Underscreened Kondo effect in $S=1$ magnetic quantum dots: Exchange, anisotropy, and temperature effects

We present a theoretical analysis of the effects of uniaxial magnetic anisotropy and contact-induced exchange field on the underscreened Kondo effect in $S=1$ magnetic quantum dots coupled to ferromagnetic leads. First, by using the second-order perturbation theory we show that the coupling to spin-polarized electrode results in an effective exchange field $B_{\mathrm{eff}}$ and an effective magnetic anisotropy $D_{\mathrm{eff}}$. Second, we confirm these findings by using the numerical renormalization group method, which is employed to study the dependence of the quantum-dot spectral functions, as well as quantum-dot spin, on various parameters of the system. We show that the underscreened Kondo effect is generally suppressed due to the presence of effective exchange field and can be restored by tuning the anisotropy constant, when $|D_{\mathrm{eff}}|=|B_{\mathrm{eff}}|$. The Kondo effect can also be restored by sweeping an external magnetic field, and the restoration occurs twice in a single sweep. From the distance between the restored Kondo resonances one can extract the information about both the exchange field and the effective anisotropy. Finally, we calculate the temperature dependence of linear conductance for the parameters where the Kondo effect is restored and show that the restored Kondo resonances display a universal scaling of $S=1/2$ Kondo effect.

Figure 1

Schematic of a two-level magnetic quantum dot coupled to a metallic ferromagnetic electrode. The magnetic moment of the electrode (denoted by a bold arrow) is collinear with the dot's easy axis. The dot levels have energies ${\varepsilon }_1$ and ${\varepsilon }_2$, respectively, with $\delta$ being the level spacing, while $J$ denotes the exchange interaction. The tunnel matrix elements between the dot and the lead are denoted by $T_{j\sigma }$ for the dot level $j$ and spin $\sigma$.

Figure 2

Sketch showing how the uniaxial magnetic anisotropy lifts partially the degeneracy between the components of the triplet $S=1$. Note that in reality the energy of the $S_z=0$ component is independent of magnetic anisotropy $D$.

Figure 3

(a)–(c) Schematic representing level renormalization due to the effective anisotropy constant $D_{\mathrm{eff}}$ and effective exchange field $B_{\mathrm{eff}}$ for the quantum dot with $S=1$. Dashed lines in (a) and (c) correspond to energy levels without the renormalization of the uniaxial magnetic anisotropy constant, $D_{\mathrm{eff}}=D$ [see Eq. (11)]. It is assumed that $B_{\mathrm{eff}}<0$, so that the ground state [in (b) and (c)] is $|T_{+}\rangle$. (Bottom panels) The dependence of $B_{\mathrm{eff}}$ and $D_{\mathrm{eff}}$ on the anisotropy constant $D$ for $\varepsilon /U=-1.4$ (d) and on the level position $\varepsilon$ in the Coulomb blockade regime for $\left|D\right|/\Gamma =0.1$ (e). Note that in (e) for $D<0$ one also gets $D_{\mathrm{eff}}<0$, but for practical reasons $|D_{\mathrm{eff}}|$ is plotted. For comparison, in (d) we also show $\Delta D$ and $D$. The other parameters are $\varepsilon =-17.5\Gamma$, $\delta =2.5\Gamma$, $U=12.5\Gamma$, $J=-2.5\Gamma$, and $P=0.5$.

Figure 4

Total normalized spectral function $A\left(\omega \right)$ of the dot for $\omega \to 0$, $A\left(0\right)$, as a function of the uniaxial magnetic anisotropy constant $D$ for several values of the interlevel exchange interaction $J$ in the presence of (a) nonmagnetic (NM) and (b) ferromagnetic (FM) electrode. The insets in (a) and (b) display $A\left(0\right)$ as a function of $D/\Gamma$ plotted in a linear scale in the case of nonmagnetic (a) and ferromagnetic (b) electrode for different values of $J$. Note that in the inset in (a) the Kondo peaks have been shifted by $-0.01$ for better visibility. (c) Variation of the spectral function with the energy $\omega$ and uniaxial magnetic anisotropy constant $D$ for $J/\Gamma =-2.5$. (d) Cross-sections of the plot in (c) for selected values of the uniaxial anisotropy constant $D>0$. The other parameters are $\varepsilon =-17.5\Gamma$, $\delta =2.5\Gamma$, $U=12.5\Gamma$, and $P=0.5$ [except (a) where $P=0$].

Figure 5

(a) Total normalized spectral function $A\left(\omega \right)$ of the dot for $\omega \to 0$ and (b) the corresponding expectation value of the dot's spin $z$th component as functions of the uniaxial magnetic anisotropy constant $D$. Except $J/\Gamma =-2.5$, remaining parameters are the same as in Fig. 4.

Figure 6

(a) Dependence of the magnetic anisotropy constant $D_{\mathrm{res}}$ at which the Kondo resonance is restored on the interlevel exchange coupling $J$ for $P=0.5$ and (b) on the spin polarization $P$ for $J/\Gamma =-2.5$. Other parameters as in Fig. 4.

Figure 7

Total normalized spectral function $A\left(\omega \right)$ of the dot shown as a function of energy $\omega$ and the spin polarization parameter $P$ for indicated values of the uniaxial magnetic anisotropy constant $D$. Other parameters are the same as in Fig. 5.

Figure 8

Dependence of the total normalized spectral function $A\left(\omega \right)$ for $\omega \to 0$ (a),(b) and the average $z$th component of the spin $\langle S_z\rangle$ (c),(d) on an external magnetic field $B_z$ oriented along the dot's easy axis. Different lines correspond to selected values of the uniaxial magnetic anisotropy constant $D$, and the left panel corresponds to $D<0$, while the right panel refers to $D>0$. Except for $P=0.5$ and $J/\Gamma =-2.5$, all remaining parameters are the same as in Fig. 4.

Figure 9

The temperature dependence of the normalized linear conductance $G\left(T\right)/G\left(0\right)$ (a) and the universal scaling curves (b). The relevant curves correspond to (i) underscreened Kondo effect, with $D=0$, $P=0$, $B_z=0$; (ii) restored Kondo resonance when $B_{\mathrm{eff}}+D_{\mathrm{eff}}=0$, with $D/\Gamma \approx 0.1$, $P=0.5$, $B_z=0$; and (iii) restored Kondo resonance by magnetic field $B_z=D_{\mathrm{eff}}-B_{\mathrm{eff}}$, with $D/\Gamma \approx 0.2$, $P=0.5$, $B_z/\Gamma \approx 0.22$. The curves (ii) and (iii) display scaling typical for spin $S=1/2$ Kondo effect [dot-dashed line in (b)]. Other parameters the same as in Fig. 4 with $J/\Gamma =-2.5$.