Ferromagnetic Kondo Effect in a Triple Quantum Dot System

A simple device of three laterally coupled quantum dots, the central one contacted by metal leads, provides a realization of the ferromagnetic Kondo model, which is characterized by interesting properties like a nonanalytic inverted zero-bias anomaly and an extreme sensitivity to a magnetic field. Tuning the gate voltages of the lateral dots allows us to study the transition from a ferromagnetic to antiferromagnetic Kondo effect, a simple case of a Berezinskii-Kosterlitz-Thouless transition. We model the device by three coupled Anderson impurities that we study by numerical renormalization group. We calculate the single-particle spectral function of the central dot, which at zero frequency is proportional to the zero-bias conductance, across the transition, both in the absence and in the presence of a magnetic field.

Figure 1

A schematic representation of our device described by Eq. (1), with three quantum dots (in green). Only the central one, labelled as 0, is attached to metallic leads (in red).

Figure 2

NRG spectral function $A_0(\omega )$ on dot 0 as a function of $\delta ϵ$ for $U=0.3$, $t=0.03$, $\Gamma =0.02\pi$ with a flat conduction band density of states of half-bandwidth $D=1$. Note the transition from the ferromagnetic Kondo, signaled by a minimum of $A_0(\omega )$ at $\omega =0$, to the regular, antiferromagnetic Kondo, where instead $A_0(\omega )$ is maximum at $\omega =0$.

Figure 3

NRG spectral function $A_0(\omega )$ of the central dot 0, for increasing magnetic field $B$ (we take ${\mu }_B=1$ and $g=2$, hence $2B$ is the Zeeman splitting) and the same parameters of Fig. 2. Panel (a) refers to $\delta ϵ=0$, while panel (b) to $\delta ϵ=0.147$. In the ferromagnetic Kondo (FMK) regime, panel (a), the logarithmic dimple is immediately destroyed, and replaced by inelastic spin-flip excitations; in the antiferromagnetic (standard) Kondo regime, panel (b), where $T_K\simeq 7\times {10}^{-5}$, the Abrikosov-Suhl resonance is only split by a sufficiently large field $2g{\mu }_{B}B\sim k_B{T}_K$.

Figure 4

NRG spectral function $A_0(\omega )$ of the central dot 0, for increasing temperature $T$ and the same parameters of Fig. 2. Panel (a) refers to $\delta ϵ=0$, while panel (b) to $\delta ϵ=0.147$.

Figure 5

Entropy as a function of temperature with the same parameters as in Fig. 2.