Conductance and Kondo Interference beyond Proportional Coupling

The transport properties of nanostructured systems are deeply affected by the geometry of the effective connections to metallic leads. In this work we derive a conductance expression for a class of interacting systems whose connectivity geometries do not meet the Meir-Wingreen proportional coupling condition. As an interesting application, we consider a quantum dot connected coherently to tunable electronic cavity modes. The structure is shown to exhibit a well-defined Kondo effect over a wide range of coupling strengths between the two subsystems. In agreement with recent experimental results, the calculated conductance curves exhibit strong modulations and asymmetric behavior as different cavity modes are swept through the Fermi level. These conductance modulations occur, however, while maintaining robust Kondo singlet correlations of the dot with the electronic reservoir, a direct consequence of the lopsided nature of the device.

Figure 1

(a) Experimental $\mathrm{dot}+\text{cavity}$ system; the cavity has radius $\ell$ and aperture ${\theta }_C$. (b) Schematic of single-level dot (${ϵ}_d$) coupled to multimode cavity (${ϵ}_j$). The dot is connected to leads ($L$ and $R$), while the cavity is only coupled to the $R$ lead; coupling matrix elements are indicated. (c) Cavity modes for ${\theta }_C=\pi$ are described by Bessel modes ${\psi }_{n,j}(r,\theta )$. The $n=1$ modes dominate the LDOS at $r\approx 0$. (d) Kwant mode simulation for finite aperture cavity (${\theta }_C=\pi /2$) coupled to wide leads shows good agreement with Bessel modes.

Figure 2

NRG-calculated dot spectral density $A_d(\omega )$ for cavity gate voltages ${\epsilon }_c$ in the weak [$\mathrm{\Omega }=0.01D$, (a),(c)] and strong coupling [$\mathrm{\Omega }=0.15D$, (b),(d)] regimes. Panels (c) and (d) show $A_d(0)$ vs ${\epsilon }_c$ (cuts through the horizontal dashed lines). Peaks in $A_d(0)$ correspond to dips in $\mathrm{\Delta }(0)$ and vice versa [18]. Insets show typical Kondo peaks in $A_d(\omega )$, present even when cavity modes dominate $\mathrm{\Delta }(0)$ (vertical dotted lines).

Figure 3

Conductance $G/G_0$ vs cavity gate voltages ${\epsilon }_c$ with cavity-dot couplings ranging from the weak ($\mathrm{\Omega }=0.01D$) to the strong coupling regime ($\mathrm{\Omega }=0.2D$) for (a) $T=0$ and (b) $T=0.031U$ (or $T=250\mathrm{mK}$ for $U=0.7\mathrm{meV}$). Inset: Kondo temperature as a function of cavity-dot coupling $\mathrm{\Omega }$ for ${\epsilon }_c=-0.9D$.