Temperature dependence of electronic transport through molecular magnets in the Kondo regime

The effects of finite temperature in transport through nanoscopic systems exhibiting uniaxial magnetic anisotropy $D$, such as molecular magnets, adatoms, or quantum dots side coupled to a large spin, are analyzed in the Kondo regime. The linear-response conductance is calculated by means of the full density-matrix numerical renormalization group method as a function of temperature $T$, magnetic anisotropy $D$, and exchange coupling $J$ between the molecule's core spin and the orbital level. It is shown that such system displays a two-stage Kondo effect as a function of temperature and a quantum phase transition as a function of the exchange coupling $J$. These effects become, however, suppressed by finite magnetic anisotropy, provided the exchange coupling is sufficiently strong. Moreover, additional peaks are found in the linear conductance for temperatures of the order of $T\sim \left|J\right|$ and $T\sim D$. It is also shown that the conductance variation with $T$ remarkably depends on the sign of the exchange coupling $J$.

Figure 1

Schematic depiction of the system under consideration. It consists of a single orbital level (OL) tunnel coupled to external leads, with coupling strengths ${\Gamma }_L$ and ${\Gamma }_R$ for the left and right leads, and additionally exchanged coupled to a spin $S$, with $J$ denoting the strength of exchange interaction.

Figure 2

The dependence of the linear conductance $G$ on the energy of the orbital level (OL) $\varepsilon$ and the temperature $T$ in the case of (a), (c) ferromagnetic (FM) and (b), (d) antiferromagnetic (AFM) exchange coupling $J$ for $\left|J\right|/T_{\mathrm{K}}=3$ and $D=0$. The variable $Q$ represents the average number of electrons that occupy the OL. Here, $G_{T,J,D=0}$ corresponds to the conductance calculated for $T=D=J=0$. Parts (c) and (d) present the relevant cross sections of the density plots at the electron-hole symmetry point ($\varepsilon =-U/2$) for different values of the $J$ coupling, as indicated. The parameters of the system are as follows: $U=10$ meV, $\Gamma /U=0.1$, and $T_{\mathrm{K}}/U\approx 5\times {10}^{-3}$.

Figure 3

The normalized linear conductance $G$ as a function of temperature $T$ and exchange-coupling parameter $J$ in the case of (a), (c) ferromagnetic (FM) and (b), (d) antiferromagnetic (AFM) type of the coupling between magnetic core and orbital level shown in the presence of uniaxial magnetic anisotropy $D/U={10}^{-4}$ ($D/T_{\mathrm{K}}=0.2$). Curves in (c) and (d) represent cross sections of figures (a) and (b), respectively, for indicated values of $\left|J\right|$, but now each curve is normalized to the corresponding $G_{T=0}$ instead of $G_{T,J,D=0}$ as in (a) and (b). The other parameters are the same as in Fig. 2 with $\varepsilon =-U/2$.

Figure 4

The normalized linear conductance $G$ as a function of temperature $T$ and uniaxial magnetic anisotropy $D$ for $\left|J\right|/T_{\mathrm{K}}=0.4$ in (a)–(d) and $\left|J\right|/T_{\mathrm{K}}=2$ in (e)–(h). The other parameters are the same as in Fig. 2 with $\varepsilon =-U/2$. Dashed lines in (a) and (b) and (e) and (f) represent maximal value of the uniaxial magnetic anisotropy constant $D$ typical to molecular magnets (for further details, see the text).

Figure 5

The normalized linear conductance $G$ as a function of temperature $T$ and orbital level position $\varepsilon$. Parts (c) and (d) present the relevant cross sections of the density plots for different temperatures, as indicated. The parameters are the same as in Fig. 2 with $\left|J\right|/T_{\mathrm{K}}=3$ and $D/T_{\mathrm{K}}=0.2$.

Figure 6

Dependence of the normalized linear conductance $G$ on the magnetic anisotropy $D$ and the exchange coupling $J$ of (a) the ferromagnetic (FM) and (b) antiferromagnetic (AFM) types for $T/T_{\mathrm{K}}=0.01$. Parts (c) and (d) show the dependence of $G$ on (c) ferromagnetic and (d) antiferromagnetic exchange coupling $J$ for $T/T_{\mathrm{K}}={10}^{-2}$ and for a few anisotropy constants $D$, as indicated. Parts (e) and (f) show the same calculated for $T\to 0$. The other parameters are the same as in Fig. 2 with $\varepsilon =-U/2$.