Competition between Kondo Screening and Indirect Magnetic Exchange in a Quantum Box

Nanoscale systems of metal atoms antiferromagnetically exchange coupled to several magnetic impurities are shown to exhibit an unconventional reentrant competition between Kondo screening and indirect magnetic exchange interaction. Depending on the atomic positions of the magnetic moments, the total ground-state spin deviates from predictions of standard Ruderman-Kittel-Kasuya-Yosida perturbation theory. The effect shows up on an energy scale larger than the level width induced by the coupling to the environment and is experimentally accessible by studying magnetic field dependencies.

Figure 1

Competition between Kondo screening and RKKY interaction in a quantum box. Local Kondo singlets (LKS) are formed for extremely strong $J$ (regime 4). With decreasing $J$, the energy to break up a Kondo singlet becomes exponentially small (3). Below $J_D$, RKKY coupling is dominant (2). Finite-size effects set in for $J<J_{\Delta }$ defined as the point where the bulk Kondo temperature equals the finite-size gap, $T_K=\Delta$. If the system is sufficiently large, we have $J_{\Delta }<J_D$. The singlet-formation energy is linear in $J$ for “good” and vanishes for “bad” sites (1): i.e., for sites where the $k_F$ conduction-electron wave function has a finite (vanishing) weight, the Kondo effect “wins” (RKKY wins). The remaining unscreened moments are subjected to nonlocal RKKY exchange subsequently. At a still lower energy scale $\delta$, determined by the residual coupling of the box to the environment, the Kondo scale becomes exponentially small again with a screening cloud leaking out into the environment (0) [9]. Here, for $J<J_{\delta }$, the RKKY coupling is dominant again [26, 27]. The arrow marks the range covered by our numerical calculations.

Figure 2

Interimpurity and local spin correlations showing reentrant competition between Kondo screening and RKKY coupling for the $M=3$ model (“bad-good-bad”) with increasing (see arrows) system size up to $L=201$ sites. Note the log scale for $J$. With decreasing $J$, different regimes 1–4 (cf. Fig. 1) are found: (4) Strong-coupling limit with local Kondo singlets. (3) Kondo screening dominates over RKKY interaction. (2) RKKY interaction is the leading energy scale for large systems ($J_D>J_{\Delta }$). (1) The linear Kondo scale dominates and leads to a screening of the central spin (coupled to a good site, green) while in a second step RKKY couples the remaining spins (at bad sites, red) to a nonlocal triplet.