Quantum-fluctuation effects on the thermopower of a single-electron transistor

We study thermal conductance and thermopower of a metallic single-electron transistor beyond the limit of weak tunnel coupling. Employing both a systematic second-order perturbation expansion and a nonperturbative approximation scheme, we find, in addition to sequential and cotunneling contributions, terms that are associated with the renormalization of system parameters due to quantum fluctuations. The latter can be identified by their logarithmic temperature dependence that is typical for many-channel Kondo correlations. In particular, the temperature dependence of thermopower, which provides a direct measure of the average energy of transported particles, reflects the logarithmic reduction of the Coulomb-blockade gap due to quantum fluctuations.

Figure 1

Setup for thermopower measurement on a single-electron transistor. The two leads are kept at different temperatures and a voltage bias $V=V_L-V_R$ is applied, such that no electrical current flows through the device. A gate voltage sets the working point and thereby the charging-energy gap $\Delta$ of the SET.

Figure 2

Thermal conductance of an SET calculated to first- and second-order in tunneling coupling for $k_{B}T∕E_C=0.05$ and ${\alpha }_0=0.04$. The inset shows the different contributions in second-order, due to cotunneling $\left(g_T^{\cot }\right)$, renormalization of the tunnel-coupling strength $\left(g_T^{\tilde{\alpha }}\right)$ and of the charging-energy gap $\left(g_T^{\tilde{\Delta }}\right)$. Renormalization leads to a broadening and a suppression of the thermal conductance as compared to the sequential-tunneling result and cotunneling yields an algebraically decaying contribution in the Coulomb-blockade valley.

Figure 3

Thermopower within perturbative calculation for ${\alpha }_0=0.002$ and $k_{B}T∕E_C=0.1$ (solid line), 0.04 (dashed), and 0.01 (dotted).

Figure 4

Thermopower as a function of $\beta {\Delta }_0$ for ${\alpha }_0=0.02$ at low temperature. Both the curves for sequential tunneling (gray solid line) and sequential plus cotunneling (dotted line) are temperature independent. A full second-order perturbation theory, however, shows deviations from this universal scaling behavior due to charging-energy gap renormalization. The chosen temperatures are $k_{B}T∕E_C=0.01$ (dashed line) and 0.0001 (black solid line). We analyze in detail the reduction of slope at resonance (see inset) and the shift of the maximum away from resonance with decreasing temperature.

Figure 5

Average energy of the transported particles as a function of temperature for coupling ${\alpha }_0=0.01$. The logarithmic temperature dependence typical for Kondo physics indicates a renormalization of the charging-energy gap. We display the sequential-tunneling result (dashed line), the “universal” (temperature-independent) result of sequential and cotunneling (dotted) and the full next-to-leading-order perturbative result of Eq. (23) (solid). RTA (dot-dashed line) gives a similar low-temperature result for the slope in this figure, but the off-set is unknown.

Figure 6

(a) Position of the maximum of thermopower for ${\alpha }_0=0.01$. When taking only sequential and cotunneling into account, the maximum’s position approaches a constant at low temperature (dotted line). In the full next-to-leading-order perturbative calculation (solid line) and the RTA (dot-dashed line), the maximum moves away from resonance with decreasing temperature. (b) Approximative determination of the temperature dependence of the maximum’s position [thick solid line as in (a)]. The crossover between sequential and cotunneling (squares) becomes temperature independent at low-temperature. The correct temperature dependence is reproduced when the second-order terms associated with renormalization of the system parameters are added to the sequential-tunneling (circles) but not to the cotunneling (triangles) contribution.