Influence of magnetic anisotropy on the Kondo effect and spin-polarized transport through magnetic molecules, adatoms, and quantum dots

Transport properties in the Kondo regime of a nanosystem displaying uniaxial magnetic anisotropy (such as a magnetic molecule, magnetic adatom, or quantum dot coupled to a localized magnetic moment) are analyzed theoretically. In particular, the influence of spin-polarized transport through a local orbital of the system and exchange coupling of conduction electrons to the system’s magnetic core on the Kondo effect is discussed. The numerical renormalization-group method is applied to calculate the spectral functions and linear conductance in the case of the parallel and antiparallel configurations of the electrodes’ magnetic moments. It is shown that both the magnetic anisotropy as well as the exchange coupling between electrons tunneling through the conducting orbital and magnetic core play an important role in the formation of the Kondo resonance, leading generally to its suppression. Specific transport properties of such a system appear also as a nontrivial behavior of tunnel magnetoresistance. It is also shown that the Kondo effect can be restored by an external magnetic field in both the parallel and antiparallel magnetic configurations.

Figure 1

(a) Schematic representation of the system under consideration. The system consists of two ferromagnetic electrodes to which a magnetic quantum dot (MQD) exhibiting magnetic anisotropy is attached. As the MQD one can conceive either (b) a magnetic adatom (i.e., Fe, Co, Mn), (c) semiconductor quantum dot coupled to a magnetic moment, or (d) a single-molecule magnet (SMM)—here, only the magnetic core of the Fe${}_4$ molecule (Ref. 56) is schematically depicted.

Figure 2

Normalized spin-resolved orbital level (OL) spectral functions, $\pi {\Gamma }_{\sigma }{A}_{\sigma }\left(\omega \right)$, shown as a function of the OL energy $\varepsilon$ in the antiparallel (a),(b) and parallel (c)–(f) magnetic configurations. Top panel corresponds to the case of ferromagnetic ($J>0$) coupling between electrons in the OL and MQD’s magnetic core, while the bottom one presents results for the antiferromagnetic coupling. The variable $Q$ denotes the average number of electrons occupying the OL. The parameters are $U/W=0.3$, $D/U\approx 1.7\times {10}^{-4}$ ($D/T_{\mathrm{K}}=0.1$), $\Gamma /U\approx 0.075$, $\left|J\right|/\Gamma \approx 0.044$ ($\left|J\right|/T_{\mathrm{K}}\approx 2$), $B_z=0$, and $P=0.5$. Note that the spectral functions are presented on a logarithmic scale.

Figure 3

Normalized spin-resolved orbital level (OL) spectral functions shown for different values of the exchange parameter $J$ in the case of (a),(c),(e) ferromagnetic ($J>0$) and (b),(d),(f) antiferromagnetic ($J<0$) coupling, and for $\varepsilon =-U/2$. The spectral function displays two additional resonances for $J>0$, marked by dashed lines A and B, and a single resonance for $J<0$, marked by dashed line C. The bottom panel (g),(h) illustrates the dependence of several lowest energy states of a singly occupied MQD on the parameter $J$ (thick lines), and the corresponding probabilities of finding the electron in a certain spin state for $|\pm \frac{3}{2}{\rangle }^{\pm }$ (thin lines): ${\alpha }^2\equiv {\left({\mathbb{A}}^{+}\right)}^2={\left({\mathbb{B}}^{-}\right)}^2$, while ${\beta }^2\equiv {\left({\mathbb{B}}^{+}\right)}^2={\left({\mathbb{A}}^{-}\right)}^2$ [for details see Eq. (14) and the paragraph below it]. Remaining parameters as in Fig. 2.

Figure 4

Similar as in Fig. 3, but now the dependence of the normalized spin-resolved spectral function on the uniaxial anisotropy constant $D$ is presented for $\left|J\right|/T_{\mathrm{K}}\approx 2$. A monochromatic color scheme is used here to highlight how the position of the resonances changes with varying $D$.

Figure 5

Total linear conductance $g={\sum }_{\sigma }{g}_{\sigma }$ in the antiparallel (a),(b) and parallel (c),(d) magnetic configurations, and the spin-resolved linear conductance $g_{\sigma }$ in the parallel configuration (e)–(h), presented as a function of the OL energy $\varepsilon$ for indicated values of the parameter $J$ in the case of the ferromagnetic ($J>0$, top panel) and antiferromagnetic ($J<0$, bottom panel) exchange interaction between electrons in the OL and magnetic core. All other parameters are as in Fig. 2.

Figure 6

Influence of the uniaxial magnetic anisotropy $D$ on the total linear conductance $g={\sum }_{\sigma }{g}_{\sigma }$ for $\left|J\right|/T_{\mathrm{K}}\approx 2$ and the exchange $J$ coupling of either (a),(c) ferromagnetic ($J>0$) or (b),(d) antiferromagnetic ($J<0$) type. Other parameters are the same as in Fig. 2.

Figure 7

Total linear conductance $g={\sum }_{\sigma }{g}_{\sigma }$ in the parallel (P:- filled points) and antiparallel (AP:- hollow points) magnetic configuration shown as a function of the uniaxial anisotropy constant $D$ for indicated values of the $J$ coupling and two representative OL energies: (a),(b) $\varepsilon =-U/2$ and (c),(d) $\varepsilon =-U/4$. Left panel corresponds to the ferromagnetic $J$ coupling ($J>0$), right panel to the antiferromagnetic one ($J<0$). Other parameters as in Fig. 2.

Figure 8

Tunnel magnetoresistance (TMR) presented as a function of the OL energy $\varepsilon$ and the $J$ coupling (a),(b) and the magnetic uniaxial anisotropy $D$ (c),(d). Specific data points and parameters used in (a),(b) correspond to those in Fig. 5 and, analogously, in (c),(d) to Fig. 6. Dashed lines are introduced in order to facilitate comparison between TMR scales of adjacent plots for $J>0$ and $J<0$.

Figure 9

Restoration of the Kondo peak in the total spectral function $a\left(\omega \right)=\pi {\sum }_{\sigma }{\Gamma }_{\sigma }{A}_{\sigma }\left(\omega \right)$ (a),(c), and total conductance $g={\sum }_{\sigma }{g}_{\sigma }$ (b),(d) by an external magnetic field $B_z$, shown for indicated values of the antiferromagnetic $J$ coupling ($J<0$) and for $\varepsilon =-U/2$. Left panel corresponds to the antiparallel (AP) magnetic configuration, right panel to the parallel (P) one. The other parameters as in Fig. 2.

Figure 10

(a) Energies of ground states $|-\frac{5}{2}\rangle$ and $|-\frac{3}{2}{\rangle }^{-}$ shown as functions of an external magnetic field $B_z$ for indicated values of the exchange coupling parameter $J$ and for $D/U\approx 1.7\times {10}^{-4}$ ($D/T_{\mathrm{K}}=0.1$) and $\varepsilon =-U/2$. Dashed line corresponds to the analytical solution given by Eq. (20). (b) Dependence of the compensating field $B_{\mathrm{c}}$ on the value of the $J$ coupling. Points represent numerical results obtained for the antiparallel (AP) and parallel (P) magnetic configuration, while the bold line delineates the analytical solution (a.s.), Eq. (20). IP indicates the inflexion point of the curve $B_{\mathrm{c}}\left(\right|J\left|\right)$. Inset: the compensating field $B_{\mathrm{c}}$ normalized to $T_{\mathrm{K}}$, shown on the logarithmic scale. Other parameters as in Fig. 2.