Spectroscopy of the Kondo Problem in a Box

Motivated by experiments on double quantum dots, we study the problem of a single magnetic impurity confined in a finite metallic host. We prove an exact theorem for the ground state spin, and use analytic and numerical arguments to map out the spin structure of the excitation spectrum of the many-body Kondo-correlated state, throughout the weak to strong coupling crossover. These excitations can be probed in a simple tunneling-spectroscopy transport experiment; for that situation we solve rate equations for the conductance.

Figure 1

A double-dot system, coupled very weakly to leads (“L1” and “L2”). In the Coulomb blockade regime, the small dot “$S$” behaves as a spin $\mathcal{S}$ that is coupled to a finite reservoir “$R$” provided by the large dot. The weakly coupled leads are used to measure the excitation spectrum of the system.

Figure 2

Schematic illustration of the energy eigenvalues of the double-dot system Eq. (1) as a function of the coupling $\mathcal{J}$ for $N$ odd (left) and even (right). In the double-dot experiment proposed in the text, the $y$ axis is like $V_{\mathrm{BIAS}}$ and the $x$ axis is like $V_{\mathrm{PINCH}}$.

Figure 3

Universal form of the singlet-triplet gap ($N$ odd). Note the excellent data collapse upon plotting the extracted gaps as a function of $T_K/\Delta$. Inset: $P_{\mathrm{QMC}}(\beta )$ for $D/\Delta =41$. Circles are data for fixed $\mathcal{J}=0.05$, 0.10, 0.13, 0.15, 0.19, 0.23, 0.26, and 0.30 (right to left). Lines are one parameter fits to the two state single-triplet form.

Figure 4

Conductance through the $S\mathrm{\text{−}}R$ system of Fig. 1 as a function of the bias between the two leads for $B_{\mathrm{Z}}=(0,0.05,0.1,0.15,0.2)\delta E$ (from top to bottom). For clarity, all the zero field splittings $\delta E$ of the resonances shown are taken equal. See text for more details. The initial state of the reservoir has $N$ even (odd) for the left (right) panel. Both the total spin and energy splitting (as displayed schematically in Fig. 2) of the device as a function of $\mathcal{J}$ can be extracted from such data.