Gate-controlled spin splitting in quantum dots with ferromagnetic leads in the Kondo regime

The effect of a gate voltage $\left(V_g\right)$ on the spin splitting of an electronic level in a quantum dot (QD) attached to ferromagnetic leads is studied in the Kondo regime using a generalized numerical renormalization group technique. We find that the $V_g$ dependence of the QD level spin splitting strongly depends on the shape of the density of states (DOS). For one class of DOS shapes there is nearly no $V_g$ dependence; for another, $V_g$ can be used to control the magnitude and sign of the spin splitting, which can be interpreted as a local exchange magnetic field. We find that the spin splitting acquires a new type of logarithmic divergence. We give an analytical explanation for our numerical results and explain how they arise due to spin-dependent charge fluctuations.

Figure 1

(Color online) ${\mathrm{V}}_{\mathrm{g}}$ dependence of the spin splitting: Normalized spectral function $\pi {\Sigma }_{\sigma }{\Gamma }_{\sigma }\left(0\right)A_{\sigma }\left(\omega \right)$ as a function of energy $\omega$ and gate voltage ${ϵ}_0$, for the three different parabolic DOS shapes [insets: corresponding DOS for spin $\uparrow$ (red) and $\downarrow$ (blue)] characterized by a different $Q$, which modifies both the spin and $p\text{−}h$ asymmetry: (a-c) for magnetic field $B=0$, (d) $B∕U=0.017$, (e) and (f) $B∕U=0.0083$. The white dashed lines are obtained using Eq. (3). Here $U=0.12D_0$, $\pi V_0^2=UD∕6$, $\Delta =0.15D$, and $T=0$.

Figure 2

(Color online) The QD’s linear conductance $G$ as a function of gate voltage ${ϵ}_0$ and external magnetic field $B$ for the DOS shapes (a) (c) as for Figs. 1a, 1c, respectively. (d) Spin-dependent occupancy $n_{\sigma }$ of the dot level as a function of gate voltage ${ϵ}_0$ for the DOS shape as in Fig. 1c and $B=0$. (d) The ${ϵ}_0$ dependence of the total occupancy of the dot $n$ and magnetization $m$ for the situation from Fig. 1c. (e) The conductance $G$ for the situations from Figs. 1c (dashed); 1d (solid); and 1f (long dashed). Parameters $U,\Gamma$, and $T$ as in Fig. 1.