Mesoscopic Charge Relaxation

We consider charge relaxation in the mesoscopic equivalent of an $RC$ circuit. For a single-channel, spin-polarized contact, self-consistent scattering theory predicts a universal charge relaxation resistance equal to half a resistance quantum independent of the transmission properties of the contact. This prediction is in good agreement with recent experimental results. We use a tunneling Hamiltonian formalism and show in Hartree-Fock approximation that at zero temperature the charge relaxation resistance is universal even in the presence of Coulomb blockade effects. We explore departures from universality as a function of temperature and magnetic field.

Figure 1

Schematics of the mesoscopic capacitor. A cavity is connected via one lead to an electron reservoir at voltage $V(t)$ and capacitively coupled to a backgate with voltage $V_g$. The coupling matrix elements ${\Gamma }_{mn}$ are defined in the text. The inset shows the corresponding classical $RC$ circuit.

Figure 2

Magnetic field dependence of the charge relaxation resistance $R_q$. The upper panels show $R_q$ as a function of the Zeeman splitting ${\Delta }_B$ and the Fermi energy $E_F$ for weak $\pi \gamma /2E_c=0.13$ and strong $\pi \gamma /2E_c=1.4$ coupling. All energies are given in units of the bare level spacing $\Delta$. The lower panels show the corresponding total dot charge.

Figure 3

Charge relaxation resistance $R_q$ as a function of Fermi energy $E_F$ for different temperatures. The lower three curves are for a two level dot. The uppermost curve is the high temperature asymptote. The inset shows $R_q$ as a function of temperature for $E_F={ϵ}_1$.