Nonlinear Seebeck effect of $\text{SU}\left(N\right)$ Kondo impurity

We develop a theoretical framework to study the influence of coupling asymmetry on the thermoelectrics of a strongly coupled $\mathrm{SU}(N)$ Kondo impurity based on a local Fermi liquid theory. Applying a nonequilibrium Keldysh formalism, we investigate a charge current driven by the voltage bias and temperature gradient in the strong coupling regime of an asymmetrically coupled $\mathrm{SU}(N)$ quantum impurity. The thermoelectric characterizations are made via nonlinear Seebeck effects. We demonstrate that the beyond particle-hole (PH) symmetric $\mathrm{SU}(N)$ Kondo variants are highly desirable with respect to the corresponding PH-symmetric setups in order to have significantly improved thermoelectric performance. The greatly enhanced Seebeck coefficients by tailoring the coupling asymmetry of beyond PH-symmetric $\mathrm{SU}(N)$ Kondo effects are explored. Apart from presenting the analytical expressions of asymmetry-dependent transport coefficients for general $\mathrm{SU}(N)$ Kondo effects, we make a close connection of our findings with the experimentally studied SU(2) and SU(4) Kondo effects in quantum dot nanostructures. Seebeck effects associated with the theoretically proposed SU(3) Kondo effects are discussed in detail.

Figure 1

Upper panel: Schematic representation of an experimental setup for investigating Seebeck effect in nanostructures, where a $\mathrm{SU}(N)$ quantum impurity is sandwiched between two conducting reservoirs. The left (red) and right (blue) reservoirs are in thermal equilibrium, separately, at temperature $T_{\mathrm{L}}$ and $T_{\mathrm{R}}$, respectively. The tunneling-matrix elements from the impurity to the left and right reservoirs are characterized by $t_{\mathrm{L}}=tcos\theta$ and $t_{\mathrm{R}}=tsin\theta$ with $\theta \in (0,\pi /2)$. Lower panel: The asymmetry of the tunneling junction is accounted for by introducing a parameter $\mathcal{C}\equiv ({\mathrm{\Gamma }}_{\mathrm{L}}-{\mathrm{\Gamma }}_{\mathrm{R}})/({\mathrm{\Gamma }}_{\mathrm{L}}+{\mathrm{\Gamma }}_{\mathrm{R}})=cos2\theta$ with ${\mathrm{\Gamma }}_{\mathrm{L}/\mathrm{R}}=\pi {\rho }_{\mathrm{res}}{\left|t_{\mathrm{L}/\mathrm{R}}\right|}^2,{\rho }_{\mathrm{res}}$ being the density of states of the reservoirs. The magenta line represents the variation of asymmetry parameter $\mathcal{C}$ with respect to the asymmetry angle $\theta$. We choose the Fermi level in such a way that the chemical potentials of the left and right reservoirs take some specific values ${\mu }_{\mathrm{L}/\mathrm{R}}=\pm \frac{e\mathrm{\Delta }V}{2}(1\mp \mathcal{C})$. This choice of chemical potentials amounts to greatly simplifying the calculation of the charge and heat current through a strongly coupled Kondo impurity (see text for details). These chemical potentials are represented by the red and blue curves, respectively.

Figure 2

Linear (LR) and nonlinear (BLR) Seebeck coefficients with PH-symmetric SU(2) and SU(4) Kondo effects for a fixed value of the potential scattering ${\delta }_{\mathrm{P}}$.

Figure 3

Left panel: Plot of asymmetry-dependent zero current lines in PH-asymmetric SU(4) Kondo effects within the qudratic response level of calculations as a function of applied voltage bias and temperature gradient at reference temperature $\overline{T}=0.1$. Right panel: Corresponding Seebeck coefficients for given asymmetry parameter.

Figure 4

Left panel: Lines of zero charge currents in a single-electron SU(4) Kondo impurity within the cubic response level of calculations. The temperatures of the left and right reservoirs (normalized with corresponding Kondo temperature) are varied for a given asymmetry parameter at fixed voltage drop $\mathrm{\Delta }\overline{V}=0.05$. Right panel: The Seebeck coefficients as a function of asymmetry parameter with beyond half-filled SU(4) Kondo effects for fixed voltage drop $\mathrm{\Delta }\overline{V}=0.05$.