Violation of the Wiedemann-Franz Law in a Single-Electron Transistor

We study the influence of Coulomb interaction on the thermoelectric transport coefficients for a metallic single-electron transistor. By performing a perturbation expansion up to second order in the tunnel-barrier conductance, we include sequential and cotunneling processes as well as quantum fluctuations that renormalize the charging energy and the tunnel conductance. We find that Coulomb interaction leads to a strong violation of the Wiedemann-Franz law: the Lorenz ratio becomes gate-voltage dependent for sequential tunneling, and is increased by a factor $9/5$ in the cotunneling regime. Finally, we suggest a measurement scheme for an experimental realization.

Figure 1

Coulomb oscillations of thermoelectric coefficients and Lorenz ratio. For high temperatures ($k_{B}T=E_C/2$—dotted line) oscillations are washed out. In the sequential-tunneling regime ($k_{B}T=E_C/10$—dashed line) the Lorenz ratio is given by Eq. (13) around each resonance. For low temperatures ($k_{B}T=E_C/40$—solid line) the new universal Lorenz ratio $9/5L_0$ is reached in the cotunneling regime.

Figure 2

Temperature dependence of Lorenz ratio for different gate voltages. Two maxima separate different tunneling regimes. The rise to the maxima starts at $k_{B}T\simeq E_C$ and $k_{B}T\simeq {\Delta }_0$, respectively. For $n_x=0\iff {\Delta }_0=E_C$ the two maxima coincide, whereas at resonance ($n_x=0.5\iff {\Delta }_0=0$) the lower maximum is removed to the left.

Figure 3

A possible measurement scheme. In (a) the SET transistor is depicted in a configuration where the drain is thermalized at the bath temperature and the source is a small conductor whose temperature can be varied. In (b) we show a simple thermal model of the device.