Tunable charge and spin Seebeck effects in magnetic molecular junctions

We study the charge and spin Seebeck effects in a spin-1 molecular junction as a function of temperature, applied magnetic field, and magnetic anisotropy ($D$) using Wilson's numerical renormalization group. A hard-axis magnetic anisotropy produces a large enhancement of the charge Seebeck coefficient $S_c$ ($\sim$$k_B/\left|e\right|$) whose value only depends on the residual interaction between quasiparticles in the low-temperature Fermi-liquid regime. In the underscreened spin-1 Kondo regime, the high sensitivity of the system to magnetic fields makes it possible to observe a sizable value for the spin Seebeck coefficient even for magnetic fields much smaller than the Kondo temperature. Similar effects can be obtained in C${}_{60}$ junctions where the control parameter, instead of $D$, is the gap between a singlet and a triplet molecular state.

Figure 1

Schematic representations of the Co(tpy-SH)${}_2$ and C${}_{60}$ molecular junctions. In the former, a distortion of the crystal field and/or an external magnetic field induces the splitting of the $S=1$ state. In the C${}_{60}$ case there is a singlet-triplet gap given by the exchange constant $J$.

Figure 2

Left panels: Color maps of the conductance $G$ (top) and charge Seebeck coefficient $S_c$ (bottom) as a function of the temperature and the anisotropy constant $D$. For finite $D$, the conductance goes down to very small values for $T<T_K^{☆}$, while the $S_c$ reaches its maximum $(k_B/\left|e\right|)\pi /\sqrt{3(1+\gamma )}$ for $T\lesssim T_K^{☆}$. Right panels: Temperature dependence of $G$ and $S_c$ for different values of $D$, indicated in the color maps with dashed lines. The conductance is given in units of $g_0=(2e^2/h)4{\Gamma }_L{\Gamma }_R/{({\Gamma }_L+{\Gamma }_R)}^2$.

Figure 3

Spin Seebeck coefficient $S_s$ and spin current $I_s$ for an isotropic molecule ($D=0$). Top panel: $S_s=S_{\uparrow }-S_{\downarrow }$ as a function of the magnetic field for different temperatures. Note that $S_s$ is significantly large even for $g{\mu }_{B}B\ll k_B{T}_K$. The maximum of $S_s$ occurs for $g{\mu }_{B}B\sim k_{B}T$. Bottom panel: Spin current as a function of $g{\mu }_{B}B$ in the zero charge current condition ($I_c=0$).

Figure 4

Left column panels: Color maps of the Seebeck coefficient ($S_c$), the spin current ($I_s$), and the spin-dependent Seebeck coefficients ($S_{\sigma }$) as a function of the temperature and the anisotropy constant $D$. In all cases $g{\mu }_{B}B=0.01k_B{T}_K$. Note that $S_c$ is strongly reduced for $D/T_K\lesssim 0.5$ that corresponds to $g{\mu }_{B}B\lesssim k_B{T}_K^{☆}$ and that $I_s$ reaches its maximum value in the parameters region where $S_{\downarrow }$ changes sign. Notice that, as $D$ decreases, the $S_{\downarrow }$ peak moves down to very low $T$ before changing its sign. Right column panels: Temperature and anisotropy dependence of $S_c$ and $I_s$. Note that $S_c$ is suppressed in the region where $I_s$ is different from zero.