Kondo quantum dot coupled to ferromagnetic leads: Numerical renormalization group study

We systematically study the influence of ferromagnetic leads on the Kondo resonance in a quantum dot tuned to the local moment regime. We employ Wilson’s numerical renormalization group method, extended to handle leads with a spin asymmetric density of states, to identify the effects of (i) a finite spin polarization in the leads (at the Fermi surface), (ii) a Stoner splitting in the bands (governed by the band edges), and (iii) an arbitrary shape of the lead density of states. For a generic lead density of states, the quantum dot favors being occupied by a particular spin species due to exchange interaction with ferromagnetic leads, leading to suppression and splitting of the Kondo resonance. The application of a magnetic field can compensate this asymmetry, restoring the Kondo effect. We study both the gate voltage dependence (for a fixed band structure in the leads) and the spin polarization dependence (for fixed gate voltage) of this compensation field for various types of bands. Interestingly, we find that the full recovery of the Kondo resonance of a quantum dot in the presence of leads with an energy-dependent density of states is possible not only by an appropriately tuned external magnetic field but also via an appropriately tuned gate voltage. For flat bands, simple formulas for the splitting of the local level as a function of the spin polarization and gate voltage are given.

Figure 1

(Color online) (a) Example of an energy- and spin-dependent lead DOS ${\rho }_{\sigma }\left(\omega \right)$ with an additional spin-dependent shift ${\Delta }_{\sigma }$. To perform the logarithmic discretization, a generalized bandwidth $D_0$ is defined. Since we allow for bands with energy and spin dependence, the discretization is performed for each spin-component separately, see panel (b). Since ${\widehat{\mathcal{H}}}_{\ell d}$ does not include spin-flip processes, an impurity electron of spin $\sigma$ [circles in panel (b)] couples to lead electrons of spin $\sigma$, $\sigma =\uparrow (\downarrow )$, only. Panel (b) also illustrates that impurity electrons couple to leads’ electron and hole states with arbitrary energy $\omega$, $\mid \omega \mid ⩽D_0$.

Figure 2

(Color online) The lead density of states of a flat $[{\rho }_s\left(\omega \right)={\rho }_{\sigma }]$ but spin-dependent $({\rho }_{\uparrow }\ne {\rho }_{\downarrow })$ band for spin polarization $P=0.2$. The dark region marks the filled states below the Fermi energy. Since ${\Delta }_{\uparrow }={\Delta }_{\downarrow }=0$, the generalized bandwidth $D_0=D$.

Figure 3

(Color online) [(a)–(d)] Spin-dependent occupation $n_{\sigma }$ of the local level as a function of the leads’ spin polarization $P$ for ${ϵ}_d=-U∕3$ [(a) and (c)] and ${ϵ}_d=-2U∕3$ [(b) and (d)] (related by the particle-hole symmetry) for $B=0$ (left column) and $B=-0.1\Gamma$ (right column). For finite spin-polarization, $P\ne 0$, the condition $n_{\uparrow }=n_{\downarrow }$ can only be obtained by an appropriately tuned magnetic field, as shown in (c) and (d). Parameters: $U=0.12D_0$ and $\Gamma =U∕6$.

Figure 4

(Color online) Total spectral functions $A\left(\omega \right)={\sum }_{\sigma }{A}_{\sigma }\left(\omega \right)$ for various values of the spin polarization $P$. (a) For ${ϵ}_d=-U∕3$, an increase in $P$ results in a splitting and suppression of the Kondo resonance (see inset). The effect of finite $P$ on the Hubbard peaks is less significant. The spin-resolved spectral function $A_{\sigma }\left(\omega \right)$, shown in (b), reveals that $A_{\uparrow }\left(\omega \right)$ and $A_{\downarrow }\left(\omega \right)$ differ significantly from each other for $P\ne 0$. (c) Spectral function for the same values of $P$ as in (a) but for ${ϵ}_d=-2∕3U$ [due to particle-hole symmetry, the results are mirrored as compared to (a) but with inverted spins]. Parameters: $U=0.12D_0$, $B=0$, and $\Gamma =U∕6$.

