Ground state and excitations of quantum dots with magnetic impurities

We consider an “impurity” with a spin degree of freedom coupled to a finite reservoir of noninteracting electrons, a system which may be realized by either a true impurity in a metallic nanoparticle or a small quantum dot coupled to a large one. We show how the physics of such a spin impurity is revealed in the many-body spectrum of the entire finite-size system; in particular, the evolution of the spectrum with the strength of the impurity-reservoir coupling reflects the fundamental many-body correlations present. Explicit calculation in the strong- and the weak-coupling limits shows that the spectrum and its evolution are sensitive to the nature of the impurity and the parity of electrons in the reservoir. The effect of the finite-size spectrum on two experimental observables is considered. First, we propose an experimental setup in which the spectrum may be conveniently measured using tunneling spectroscopy. A rate equation calculation of the differential conductance suggests how the many-body spectral features may be observed. Second, the finite-temperature magnetic susceptibility is presented, both the impurity and the local susceptibilities. Extensive quantum Monte Carlo calculations show that the local susceptibility deviates from its bulk scaling form. Nevertheless, for special assumptions about the reservoir—the “clean Kondo box” model—we demonstrate that finite-size scaling is recovered. Explicit numerical evaluations of these scaling functions are given, both for even and odd parities and for the canonical and the grand-canonical ensembles.

Figure 1

(Color online) A double-dot system, coupled very weakly to leads (L1, L2). In the Coulomb blockade regime, the small dot S behaves like a magnetic impurity that is coupled to a finite reservoir R provided by the large dot. The leads are used to measure the excitation spectrum of the system.

Figure 2

(Color online) Mapping of the Kondo problem into a spin chain with an impurity. The site “0” is the point in the bath that interacts with the impurity and is used as the site to begin the tridiagonalization of the one-body Hamiltonian of the bath. The sites are labeled by the integer $i$; in the sequence they are generated by the tridiagonalization procedure. After tridiagonalization, we are left with noninteracting electrons that feel an on-site potential ${\alpha }_i$ and can hop only to the neighboring sites with amplitude $t_{i,i+1}$.

Figure 3

(Color online) Weak-coupling perturbation theory: schematic illustration of the unperturbed system for (a) $N^{\text{R}}$ odd and (b) $N^{\text{R}}$ even. The spin marked $\mathcal{S}$ is that of the small quantum dot while the solid lines represent the spectrum of the (finite) reservoir. The dashed lines show the lowest energy orbital excitation in each case; note that in the even case, any excitation requires promoting an electron to the next level and so involves a minimum energy of order ${\Delta }_{\text{R}}$.

Figure 4

(Color online) Strong-coupling perturbation theory: schematic illustration of the unperturbed system for (a) $N^{\text{R}}$ odd and (b) $N^{\text{R}}$ even. For $\mathcal{S}=1/2$, a conduction electron is bound to the impurity at strong coupling. This leaves effectively a gas of even (odd) weakly interacting quasiparticles. We note here that in the usual case of $T_{\text{K}}⪡D$ (bandwidth of the reservoir), the formation of a singlet between the impurity spin and the reservoir for $T_{\text{K}}⪢{\Delta }_{\text{R}}$ is a complicated many-body effect: it is not just a singlet between the spin and the topmost singly occupied level, as is the case for very weak coupling.

Figure 5

(Color online) Schematic of the energy eigenvalues of the double-dot system as a function of the coupling $J$ for (a) $N$ odd and (b) $N$ even in the $\mathcal{S}=1/2$ antiferromagnetic coupling case. Energy differences are shown with respect to the ground state which therefore appears on the $x$ axis. The relation to the double-dot experiment in Fig. 1 is that the $y$ axis here is like $V_{\text{BIAS}}$ and the $x$ axis is like $V_{\text{PINCH}}$. The excitations will show up as peaks in the differential conductance $G$.

Figure 6

(Color online) Schematic illustration of the finite-size spectrum of the underscreened Kondo problem $\mathcal{S}=1$, with antiferromagnetic coupling, for (a) $N$ even and (b) $N$ odd. In each panel we show all the excitations up to order $\Delta$, both at strong and weak couplings. When it seems plausible, we have connected the strong- and the weak-coupling limits; note the necessity of crossings among the excited states in (a).

Figure 7

(Color online) Illustrative calculation of transport spectroscopy starting from the ground state of a $N$ even Kondo $R\text{−}S$ system. Transitions are $S=1/2\to S=0,1$ as marked. Top panel: the differential conductance as a function of bias voltage for different values of the asymmetry between $L,R$ tunneling rates. Bottom panel: the probability of occupation of the two states forming the $S=1/2$ ground state. For large asymmetries $P_{\uparrow }+P_{\downarrow }\approx 1$, as expected. Even though $k_{\text{B}}T/g{\mu }_B{B}_{\text{Z}}=0.2⪡1$, $P_{\downarrow }$ is large because this calculation neglects inelastic relaxation on the $R\text{−}S$ system, allowing it to stay well out of equilibrium. ${\Gamma }_1=0.01$, $g{\mu }_B{B}_{\text{Z}}=0.4$, and ${\Gamma }_2/{\Gamma }_1=10$, 3, and 1 from top to bottom (Ref. 55).

