# Coulomb interaction and transient charging of excited states in open nanosystems 

Valeriu Moldoveanu, 1 Andrei Manolescu, 2 Chi-Shung Tang, 3 and Vidar Gudmundsson 4 

$^{1}$National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele, Romania 

$^{2}$Reykjavik University, School of Science and Engineering, Kringlan 1, IS-103 Reykjavik, Iceland 

$^{3}$Department of Mechanical Engineering, National United University, Lienda, Miaoli 36003, Taiwan 

$^{4}$Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland 

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 1D chain, 2D lattice, and 2D parabolic quantum wire. 

Due to the increasing interest in ultra-fast electron dynamics considerable progress occurred recently in the theoretical description of time dependent mesoscopic transport. New methods and numerical implementations are rapidly evolving. Transient currents in open nanos- tructures are studied with Green-Keldysh formalism, 1,2,3 scattering theory, 4 and quantum master equation. 5,6,7,8 Most of the results were obtained for noninteracting elec- trons due to the well known computational difficulties to include time-dependent Coulomb effects. 

It is nevertheless clear that the electron-electron inter- action is important in such problems. An effort to incor- porate it has been recently done by Kurth et al. 9 followed by My¨ oh¨ anen et al. 10 who have described correlated time- dependent transport in a short 1D chain defined by a lattice Hamiltonian. The 1D sample was connected to external leads and the current was driven by a time- dependent bias. Those authors used a method based on the Kadanoff-Baym equation for the non-equilibrium Green’s function combined with the time-dependent den- sity functional theory to include the Coulomb interac- tion in the sample. Once the Green’s functions were calculated total average quantities of interest could be obtained, like charge density or current, both in the tran- sitory and in the steady state. However this method does not say much about the dynamics of specific internal states of the sample system. In view of the spectroscopy of excited states 11 it is important to have a theoretical tool for understanding separately the charging and re- laxation of the ground states and excited states in meso- scopic systems in time-dependent conditions. 

Our alternative is to use the statistical, or density op- erator. The complete information about the time evo- lution of each quantum state of the sample is captured in the reduced density operator (RDO), which is the so- 

PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La 

## I. INTRODUCTION 

lution of the generalized master equation (GME). Once the RDO is defined in the Fock space the inclusion of the Coulomb interaction becomes a known computational problem: obtaining the many-electron states (MES) of the sample. The RDO matrix is then calculated in the basis of the interacting MES. 

Let us enumerate some of the previous theoretical schemes to treat transport and electron-electron inter- action with the master equation. One of the first at- tempts to derive a master equation for an interacting sys- tem with time-dependent perturbations belongs to Lan- greth and Nordlander for the Anderson model. 12 Gurvitz and Prager started from the time-dependent Shr¨ odinger equation for the MES wave functions and ended up with Bloch-like rate equations for the density matrix of a quan- tum dot. 13 The electronic currents were calculated in the steady state and it was shown that the Coulomb interac- tion renormalizes the tunneling rates between the leads and the system. In the same context K¨ onig et al. 14 de- veloped a powerful diagrammatic technique by expand- ing the RDO of a mesoscopic system in powers of the tunneling Hamiltonian. The time-dependence of the sta- tistical operator of the coupled and interacting system implies a quantum master equation for the so called pop- ulations. In this method the Coulomb interactions are treated exactly, which makes it appealing for studying various correlation effects like cotunneling. 15 The con- nection between the real-time diagrammatic approach of K¨ onig et al. 14 and the Nakajima-Zwanzig approach 16,17 to the generalized master equation (GME) approach was made transparent by Timm. 18 

More recently Li and Yan 19 combined the n -resolved master equation and the time dependent density- functional method to write down a Kohn-Sham master equation for the reduced single-particle density matrix. Also, Esposito and Galperin, 20 using the equation of mo- tion for the Hubbard operators, have obtained a many- 

