Universality of the Kondo Effect in Quantum Dots with Ferromagnetic Leads

We investigate quantum dots in clean single-wall carbon nanotubes with ferromagnetic PdNi-leads in the Kondo regime. Most of the Kondo resonances exhibit a splitting, which depends on the tunnel coupling to the leads and an external magnetic field $B$, but only weakly on the gate voltage. Using numerical renormalization group calculations, we demonstrate that all salient features of the data can be understood using a simple model for the magnetic properties of the leads. The magnetoconductance at zero bias and low temperature depends in a universal way on $g{\mu }_B(B\mathrm{\text{−}}{B}_c)/k_B{T}_K$, where $T_K$ is the Kondo temperature and $B_c$ the external field compensating the splitting.

Figure 1

Differential conductance $dI/d{V}_{\mathrm{sd}}$ versus bias voltage $V_{\mathrm{sd}}$ and gate voltage $V_{\mathrm{gate}}$ at $T=25\mathrm{mK}$ and $B=0\mathrm{T}$. The CB regions are numbered for future reference.

Figure 2

Differential conductance versus source-drain and gate voltage, in CB region 1 of Fig. 1, for six values of $B$. We compare here (a) experimental data with (b) NRG results for the normalized equilibrium zero-temperature spectral function $\mathcal{G}=\sum _{\sigma }\pi {\Gamma }_{\sigma }(0)A_{\sigma }(\omega )$, obtained using the model and parameters described in the text. $n_g=1/2-{\epsilon }_d/U$ is the dimensionless gate potential. The dashed line in the panel for $B=2\mathrm{T}$ in (a) indicates the gate-dependent contribution from the polarization for $\mathcal{P}=10\%$ (see text).

Figure 3

Schematic of the level renormalization process. (a) We assume flat bands ${\rho }_{\sigma }(\omega )={\rho }_0$ with bandwidth $D_0$, shifted with respect to each other by a constant Stoner splitting ${\Delta }_{\mathrm{St}}$. The filling fraction $\mathcal{F}$ determines the Fermi energy ${\epsilon }_F$. (b) Charge fluctuations for a spin-up electron on the dot, involving the empty states of the spin-up band and the occupied states of the spin-down band (hatched areas). (c) Analogous situation for a spin-down electron on the dot; the available phase space (hatched) is larger than in (b).

Figure 4

(a)–(c) Magnetic field dependence of the splitting for different charge states. Vertical arrows denote the compensation field $B_c$. (d) Scaling plot of $G(T)/G(T=0)$ vs $T/T_K$ (symbols), at $B=B_c$, for the charge state shown in Fig. 4c; the solid line displays standard NRG data for normal leads (NM) at $B=0$ while the dashed line displays the results of our NRG calculations with ferromagnetic leads (FM) at $B=B_c$. (e) Scaled zero-bias conductance $G(B)/G(B_c)$ plotted against the effective normalized field $\delta \tilde{B}$, at fixed $T=50\mathrm{mK}\ll T_K$ (100 mK for ■). Lines represent NRG calculations for several parameter sets. Inset: $G(B)$ vs $(B\mathrm{\text{−}}{B}_c)$ before scaling.