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arXiv:0908.0589 · Physical Review B

Kondo effect in a side-coupled double quantum dot system embedded in a mesoscopic ring

I-Ling Tsai, Chung‐Hou Chung

Published May 4, 2010• 0 citations
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Abstract

We study finite-size effect of the Kondo screening cloud in a double quantum dot setup via a large-$N$ slave-boson mean-field theory. In this setup, one of the dots is embedded in a close metallic ring with a finite size $L$ and the other dot is side coupled to the embedded dot via an antiferromagnetic spin-spin-exchange coupling with a strength $K$. The antiferromagnetic coupling favors local spin-singlet state and suppresses the Kondo screening. The mean-field phase diagram as a function of $1/L$ and $K$ shows a crossover to the local spin-singlet ground state from the Kondo phase to occur at $K<{K}_{c}$ $(L=4n,4n+1,4n+3)$ and at $K>{K}_{c}$ $(L=4n+2)$. The effective Kondo temperature ${T}_{k}$ (proptotional to inverse of the Kondo screening cloud size) shows the Kosterlitz-Thouless (KT) scaling at finite sizes for $L=4n,4n+2$, indicating quantum transition of the KT type between the Kondo screened phase for $K\ensuremath{\le}{K}_{c}$ and a local spin-singlet phase for $K\ensuremath{\ge}{K}_{c}$ in the thermodynamic limit with ${K}_{c}$ being the critical value. The local density of states on the embedded quantum dot and the persistent current at finite sizes with different values of $K$ are calculated in the crossover region.

DOI
10.1103/physrevb.81.195305

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We study finite-size effect of the Kondo screening cloud in a double quantum dot setup via a large-NNN slave-boson mean-field theory. In this setup, one of the dots is embedded in a close metallic ring with a finite size LLL and the other dot is side coupled to the embedded dot via an antiferromagnetic spin-spin-exchange coupling with a strength KKK. The antiferromagnetic coupling favors local spin-singlet state and suppresses the Kondo screening. The mean-field phase diagram as a function of 1/L1/L1/L and KKK shows a crossover to the local spin-singlet ground state from the Kondo phase to occur at K<KcK<{K}_{c}K<Kc​ (L=4n,4n+1,4n+3)(L=4n,4n+1,4n+3)(L=4n,4n+1,4n+3) and at K>KcK>{K}_{c}K>Kc​ (L=4n+2)(L=4n+2)(L=4n+2). The effective Kondo temperature Tk{T}_{k}Tk​ (proptotional to inverse of the Kondo screening cloud size) shows the Kosterlitz-Thouless (KT) scaling at finite sizes for L=4n,4n+2L=4n,4n+2L=4n,4n+2, indicating quantum transition of the KT type between the Kondo screened phase for K\ensuremath≤KcK\ensuremath{\le}{K}_{c}K\ensuremath≤Kc​ and a local spin-singlet phase for K\ensuremath≥KcK\ensuremath{\ge}{K}_{c}K\ensuremath≥Kc​ in the thermodynamic limit with Kc{K}_{c}Kc​ being the critical value. The local density of states on the embedded quantum dot and the persistent current at finite sizes with different values of KKK are calculated in the crossover region.
We propose that the finite size of the Kondo screening cloud, ${\ensuremath{\xi}}_{K}$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\ensuremath{\mu}$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\ensuremath{\mu}$ that are controlled by the Kondo effect when the system size LLL is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.
By means of the numerical renormalization group method, I study the Kondo effect and quantum phase transitions in a double quantum dot system with on-site repulsion UUU, interdot repulsion V,V,V, and magnetic field BBB. At zero temperature and B=0B=0B=0, there is a local spin triplet-singlet transition of the Kosterlitz-Thouless type when VVV increases to a critical value ${V}_{c}\ensuremath{\approx}U$. With increasing BBB, a singlet-triplet transition of the second order occurs at a critical magnetic field ${B}_{c}\ensuremath{\approx}V\ensuremath{-}U$. The magnetic susceptibility and the first derivatives of the spin correlation and double occupation are expected to diverge at Bc{B}_{c}Bc​ and to have different critical exponents, which are universal for different parameters. An appropriate magnetic field can restore the Kondo effect.
Measurements of the persistent current in a ring containing a quantum dot would afford a unique opportunity to finally detect the elusive Kondo screening cloud. We present the first large-scale numerical results on this controversial subject using exact diagonalization and density matrix renormalization group (RG). These extremely challenging numerical calculations confirm RG arguments for weak to strong coupling crossover with varying ring length and give results on the universal scaling functions. We also study, analytically and numerically, the important and surprising effects of particle-hole symmetry breaking.
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.
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