Current deformation and quantum inductance in mesoscopic capacitors

We present a theoretical analysis of low frequency dynamics of a single-channel mesoscopic capacitor, which is composed by a quantum dot connected to an electron reservoir via a single quantum channel. At low frequencies, it is known that the Wigner-Smith delay time $\tau_W$ plays a dominant role and it can be interpreted as the time delay between the current leaving the dot and the current entering the dot. At higher frequencies, we find that another characteristic time $\tau_S$ can also be important. It describes the deformation of the leaving current to the entering one and hence can be referred as the deformation time. At sufficient low temperatures, the deformation time $\tau_S$ can be approximated from the second-order derivative of $\tau_W$ via a simple relation $\tau"_W/\tau^3_S=24/\hbar^2$. As the temperature increases, this relation breaks down and one has instead $\tau"_W/\tau^3_S \to 0$ in the high temperature limit. We further show that the deformation time $\tau_S$ can have a pronounced influence on the quantum inductance $L_q$ of the mesoscopic capacitor, leading to features different from the ones of the quantum capacitance. The most striking one is that $L_q$ can change its sign as the temperature increases: It can go from positive values at low temperatures to large negative values at high temperatures. The above results demonstrate the importance of the deformation time $\tau_S$ on the ac conductance of the mesoscopic capacitor.