Thermoelectric response of a correlated impurity in the nonequilibrium Kondo regime

We study nonequilibrium thermoelectric transport properties of a correlated impurity connected to two leads for temperatures below the Kondo scale. At finite bias, for which a current flows across the leads, we investigate the differential response of the current to a temperature gradient. In particular, we compare the influence of a bias voltage and of a finite temperature on this thermoelectric response. This is of interest from a fundamental point of view to better understand the two different decoherence mechanisms produced by a bias voltage and by temperature. Our results show that in this respect the thermoelectric response behaves differently from the electric conductance. In particular, while the latter displays a similar qualitative behavior as a function of voltage and temperature, both in theoretical and experimental investigations, qualitative differences occur in the case of the thermoelectric response. In order to understand this effect, we analyze the different contributions in connection to the behavior of the impurity spectral function versus temperature. Especially in the regime of strong interactions and large enough bias voltages, we obtain a simple picture based on the asymmetric suppression or enhancement of the split Kondo peaks as a function of the temperature gradient. Besides the academic interest, these studies could additionally provide valuable information to assess the applicability of quantum dot devices as responsive nanoscale temperature sensors.

Figure 1

Thermoelectric response for $V_G=\mathrm{\Gamma }$ and for various values of the interaction strength $U$. The first panel depicts $\partial j/\partial \left(\mathrm{\Delta }T\right)$, Eq. (14), and the second and third panel the two parts of Eqs. (15) and (16). See Appendix for a discussion on the accuracy of the numerical derivative.

Figure 2

Spectral functions $A\left(\omega \right)$ for $\mathrm{\Delta }T=0$ on top and of the integrand of Eq. (15) $\partial j\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)$ at the bottom, which is proportional to $\partial A\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)$. The results are shown for the same parameters as in Fig. 1, i.e., on the left for $U=8\mathrm{\Gamma }$, in the center for $U=6\mathrm{\Gamma }$, and on the right for $U=4\mathrm{\Gamma }$, with $V_G=\mathrm{\Gamma }$ in all cases.

Figure 3

Thermoelectric response $\partial j/\partial \left(\mathrm{\Delta }T\right)$ as a function of bias voltage $\varphi$ in the first panel (in the second on a semilogarithmic scale) and in the third for $\varphi =0$ as a function of $T$. All calculations are for $U=6\mathrm{\Gamma }$ and for various $V_G$. The equilibrium values on the right are obtained from NRG [118]. The inset depicts the behavior for a larger temperature range.

Figure 4

Numerically estimated differentials $\partial {\underline{\mathrm{\Delta }}}_{\mathrm{aux}}\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)$, labeled by ${\mathrm{\Delta }}^{\alpha }\left(\omega \right)$ with $\alpha =R,K$, together with the analytical expressions given in Eqs. (6) and (11). Only the exact $\partial {\mathrm{\Delta }}^K\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)$ is plotted since $\partial {\mathrm{\Delta }}^R\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)=0$. Especially for $\varphi =0.2\mathrm{\Gamma }$ larger errors occur and $\partial {\mathrm{\Delta }}^R\left(\omega \right)/\partial \left(\mathrm{\Delta }T\right)$ is clearly nonzero.