Spin thermoelectric effects in Kondo quantum dots coupled to ferromagnetic leads

Thermoelectric effects in transport through a quantum dot coupled to external ferromagnetic leads are investigated theoretically. The basic thermoelectric transport characteristics, such as thermopower, electronic contribution to the heat conductance, and the corresponding figure of merit, are calculated in the linear response regime by means of the density-matrix numerical renormalization group method. The case of a nonzero spin splitting of the electrochemical potential in the electrodes is also considered and the associated spin thermoelectric effects are analyzed. It is shown that the spin-dependent thermoelectric phenomena in the local moment regime depend generally on the exchange field induced by ferromagnetic contacts. In addition, the temperature dependence of the Seebeck coefficient is rather nontrivial, and depends on the spin polarization and spin relaxation in the leads. In the presence of ferromagnetic leads, the thermopower as a function of temperature may change sign more times than the thermopower for nonmagnetic leads. These changes can be thus used to determine the relevant Kondo behavior and Kondo energy scale in the system. Moreover, the effects of external magnetic field and different spin polarization of ferromagnetic leads are also analyzed.

Figure 1

Schematic of a single-level quantum dot attached to the left and right ferromagnetic leads with coupling strengths ${\Gamma }_{L\sigma }$ and ${\Gamma }_{R\sigma }$, respectively. The dot level energy is denoted by ${\varepsilon }_{\sigma }$, while $U$ is the Coulomb correlation parameter. The magnetizations of the leads are collinear and form a parallel magnetic configuration. There is a temperature gradient $\delta T$ applied to the leads. In the limit of fast spin relaxation in the leads, the electrochemical potentials do not depend on spin. However, in the case of slow spin relaxation $\delta T$ can generate a spin bias leading to spin thermoelectric effects.

Figure 2

The linear conductance $G$ [(a), (b)], thermal conductance $\kappa$ [(c), (d)], thermopower $S$ [(e), (f)], and the figure of merit $ZT$ [(g), (h)] as a function of the level position $\varepsilon$ and temperature $T$ (left column) together with the relevant cross sections (right column) for the case of nonmagnetic leads. The parameters are $\Gamma =0.01$, $U=0.12$, $B=0$ (in units of band half-width $W$), and $p=0$. Since the Kondo temperature depends on the dot level position, the temperature is normalized here to the coupling parameter $\Gamma$. The corresponding Kondo temperature for the symmetric Anderson model, $\varepsilon =-U/2$, is $T_K=0.02\Gamma$. Accordingly, the curves in the right column correspond respectively to $T/T_K=0.05,0.5,5,50,500$.

Figure 3

The same as in Fig. 2 calculated for ferromagnetic leads with the corresponding spin polarization factor $p=0.5$ in the limit of no spin accumulation in the leads. The other parameters as in Fig. 2.

Figure 4

The temperature and level position dependence of thermal conductance $\kappa$ (a), the thermopower $S$ (c) and spin thermopower $S_S$ (e), and the respective cross-sections for different temperatures [(b), (d), (e)] in the case of finite spin accumulation in the leads. The other parameters are the same as in Fig. 3.

Figure 5

The level position dependence of the electric thermopower $S$ for different spin polarization of the leads $p$ for three temperatures: $T=\Gamma \gg T_K$ (a), $T=0.1\Gamma >T_K$ (b), and $T=0.01\Gamma \lesssim T_K$ (c) in the limit of no spin relaxation in the leads. The inset in (c) shows the dependence of $S$ on $p$ for $\varepsilon /U=-0.4,-0.12$ for $T/\Gamma =0.1$. The arrows indicate the direction of change in thermopower with increasing $p$. The other parameters are the same as in Fig. 2.

Figure 6

The same as in Fig. 5 calculated for the spin thermopower $S_S$ in the limit of no spin relaxation in the leads. The inset in (c) shows the dependence of $S_S$ on $p$ for two characteristic values of $\varepsilon$, where a large enhancement of $S_S$ with $p$ occurs and where $S_S$ displays a nonmonotonic dependence on $p$.

Figure 7

The level position dependence of the thermopower $S$ and spin thermopower $S_S$ for three different temperatures, $T\gg T_K$ (a), $T>T_K$ (b), and $T\lesssim T_K$ (c). The temperature dependence of $S$ and $S_S$ for the local moment regime, $\varepsilon /U=-0.3$ (d), the mixed valence regime, $\varepsilon /U=0$ (e), and the empty orbital regime, $\varepsilon /U=0.3$ (f). The cases of both no spin accumulation as well as finite spin accumulation in the leads are presented. The vertical dotted line in (d) indicates the Kondo temperature. The parameters are the same as in Fig. 3.

Figure 8

The magnetic field dependence of the thermopower $S$ and spin thermopower $S_S$ in the local moment regime for three different temperatures, $T\gg T_K$ [(a)–(c)], $T>T_K$ [(d)–(f)], and $T\lesssim T_K$ [(g)–(i)]. The cases of both absence and presence of spin accumulation in the leads are presented. Each column corresponds to different level position $\varepsilon$, as indicated. The vertical dotted line shows the energy scale related to the Kondo temperature $T_K$, while the vertical dashed line corresponds to the exchange field $|\Delta {\varepsilon }_{\mathrm{exch}}|$. Note that the exchange field is negative for $\varepsilon >-U/2$. The three different values of the level position correspond to $\Delta {\varepsilon }_{\mathrm{exch}}=0$ ($\varepsilon /U=-0.5$), $|\Delta {\varepsilon }_{\mathrm{exch}}|<T_K$ ($\varepsilon /U=-0.49$), and $|\Delta {\varepsilon }_{\mathrm{exch}}|>T_K$ ($\varepsilon /U=-0.4$), respectively. The other parameters are the same as in Fig. 3.

Figure 9

The magnetic field dependence of the thermopower $S$ and spin thermopower $S_S$ in the mixed valence (left column) and empty orbital (right column) regime for three different temperatures, as indicated. The cases of both no spin accumulation as well as finite spin accumulation in the leads are presented. The other parameters are the same as in Fig. 3.