Interacting Quantum Dot Coupled to a Kondo Spin: A Universal Hamiltonian Study

We study a Kondo spin coupled to a mesoscopic interacting quantum dot that is described by the “universal Hamiltonian.” The problem is solved numerically by diagonalizing the system Hamiltonian in a good-spin basis and analytically in the weak and strong Kondo coupling limits. The ferromagnetic exchange interaction within the dot leads to a stepwise increase of the ground-state spin (Stoner staircase), which is modified nontrivially by the Kondo interaction. We find that the spin-transition steps move to lower values of the exchange coupling for weak Kondo interaction, but shift back up for sufficiently strong Kondo coupling. The interplay between Kondo and ferromagnetic exchange correlations can be probed with experimentally tunable parameters.

Figure 1

(a) Schematic diagram of a small quantum dot with spin ${\mathbf{S}}_K$ (Kondo spin) that is coupled antiferromagnetically (coupling constant $J_k$) to a large quantum dot with spin ${\mathbf{S}}_d$. The large dot, described by the universal Hamiltonian, is characterized by a ferromagnetic exchange interaction (coupling constant $J_s$). We assume the large dot to have $N$ equally spaced single-particle levels (with spacing $\delta$) in a band of width $2D$ (half-filling). (b) The large dot is represented in the site basis (squares), in which ${\mathbf{S}}_K$ couples only to site 0.

Figure 2

Ground-state spin $S_{\mathrm{tot}}$ of the system in Fig. 1 at finite exchange $J_s$ and Kondo coupling $J_k$ for an odd (left column) and even (right column) number of electrons $N$. (a), (b) Transition curves for $N=11$ (left) and $N=12$ (right), separating regions of fixed $S_{\mathrm{tot}}$. Numerical results (solid lines) are compared with analytical estimates in the weak- and strong-coupling limits (dotted lines). (c), (d) Spin-transition curves for fixed bandwidth $2D$ but different $N$ (top to bottom): $N=5$, 7, 9, 11 (left) and $N=6$, 8, 10, 12 (right). For increasing $N$ (at $J_k\rho \gg 1$), the curves converge to the Stoner staircase of a dot with $N-1$ electrons (vertical arrows).

Figure 3

Spin-transition curves for $N=12$ at Zeeman field $B$ and Kondo coupling $J_k$. The exchange coupling is (a) $J_s=0$ and (b) $J_s/\delta =0.65$. Solid (dotted) lines show the numerical (perturbative) solutions.