Low-temperature behavior of transmission phase shift across a Kondo correlated quantum dot

We study the transmission phase shift across a Kondo correlated quantum dot in a GaAs heterostructure at temperatures below the Kondo temperature $\left(T<T_{\mathrm{K}}\right)$, where the phase shift is expected to show a plateau at $\pi /2$ for an ideal Kondo singlet ground state. Our device is tuned such that the ratio $\mathrm{\Gamma }/U$ of level width $\mathrm{\Gamma }$ to charging energy $U$ is quite large ($\lesssim 0.5$ rather than $\ll 1$). This situation is commonly used in GaAs quantum dots to ensure Kondo temperatures large enough ($\simeq 100$ mK here) to be experimentally accessible; however, it also implies that charge fluctuations are more pronounced than typically assumed in theoretical studies focusing on the regime $\mathrm{\Gamma }/U\ll 1$ needed to ensure a well-defined local moment. Our measured phase evolves monotonically by $\pi$ across the two Coulomb peaks, but without being locked at $\pi /2$ in the Kondo valley for $T\ll T_{\mathrm{K}}$, due to a significant influence of large $\mathrm{\Gamma }/U$. Only when $\mathrm{\Gamma }/U$ is reduced sufficiently does the phase start to be locked around $\pi /2$ and develops into a plateau at $\pi /2$. Our observations are consistent with numerical renormalization group calculations, and can be understood as a direct consequence of the Friedel sum rule that relates the transmission phase shift to the local occupancy of the dot, and thermal average of a transmission coefficient through a resonance level near the Fermi energy.

Figure 1

(a) A scanning electron micrograph of the device including the measurement setup. The dotted green lines mimic the electron trajectories. The currents flowing through the lower contacts are measured through voltage measurements $[I_{1\left(2\right)}=V_{1\left(2\right)}/R,R=10\mathrm{k}\mathrm{\Omega }]$. (b) Typical quantum interference as a function of the magnetic field applied perpendicularly to the surface. Nonoscillating background current is subtracted from the raw data and only the oscillating component is shown. The visibilities of the oscillations are $4\%\text{–}8\%$.

Figure 2

(a) Temperature dependence of the Coulomb peaks. The conductance at different temperatures is plotted with different colors. (b) Coulomb diamond for the Coulomb peaks shown in (a) at $T=70\mathrm{mK}$.

Figure 3

(a) Magneto-oscillations of $\left(I_1-I_2\right)$ as a function of $V_{\mathrm{p}}$ across the two Coulomb peaks showing Kondo correlations. The nonoscillating background as a function of the magnetic field is subtracted. (b)–(d) The transmission phase shift (red circle, left axis), determined by a complex FFT of the magneto-oscillations, together with $I_1$ (black line, right axis). $I_1$ is averaged over one oscillation period of the magnetic field. From (b) to (d) the coupling between the QD and the leads is reduced.

Figure 4

Transmission phase (red line, left axis) and transmission amplitude (black line, right axis) calculated by a two-level Anderson impurity model. The single-level spacing is fixed to $0.3U$ and the orbital parity relation between two single-particle levels is chosen to be same. The phase shift across two CPs, which originates from the upper energy level, is shown here. The parameters used for the calculations are indicated in the graphs. $T_{\mathrm{K}}^{\mathrm{c}}$ indicates the Kondo temperature at the center of the Kondo valley.