Coulomb interaction and transient charging of excited states in open nanosystems

We obtain and analyze the effect of electron-electron Coulomb interaction on the time-dependent current flowing through a mesoscopic system connected to biased semi-infinite leads. We assume the contact is gradually switched on in time and we calculate the time-dependent reduced density operator of the sample using the generalized master equation. The many-electron states (MES) of the isolated sample are derived with the exact-diagonalization method. The chemical potentials of the two leads create a bias window which determines which MES are relevant to the charging and discharging of the sample and to the currents, during the transient or steady states. We discuss the contribution of the MES with fixed number of electrons $N$ and we find that in the transient regime there are excited states more active than the ground state even for $N=1$. This is a dynamical signature of the Coulomb-blockade phenomenon. We discuss numerical results for three sample models: short one-dimensional chain, two-dimensional (2D) lattice, and 2D parabolic quantum wire.

Figure 1

(Color online) The equilibrium chemical potentials ${\mu }_N^{\left(0\right)}$ for $1\le N\le 5$ as a function of the interaction strength $U$. The dotted lines mark the chemical potentials of the leads selected in the transport simulations shown in the next figure, i.e., ${\mu }_{\text{L}}=5.25$ and ${\mu }_{\text{R}}=4.75$.

Figure 2

(Color online) The total current entering the $5\times 1$ sample from the left lead as a function of time for the different values of the interaction strength $U$. The chemical potentials of the leads ${\mu }_{\text{L}}=5.25$ and ${\mu }_{\text{R}}=4.75$.

Figure 3

(Color online) The charge accumulated on $N$-particle states $\left(N=1,..,4\right)$ and the total charge for biased (solid line) and unbiased system (dashed line). Parameters: $U=0.5$, ${\mu }_{\text{L}}=6$, and ${\mu }_{\text{R}}=5.5$ for the biased system and ${\mu }_{\text{L}}={\mu }_{\text{R}}=5.75$ for the unbiased system.

Figure 4

(Color online) The time-dependent total currents in the left and right leads at different values of the chemical potential ${\mu }_{\text{R}}$. The current in the right lead starts at negative values. Other parameters: $V_{\text{L}}=V_{\text{R}}=0.750$ and $U=1.0$.

Figure 5

(Color online) The separate contributions to the current of the ground state with $N$ particles and of all excited states with $N$ particles, for different values of ${\mu }_{\text{R}}$. For completeness we also include the total currents $J_{\text{L}}$ for the same configurations. The discussion is made in the text. Other parameters: $V_{\text{L}}=V_{\text{R}}=0.750$ and $U=1.0$.

Figure 6

(Color online) The current carried by three-particle configurations for different number of MES included in the basis used to compute the reduced density operator. Other parameters $U=1.0$, ${\mu }_{\text{L}}=3.5$, and ${\mu }_{\text{R}}=1.75$.

Figure 7

(Color online) (a) The coupling amplitudes ${\left|T_{qn}\right|}^2$ for $n=1,..,5$ between single-particle states in the leads with momentum $q$ and the lowest five single-particle states of the isolated dot. (b) The generalized chemical potentials for $N$-particle interacting configurations. The red + symbols mark ${\mu }_N^{\left(0\right)}$ while the other ones correspond to generalized potentials ${\mu }_N^{\left(i\right)}$ related to the $i\text{th}$ excited state of the $N$-particle system.

Figure 8

(Color online) (a) The total currents in the left and right leads for ${\mu }_{\text{L}}=0.6$ and ${\mu }_{\text{L}}=0.4$ while keeping ${\mu }_{\text{R}}=0.2$. (b) The partial currents in the left lead for single-particle states when ${\mu }_{\text{L}}=0.4$ and ${\mu }_{\text{R}}=0.2$. (c) The partial currents in the left lead for single-particle states when ${\mu }_{\text{L}}=0.6$ and ${\mu }_{\text{R}}=0.2$.

Figure 9

(Color online) (a) The total current in the left lead carried by all many-body configurations with $N=2$ at ${\mu }_{\text{L}}=0.6$. (b) The same for ${\mu }_{\text{L}}=0.9$. Other parameters ${\mu }_{\text{R}}=0.35$.

Figure 10

(Color online) (a) The total current in the left lead carried by $N$-particle states and the total charge. for ${\mu }_{\text{L}}=1.5$ and for ${\mu }_{\text{R}}=0.65$. (b) The occupation number of the $N$-particle states.

Figure 11

(Color online) The total charge residing in one- and two-electron states as a function of time for two different values of the bias $\Delta \mu$. $B=1.0\text{T}$, $L_x=300\text{nm}$, and $\hslash {\Omega }_0=1.0\text{meV}$.

Figure 12

(Color online) The total steady-state current for interacting ten SES, and the total charge at $t=270\text{ps}$, for different values of the bias $\Delta \mu$. $B=1.0\text{T}$, $L_x=300\text{nm}$, and $\hslash {\Omega }_0=1.0\text{meV}$.

Figure 13

(Color online) The total current in the right lead for interacting and noninteracting ten SES as a function of the bias $\Delta \mu$ and time. $B=1.0\text{T}$, $L_x=300\text{nm}$, and $\hslash {\Omega }_0=1.0\text{meV}$.