Figure 5

(Color online) (a) Compensation field $B_{\mathrm{comp}}\left(P\right)$ for different values of ${ϵ}_d$. For flat bands, $B_{\mathrm{comp}}$ depends linearly on the lead polarization. At the point where there is particle-hole symmetry, so for gate voltage $({ϵ}_d=-U∕2)$, $B_{\mathrm{comp}}\left(P\right)=0$ for any value of $P$, i.e., the spectral function is not split for any value of $P$ even though $B=0$. (b) Spectral function for various values of $P$ for $B=B_{\mathrm{comp}}$. Note the sharper resonance in the spectral function, i.e., a reduced Kondo temperature $T_K$.

Figure 6

(Color online) Spin-dependent spectral function $A_{\sigma }\left(\omega \right)$ for various values of ${ϵ}_d$, fixed $P=0.2$, and $B=0$ [(blue) dashed: $A_{\uparrow }\left(\omega \right)$; (green) long dashed: $A_{\uparrow }\left(\omega \right)$; (red) dotted: $A\left(\omega \right)={\sum }_{\sigma }{A}_{\sigma }\left(\omega \right)$]. The splitting between $A_{\uparrow }$ and $A_{\downarrow }$ changes its sign at ${ϵ}_d=-U∕2$. The spectral function $A\left(\omega \right)$ is plotted for several values of $P$ for ${ϵ}_d=-U∕3$ in Fig. 4a. The splitting of the spectral function $A\left(\omega \right)$ depends on $P$ as well on ${ϵ}_d$. Parameters: $U=0.12D_0$ and $\Gamma =U∕6$.

Figure 7

(Color online) The isotropic Kondo effect, accompanied by universal scaling, can be recovered even for $P\ne 0$ and $B=B_{\mathrm{comp}}$ since both (a) the spin-susceptibility $\mathrm{Im}{\chi }_S^z$ and (b) the spectral function collapse onto a universal curve. This statement holds for any value of ${ϵ}_d$ with some corresponding compensation fields, as given in Fig. 5a. Parameters: $U=0.12D_0$, ${ϵ}_d=-U∕3$, and $\Gamma =U∕6$.

Figure 8

(Color online) (a) Spin polarization dependence of the Kondo temperature $T_K\left(P\right)$ [obtained by applying an appropriate $B=B_{\mathrm{comp}}(P,{ϵ}_d)$]. The functional dependence of $T_K\left(P\right)$ found via a poor man’s scaling analysis (dotted line) [see Eq. (6) of Ref. 6] is confirmed by the NRG analysis. The plot confirms that a dot tuned to the local moment regime, ${\sum }_{\sigma }{n}_{\sigma }\simeq 1$, shows the universal $P$ dependence of $T_K\left(P\right)$ for arbitrary values of ${ϵ}_d$. (b) The ${ϵ}_d$ dependence of $T_K$ for fixed $P=0.2$ at $B=B_{\mathrm{comp}}$. As long as the impurity remains in the local moment regime, Haldane’s formula (Ref. 16) for $T_K$ (dotted line) properly describes $T_K\left({ϵ}_d\right)$ even though the leads have a finite spin polarization. Parameters: $U=0.12D_0$ and $\Gamma =U∕6$.

Figure 9

(Color online) (a) Height of the spin-resolved spectral function $A_{\sigma }(\omega =0)$ as a function of $P$. The dashed line shows the $1∕(1\pm P)$ dependence as expected from the Friedel sum rule [Eq. (21)]. (b) Spin-resolved conductance $G_{\sigma }$ as obtained from Eq. (23). This plot confirms the expectation based on the Friedel sum rule that the spin-resolved conductance should be independent of the spin species, consequently, it serves as a consistency check of our numerics.

Figure 10

(Color online) The DOS in the case of flat bands with the Stoner splitting $\Delta$ and a finite spin polarization $P$. In general, the shifts of the ↑ and ↓ bands are unrelated to each other, ${\Delta }_{\downarrow }\ne -{\Delta }_{\uparrow }$. In this section, however, we assume ${\Delta }_{\downarrow }=-{\Delta }_{\uparrow }$.

Figure 11

(Color online) Spin polarization $P$ dependence of the local occupation $n_{\sigma }$ for (a) $B=0$ and (b) $B=0.1\Gamma$ with ${ϵ}_d=-U∕3$. The Stoner splitting $\Delta$ even in the absence of spin polarization of the leads, $P=0$, introduces an effective exchange field which splits the local level $n_{\downarrow }>n_{\uparrow }$ in the absence of an external magnetic field $B=0$. The spin polarization dependence of $B_{\mathrm{comp}}$ for various values of ${ϵ}_d$ is shown in (c). To a good approximation, the Stoner splitting introduces a constant exchange field (here, $B_{\mathrm{comp}}∕\Gamma \approx 0.061$); therefore, it does nothing else but shift the $B_{\mathrm{comp}}$ dependence of Sec. 5. Parameters: $U=0.12\left(\frac{D}{D_0}\right)D_0$, $\Gamma =U∕6$, and ${\Delta }_{\downarrow }=-{\Delta }_{\uparrow }=0.10\left(\frac{D}{D_0}\right)D_0$.

