Kondo cloud conductance in cavity-coupled quantum dots with asymmetric barriers

TL;DR

The study addresses the spatial extent of the Kondo screening cloud in a quantum dot coupled to a finite-length Fabry-Pérot cavity, enabling controlled density of states modulation at the Fermi level. It reports Kondo temperature oscillations with cavity resonances, implying a Kondo length of order a micron, with $ξ_K = ħ v_F / (k_B T_K)$. Barrier asymmetry between the dot reservoirs controls the phase and amplitude of the conductance oscillations: the zero-bias conductance can be in-phase or out-of-phase with the $T_K$ oscillations depending on whether $Γ_R$ exceeds $Γ_L$. A simple numerical model with $Γ_R = Γ_{R∞} (1 + α \, cos(2 k_F L))$ reproduces the observed patterns and explains the π phase shift when asymmetry is reversed; bias-scaling of the conductance provides a robust measurement of $T_K$ in the presence of a structured DOS.

Abstract

The Kondo effect emerges when a localized spin is screened by conduction electrons, giving rise to a strongly-correlated many-body ground state. In this work, we investigate this phenomenon in a GaAs/AlGaAs quantum dot, focusing on the spatial extension of the Kondo screening cloud in the electron reservoirs. To probe its properties, the dot is coupled to an electronic Fabry-Pérot interferometer, enabling controlled modulation of the density of states at the Fermi level. The observation of Kondo temperature oscillations indicates a Kondo screening length comparable to the cavity size. Furthermore, we explore how the coupling asymmetry with the two reservoirs affects both the amplitude and the phase of the conductance oscillations, revealing a subtle interplay between coherent transport and Kondo effect.

Figures (4)

• Figure 1: Principle of the Kondo cloud investigation in a dot-cavity device. (a) The gate voltages $V_{\rm DL}$, $V_{\rm P}$, $V_{\rm UL}$, $V_{\rm UR}$ and $V_{\rm DR}$ control the QD potential and tunnel barriers, while $V_{\rm mirror}$ and $V_{\rm cav}$ control the cavity. Other gates are kept at zero voltage. One of the QD reservoirs is connected to the RF measurement setup, while the reservoir on the right is a grounded FP cavity. An electron microscope image of the device is shown in SM Fig. S17. (b) Simplified schematic showing the three regions of interest and their energy diagrams. $\Gamma_{\rm L}$ and $\Gamma_{\rm R}$ represent the tunneling rates between the QD and the reservoirs, $U$ the QD charging energy, $T_{\rm K}$ the Kondo temperature and $\Delta$ the cavity level spacing. The DOS in the FP cavity is shown for two different values of $V_{\rm cav}$ corresponding to large (max) and small (min) tunneling rate $\Gamma_{\rm R}$. (c) Stability diagram obtained by measuring the differential conductance as a function of gate and bias voltages (other parameters are given in SM Table S1). Coulomb blockade diamonds are highlighted by dashed lines. The arrow indicates the Kondo ridge around zero bias. (d) Kondo peak in the center of the Kondo valley indicated in (c), plotted as a function of the cavity gate voltage which generates interference fringes. (e) Kondo peak along the diagonal dashed lines in (d) after rescaling the bias voltage to extract $T_{\rm K}$ from the width of the conductance peak.
• Figure 2: Kondo temperature oscillations for opposite barrier asymmetries. (a) Zero-bias conductance map of the QD as a function of two gate voltages tuning both the charge number and the relative coupling to the two reservoirs. The white stripes correspond to Coulomb blockade peaks (the stability diagram in absence of cavity is shown in SM Fig. S9b). The red dashed line indicates the Kondo valley where the investigation is performed. Along this line, from left to right, the barrier asymmetry changes from a smaller to a larger coupling to the cavity. (b) Evolution of the zero-bias conductance (blue dots) along the dashed line in (a). The tunneling rate ratio (green dots) is equal to 1 when the conductance is maximum (dashed line). (c,d) Kondo temperature (blue dots) and zero-bias conductance (red dots) as a function of the gate voltage tuning the interference in the cavity. These two quantities oscillate out of phase in (c) and in phase in (d) due to opposite barrier asymmetries, as represented in the schematics.
• Figure 3: Sign inversion of conductance oscillations for opposite barrier asymmetry. (a) Zero-bias conductance map of the QD as a function of two gate voltages tuning both the charge number and the relative coupling to the two reservoirs. The stability diagram along the grey dashed line is shown in Fig. \ref{['fig:fig1']}c. (b) Evolution of the zero-bias conductance oscillations (tuned by $V_{\rm cav}$) as the barrier asymmetry is varied along the red dashed line in (a). (c) Blue dots: zero-bias conductance averaged over the oscillations shown in (b). Red squares: phase of the conductance oscillations shown in (b) after subtraction of the slope indicated by the orange dashed line. A phase shift by $\pi$ is observed when the conductance is maximum for symmetric barriers (black dashed line).
• Figure 4: Numerical modeling of $G$ and $T_{\rm K}$ oscillations. (a) Kondo temperature normalized to its value in absence of cavity, as a function of tunneling rate ratio and cavity length normalized to the Fermi wavelength. (b) Conductance of the dot-cavity system normalized to its value in absence of cavity, as a function of the same parameters. (c,d) Conductance (blue line) and Kondo temperature (red line) versus cavity length, for barrier asymmetries indicated by dashed lines in (a,b).