ are caused by the Kondo effect. The valleys of the conductance emerge for even ${N}_{\mathrm{el}}$, 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.

DOI
10.1103/physrevb.73.125108

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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 \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ between the chain and leads. We also discuss the parallel conductance for the array connected to four leads.
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.
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 study transport through a double quantum dot with interdot hopping ttt, intradot repulsion UUU and interdot repulsion U′U^{\prime}U′, using the numerical renormalization group (NRG) method. At half-filling, the conductances in two-terminal series and four-terminal parallel configuration are calculated via two phase shifts for quasi-particles of double quantum dot connected to two noninteracting leads with hybridization strength ΓΓΓ. For small values of t/Γt/Γt/Γ and U′/UU^{\prime}/UU′/U, conductance in the two-terminal series configuration is suppressed to almost zero. In this region, plateau of conductance in the four-terminal parallel configuration appears and almost reaches a unitary limit value 4e2/h4e^{2}/h4e2/h of two conducting modes. For large values of t/Γt/Γt/Γ or U′/UU^{\prime}/UU′/U, both conductances are suppressed to almost zero. The conductance in the two-terminal series configuration almost reaches a unitary limit value 2e2/h2e^{2}/h2e2/h only around cross-over regions of electron-configuration in double quantum dot. Through the behavior of the local charge and some thermodynamic quantities, we discuss the relation between transport and electron-configuration.
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 study the low-energy properties of a Hubbard chain of finite size N_C connected to two noninteracting leads using the numerical renormalization group (NRG) method. The results obtained for N_C = 3 and 4 show that the low-lying eigenstates have one-to-one correspondence with the free quasi-particle excitations of a local Fermi liquid. It enables us to determine the transport coefficients from the fixed-point Hamiltonian. At half-filling, the conductance for even N_C decreases exponentially with increasing U showing a tendency towards the development of a Mott-Hubbard gap. In contrast, for odd N_C, the Fermi-liquid nature of the low-energy states assures perfect transmission through the Kondo resonance. Our formulation to deduce the conductance from the fixed-point energy levels can be applied to various types of interacting systems.
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.