Nonadiabatic features of electron pumping through a quantum dot in the Kondo regime

We investigate the behavior of the dc electronic current $J^{\text{dc}}$ in an interacting quantum dot driven by two ac local potentials oscillating with a frequency ${\Omega }_0$ and a phase lag $\phi$. We provide analytical functions to describe the fingerprints of the Coulomb interaction in an experimental $J^{\text{dc}}$ vs $\phi$ characteristic curve. We show that the Kondo resonance, at low temperatures, reduces the frequency range for the linear behavior of $J^{\text{dc}}$ in ${\Omega }_0$ to take place and determines the evolution of the dc current as the temperature increases.

Figure 1

(Color online) Sketch of the setup. The two external ac potentials and the induced potential at the interacting site are indicated with solid and dashed arrows, respectively.

Figure 2

(Color online) $J^{\text{el}}$ (solid line), $J^{\text{in}}$ (dashed line), and $J^{\text{dc}}$ (circles) for $\phi =\pi /2$, as functions of the driving frequency ${\Omega }_0$ for $U=1$. The inelastic component is $J^{\text{in}}\sim 0$. The dc current for the noninteracting system $\left(U=0\right)$ is also shown as a dotted line for comparison. The corresponding local densities of states at the interacting site ${\rho }_0\left(\omega \right)$ are indicated in the inset (solid lines), along with the noninteracting one (dotted line). Other parameters are $E=2$, $w=2$, $w_c=8$, $V=0.4$, and $\mu =2$. (Energies, frequencies, and currents are expressed in units of ${\Gamma }_0$.)

Figure 3

(Color online) $J^{\text{el}}$ (solid line), $J^{\text{in}}$ (dashed line), and $J^{\text{dc}}$ (circles) for $\phi =\pi /2$, as functions of the driving frequency ${\Omega }_0$ for $U=10$. The details are the same as in Fig. 2. The local density of states ${\rho }_0\left(\omega \right)$ at the interacting site for ${\Omega }_0=0.1$ is shown as a solid line in the inset, where the corresponding density of states for the noninteracting system is also shown as a dashed line for comparison.

Figure 4

(Color online) Transmission function $T\left(\omega \right)$ for a noninteracting structure with ${\Omega }_0=0.1$ and $w_c=0.64$ (upper left panel) and $w_c=8$ (lower left panel). Other details are the same as in Fig. 2.

Figure 5

(Color online) Integrand $I\left(\omega \right)$ for ${\Omega }_0=0.1$ and $U=10$. Different plots correspond to different equidistant values of the chemical potential $-2\le \mu \le 0.4$. In the inset, the mean occupation per spin of the dot is shown as a function of the chemical potential. Other parameters are as in Fig. 2.

Figure 6

(Color online) $J^{\text{dc}}$ for ${\Omega }_0=0.1$ as a function of the phase lag $\phi$ for $U=1$ (thick solid line), and $10$ (thick dashed line). The current corresponding to the noninteracting dot (with $U=0$) is shown as a dotted line, for comparison. The symbols are fits with the function $A_0\text{sin}\left(\phi \right)+A_1{\text{sin}}^2\left(\phi \right)+A_2\text{sin}\left(\phi \right)\left[1+\text{cos}\left(\phi \right)\right]$, suggested by Eq. (22). Inset: Effective potential $V_{\text{eff}}=V_{\text{eff}}^{\prime }+i{V}_{\text{eff}}^{″}$ along with fits as symbols with the function $A_0\text{sin}\left(\phi \right)+A_1\left[1+\text{cos}\left(\phi \right)\right]$. Other details are as in Fig. 2.

Figure 7

(Color online) $J^{\text{el}}/{\Omega }_0$ (top left), $J^{\text{in}}/{\Omega }_0$ (bottom left), and $J^{\text{dc}}/{\Omega }_0$ (right) as functions of the pumping frequency ${\Omega }_0$ for 20 values of equally spaced temperatures in the range $0.04\le T/T_K\le 0.8$ and Coulomb interaction $U=10$. The lowest and highest temperatures are indicated as thick solid and dashed lines, respectively. The local density of states of the dot, ${\rho }_0\left(\omega \right)$, is depicted in the inset for the lowest and highest temperatures. Other details are as in Fig. 2.