Figure 8

(Color online) Transport spectroscopy with energy relaxation included. ${\Gamma }_2/{\Gamma }_1=10$ (fixed), the parameter ${\lambda }_{\text{rel}}$ that models energy relaxation varies, and other parameters are as in Fig. 7. Top panel: the differential conductance as a function of bias voltage. Note how the symmetry of the $S=1/2\to S=1$ peaks is unaffected while the $S=1/2,S_z=-1/2\to S=0$ transition is suppressed (this transition would completely vanish if the $R\text{−}S$ system were in thermal equilibrium). Bottom panel: the probability of occupation of the two states forming the $S=1/2$ ground state. As ${\lambda }_{\text{rel}}$ is increased, the $R\text{−}S$ system has a larger probability to occupy its ground state $\left(S_z=1/2\right)$.

Figure 9

(Color online) Universal form of the singlet-triplet gap ($N$ odd) in the absence of mesoscopic fluctuations (Kondo in a box model). Note the excellent data collapse for a wide range of bare parameters upon plotting the extracted gaps as a function of $T_{\text{K}}/{\Delta }_{\text{R}}$. Inset: $P_{\text{QMC}}\left(\beta \right)$ for $D/{\Delta }_{\text{R}}=41$. Circles are data for fixed $\mathcal{J}=0.05$, 0.10, 0.13, 0.15, 0.19, 0.23, 0.26, and 0.30 (from right to left). Lines are one parameter fits to the two state singlet-triplet form used to extract the values of the gap.

Figure 10

(Color online) Rescaled susceptibility ${\chi }_{\text{univ}}=z_{\text{loc}}^{-1}{\chi }_{\text{loc}}$ [in units of ${\left(g{\mu }_B\right)}^2/4$] in the grand-canonical ensemble as a function of $T/T_{\text{K}}$ while keeping the ratio $T/{\Delta }_{\text{R}}$ fixed. The four panels correspond to $T/{\Delta }_{\text{R}}=0.5$, 0.2, 0.1, and 0.05. In each panel, the light (yellow online) and dark symbols correspond, respectively, to odd and even mean numbers of particles in the bath. Note the collapse of the data on universal curves and the substantial difference between the even and the odd cases at low temperature. Inset: comparison between the Kondo temperatures extracted from our fits and the two-loop perturbative formula. As expected, the agreement is excellent for $\mathcal{J}⪡1$. At larger values of $\mathcal{J}$, the two-loop formula tends to underestimate $T_{\text{K}}$. In each panel there are data for $D/{\Delta }_{\text{R}}=20$ (21), 40 (41), 80 (81), 160 (161), and 320 (321) for the odd (even) case, and for each case we have used $\mathcal{J}=0.19$, 0.23, 0.26, and 0.30.

Figure 11

(Color online) Rescaled susceptibility ${\chi }_{\text{univ}}$ in the grand-canonical ensemble. Same data as in Fig. 10 but now plotted for fixed values of $T_{\text{K}}/{\Delta }_{\text{R}}$, i.e., a fixed Hamiltonian. Points connected by solid (dotted) lines have 161 (160) sites in the bath, resulting in an odd (even) value for $⟨N⟩$. Results are plotted in units of (a) ${\Delta }_{\text{R}}$ and (b) $T_{\text{K}}$. Magnitude of coupling: from top to bottom in (a) and outside to inside in (b), $\mathcal{J}=0.19$, 0.23, 0.26, and 0.30; these correspond to $T_{\text{K}}/{\Delta }_{\text{R}}=0.18$, 0.54, 1.03 and 2.02 ($T_{\text{K}}$ used here is determined as in Fig. 10).

Figure 12

(Color online) Comparison of canonical and grand-canonical ensemble results for the rescaled susceptibility ${\chi }_{\text{univ}}=z_{\text{loc}}^{-1}{\chi }_{\text{loc}}$ [in units of ${\left(g{\mu }_B\right)}^2/4$] at the fixed temperature $T/{\Delta }_{\text{R}}=0.1$ for (a) even and (b) odd numbers of particles. Symbols: quantum Monte Carlo results for the canonical (circles) and the grand-canonical (triangles) ensembles. Solid line: bulk zero-temperature limit. Dashed lines: weak-coupling expression (22), for odd case, uses QMC values of $\delta E_{\text{ST}}$. Dotted line: weak-coupling expression (22) using the asymptotic expression (24) for $\delta E_{\text{ST}}$. Dashed-dotted lines: average of the even and the odd canonical results to compare with the grand-canonical result [using the numerical quantum Monte Carlo data in (a) but the asymptotic expressions (22) in (b)].