Kondo screening cloud and charge staircase in one-dimensional mesoscopic devices

We propose that the finite size of the Kondo screening cloud, ${\xi }_K$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\mu$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\mu$ that are controlled by the Kondo effect when the system size $L$ is comparable to ${\xi }_K$. 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 ${\xi }_K∕L$. 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.

Figure 1

Possible experimental setup. $V_{dw}$ controls the tunneling $t^{\prime }$ between the small dot (on the left) and the wire and $V_g$ varies the chemical potential in the wire.

Figure 2

(Color online) Charge quantization steps for the wire coupled to a small quantum dot. The arrows indicate the direction in which the single steps move as $\lambda \left(L\right)$ grows.

Figure 3

(Color online) Universal ratio $R=\delta {\mu }^o∕\delta {\mu }^e$ as a scaling function of ${\xi }_K∕L$. Bethe ansatz results obtained for various systems sizes $(N_0=51,101,201,501,1001,2001)$ and 13 different values of the Kondo exchange $J$, indicated by different symbols. For each value of $J$, the system lengths $L$ have been rescaled $L\to L∕{\xi }_K\left(J\right)$ in order to obtain the best collapse of the data using the strong coupling curve [Eq. (28)] (dashed red line) as a support for the rest of the collapse. The weak coupling regime for $L⪡{\xi }_K$, enlarged in inset (a), is described by the weak coupling expansion [Eq. (21)] with $\text{constant}\simeq 0.33$ (continuous blue curve). Inset (b): The Kondo length scale, extracted from the universal data collapse of the main panel (black squares), is described by the exponential fit [Eq. (31)] with ${\xi }_0\simeq 0.31$ (dashed green line).

Figure 4

Schematic picture for an open Heisenberg chain coupled to a spin impurity (arrow) at the left boundary via a weak antiferromagnetic exchange $J_K^{\prime }$ (dashed bond).

Figure 5

(Color online) Effect of the marginally irrelevant bulk operator on the scaling function $f\mathbf{(}{\xi }_K∕L,g\left(L\right)\mathbf{)}$ calculated from the Bethe ansatz. The symbols correspond to the noninteracting case, $g\left(L\right)=0$, as in Fig. 3. The black circles are obtained with the Bethe ansatz solution of the spin chain Kondo model (60) for $L=1000$ $[g\left(1000\right)\simeq 0.115]$.

Figure 6

(Color online) Flux dependence of the charge steps in the ring with an embedded quantum dot (inset), assuming no potential scattering $(V=0)$. Dashed lines: weak coupling regime $({\xi }_K⪢L)$; solid lines: strong coupling regime $({\xi }_K⪡L)$.

Figure 7

(Color online) Persistent current for the embedded quantum dot with $N$ electrons, $N\mathrm{mod}4=0$, in the strong coupling limit. The dashed line represents the sawtooth shape expected for periodic boundary conditions at the strong coupling fixed point. The solid line is the result of Eq. (94) for $L∕{\xi }_K=20$.

Figure 8

Occupation of the energy levels of the embedded quantum dot in the strong coupling limit for $N=4(p+n)+3$ and $\alpha \approx 0$. The pair of states with energy ${ϵ}_n^R\approx {ϵ}_n^L$ is partially occupied by a single electron.