Figure 12

(Color online) Spin-resolved equilibrium spectral function $A_{\sigma }(\omega ,T,V=0)$ with parameters extracted from Ref. 11. The (blue) dashed lines correspond to $A_{\uparrow }(\omega ,T,V=0)$, the (green) long dashed lines to $A_{\downarrow }(\omega ,T,V=0)$, and the (red) solid ones to the sum of both, i.e., $A(\omega ,T,V=0)$. [(a)–(e)] A gradual increase in the temperature for those temperatures used in Ref. 11. For comparison, we plot the $T=0$ spectral function $A(\omega ,T=0,V=0)$ in panel (a) as well. Note that the splitting of the Kondo resonance disappears upon increasing $T$.

Figure 13

(Color online) Linear conductance as a function of temperature (a) for the model studied in this section and (b) for the case of a dot coupled to normal spin unpolarized leads ($P=0$ and $\Delta =0$) but in the presence of a magnetic field $B=0.02U$. In both cases, a plateau in the linear conductance around $T_K$ can be observed. Whereas the spin-resolved linear conductance is degenerate in case (b) (since it is the symmetric Anderson model with particle-hole symmetry), this is not the case for (a). Parameters are as in Fig. 12 except for $P=\Delta =0$ in case (b).

Figure 14

(Color online) The parabolic density of states (typical for $s$ electron band) given by Eq. (24) with the same Stoner splitting $\Delta =0.3D$ but with an additional spin asymmetry $Q$ defined in the text. Here, $Q=\left(\mathrm{a}\right)$ 0.0, (b) 0.1, and (c) 0.3.

Figure 15

(Color online) Color scale plot of the (rescaled) spectral function ${\sum }_{\sigma }{\Gamma }_{\sigma }{A}_{\sigma }\left(\omega \right)$: the dashed lines mark the Kondo resonance for the lead band structure as shown in Fig. 14 in the absence of an external magnetic field. Whereas panels (a)–(c) correspond to a quantum dot in the absence of an external magnetic field, in panels (d)–(f), an external field is applied.

Figure 16

(Color online) Color scale plot of the (rescaled) spectral function ${\sum }_{\sigma }{\Gamma }_{\sigma }{A}_{\sigma }\left(\omega \right)$: the dashed lines mark the Kondo resonance for the lead band structure as shown in Fig. 14. In contrast to Fig. 15, an external field is applied. As explained in Sec. 5, a local magnetic field shifts the spin-resolved local levels relative to each other, leading to a change in the gate voltage where the compensation $(n_{\uparrow }=n_{\downarrow })$ takes place.

Figure 17

(Color online) Color scale plot of the linear conductance $G$ as a function of gate voltage ${ϵ}_d$ and magnetic field $B$ for the band structures corresponding to Figs. 14a, 14b, 14c [i.e., corresponds (a) to $Q=0$, (b) to $Q=0.1$, and (c) to $Q=0.3$]. The black dashed line indicates cross sections for which the amplitude is plotted in Fig. 19b. Note the charging resonances around ${ϵ}_d\approx 0$ and ${ϵ}_d\approx -U$ and the tilted resonance (due to the Kondo effect) for $-U\lesssim {ϵ}_d\lesssim 0$.

Figure 18

(Color online) Logarithmic-linear version of Fig. 17c. Color scale plot of the linear conductance $G$ as a function of gate voltage ${ϵ}_d$ and magnetic field $B$ for the band structures corresponding to Fig. 14c.

Figure 19

(Color online) The spin-resolved impurity occupation $n_{\sigma }$ for the DOS reflecting Figs. 14a, 15a is plotted in panel (a). It nicely illustrates that ${ϵ}_d$ can be employed to tune the average impurity magnetization. Since the Kondo resonance is only fully recovered for $n_{\uparrow }=n_{\downarrow }$, the linear conductance $G$ shows three peaks in panel (b) for three different situations corresponding to black dashed lines from Fig. 17.