Local Kondo temperatures in atomic chains

We study the effect of disorder in strongly interacting small atomic chains. Using the Kotliar-Ruckenstein slave-boson approach, we diagonalize the Hamiltonian via scattering matrix theory. We numerically solve the Kondo transmission and the slave-boson parameters that allow us to calculate the Kondo temperature. We demonstrate that in the weak disorder regime, disorder in the energy levels of the dopants induces a nonscreened disorder in the Kondo couplings of the atoms. We show that disorder increases the Kondo temperature of a perfect chain. We find that this disorder in the couplings comes from a local distribution of Kondo temperatures along the chain. We propose two experimental setups where the impact of local Kondo temperatures can be observed.

Figure 1

Tight-binding model for a chain of QDs coupled to conducting leads.

Figure 2

Averaged Kondo conductance over 2000 random configurations vs disorder for a chain of $N=3$ QDs for different strengths of the electron-electron interaction. The inset shows the curves lying on top of each other when disorder is renormalized by $U$.

Figure 3

Calculations were performed using a chain of three QDs with a Coulomb interaction of $U=10$. (a) Renormalized energy level of the QDs in the chain averaged over 2000 configurations. (b)–(d) Distribution of the quasiparticle weight $z_i^2$ for $i=1$ and $i=2$ for all the different system configurations using different values of disorder such that $W<U$.

Figure 4

(a) Comparison of the single impurity Kondo temperature calculated between the well known relation proposed by Haldane $T_{K,H}=\sqrt{2U\Gamma }{e}^{-\frac{\pi U}{8\Gamma }}$ (magenta dashed-dotted line) and the one for the KR slave-boson method used in our study $T_{K,\text{SB}}=z^2\Gamma$ (black solid line). (b)–(d) Local Kondo temperatures reported as $T_{K,i}=z_i^2\Gamma$ for a chain of $N=3$ dopants. (b) Symmetric nondisordered system. (c) Nondisordered chain attached to one lead. (d) Disordered chain attached to one lead. Energy levels were set at ${\epsilon }_F-{\epsilon }_1=0.5U,{\epsilon }_F-{\epsilon }_2=0.2U$, and ${\epsilon }_F-{\epsilon }_3=0.8U$.

Figure 5

(a) Scheme of the first proposed setup with a strong coupled lead (${\Gamma }_L=7$ meV) and a STM tip that has a tunable coupling to the chain (${\Gamma }_R$). (b) Scheme of the second setup with two equally coupled leads and a STM tip that serves as a local gate to the dopants. (c) Calculations using setup (a), $T_K$ as a function of the degree of symmetry coupling. The tip moves from site $i=3$ to $i=5$ and we model a chain without disorder and one with small disorder were ${\epsilon }_1\approx -0.6U,{\epsilon }_1\approx -0.51U,{\epsilon }_1\approx -0.52U,{\epsilon }_1\approx -0.52U$, and ${\epsilon }_1\approx -0.63U$. (d) Calculations using setup (b), $T_K$ vs a single dopant energy level.