1001.0047.txt

Coulomb interaction and transient charging of excited states in open nanosystems
Valeriu Moldoveanu,1Andrei Manolescu,2Chi-Shung Tang,3and Vidar Gudmundsson4
1National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele, Romania
2Reykjavik University, School of Science and Engineering, Kringlan 1, IS-103 Reykjavik, Iceland
3Department of Mechanical Engineering, National United University, Lienda, Miaoli 36003, Taiwan
4Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
We obtain and analyze the eect of electron-electron Coulomb interaction on the time dependent
current 
owing through a mesoscopic system connected to biased semi-innite 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 xed number of electrons Nand we nd 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.
PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
I. INTRODUCTION
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,4and quantum master equation.5,6,7,8
Most of the results were obtained for noninteracting elec-
trons due to the well known computational diculties to
include time-dependent Coulomb eects.
It is nevertheless clear that the electron-electron inter-
action is important in such problems. An eort to incor-
porate it has been recently done by Kurth et al.9followed
by My oh anen et al.10who have described correlated time-
dependent transport in a short 1D chain dened 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 Kadano-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 specic internal
states of the sample system. In view of the spectroscopy
of excited states11it 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-lution of the generalized master equation (GME). Once
the RDO is dened 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 rst 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.12Gurvitz
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.13The 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.14de-
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 eects like cotunneling.15The con-
nection between the real-time diagrammatic approach of
K onig et al.14and the Nakajima-Zwanzig approach16,17
to the generalized master equation (GME) approach was
made transparent by Timm.18
More recently Li and Yan19combined 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,20using the equation of mo-
tion for the Hubbard operators, have obtained a many-arXiv:1001.0047v1  [cond-mat.mes-hall]  30 Dec 20092
body description of quantum transport in an open sys-
tem and established a connection between the GME and
non-equilibrium Greeen's functions. They studied simple
systems in the steady state regime: a resonant level cou-
pled to a a single vibration mode, an interacting dot with
two spins, and a two-level bridge. Another recent work
by Darau et al.21implemented the GME for a benzene
single-electron transistor and used exact MES to compute
steady state currents within the Markov approximation.21
The stability diagram and the conductance peaks were
obtained and a current blocking due to interferences be-
tween degenerated orbitals was noticed.
In our previous papers7,8we considered the GME
method for the RDO of independent electrons in the Fock
space. We discussed the transient transport through
quantum dots and quantum wires. The contact between
the leads and the sample was switched on at a certain ini-
tial moment t0. We discussed extensively the occupation
of the states within the bias window and the geometrical
eects on the transient currents. We described the cou-
pling between the sample and the leads via a tunneling
Hamiltonian in which we took into account the spatial
extension of the wave functions of both subsystems in
the contact region.
In spite of earlier or more recent attempts a complete
description of the Coulomb eects in the time-dependent
transport is still missing, especially in sample models
larger than a few sites. In the present work we com-
bine the GME method with the Coulomb interaction in
the sample and we analyze the dynamics of the electrons
starting with the moment when the leads are coupled to
the sample until a steady state is reached. The Coulomb
interaction is included in the Hamiltonian of the isolated
sample and the interacting MES are calculated with the
exact diagonalization method. This means the Coulomb
interaction is fully included with no mean eld assump-
tion or density-functional model. The number of single-
electron states (SES) used to dene the matrix elements
of the Hamiltonian of interacting electrons is suciently
large such that the MES of interest are convergent. Due
to the nite bias window only a limited number of MES
participate to the charge transport through the sample,
i. e.only those energetically compatible with the elec-
trons in the leads. Hence the MES of interest are selected
by the chemical potentials in the leads. We calculate the
RDO matrix elements in the subspace of these MES using
the GME. The electron-electron interaction in the leads
is neglected.
It is well known that the Fock space increases expo-
nentially with the number of SES. In addition the time
dependent numerical solution of the GME is also com-
putational expensive. So at this stage we are limited to
describe only few electrons in the system: up to ve in a
small system, but only up to three in a larger one.
The paper is organized as follows. In Section 2 we
brie
y describe the GME, the inclusion of the Coulomb
interaction, and the selection of the MES. Next, in Sec-
tion 3, we show results for three models: a short 1Dchain, a 2D lattice of 12 10 sites, and a nite quantum
wire with parabolic lateral connement. Conclusions and
discussions are presented in Section 4.
II. GME METHOD AND COULOMB
INTERACTION
In this section we summarize the main lines of our
method. The equations apply both to the lattice and
continuous models. The time-dependent transport prob-
lem is considered within the partitioning approach which
is known both from the pioneering work of Caroli22and
from the derivation of the GME. Prior to an initial time
t0the left lead (L) having a \source" role, and the right
lead (R) having a \drain" role, are not connected to the
sample and therefore can be characterized by equilibrium
states with chemical potentials LandRrespectively.
Our aim is to compute the time dependent currents 
ow-
ing through the sample and leads starting at moment t0,
when the three subsystems are connected, until a station-
ary state is reached.
The generic Hamiltonian of the total system consisting
of the sample plus the leads is:
H(t) =HL+HR+HS+HT(t): (1)
Hlwithl=L;R are the Hamiltonians of the leads. We
denote by "qland qlthe single-particle energies and
wave functions respectively, for each lead. Using the cre-
ation and annihilation operators associated to the single-
particle states, cy
qlandcql, we can write
Hl=Z
dq"qlcy
qlcql: (2)
HSis the Hamiltonian of the sample. In the absence
of the interaction the SES have discrete energies denoted
asEnand corresponding one-body wave functions n(r).
Using now the creation and annihilation operators for the
sample SES, dy
nanddn, we can write
HS=X
nEndy
ndn+1
2X
nm
n0m0Vnm;n0m0dy
ndy
mdm0dn0:(3)
The second term in Eq. (3) is the Coulomb interaction.
In the SES basis the two-body matrix elements are given
by:
Vnm;n0m0=Z
drdr0
n(r)
m(r0)u(rr0)n0(r)m0(r0);
(4)
whereu(rr0) is the Coulomb potential.
The third term of Eq. (1) is the so-called tunneling
Hamiltonian describing the transfer of particles between
the leads and the sample:
HT(t) =X
l=L;RX
nZ
dql(t)(Tl
qncy
qldn+h:c:):(5)3
HTcontains two important elements: (1) The time de-
pendent switching functions l(t) which open the contact
between the leads and the sample; these functions mimic
the presence of a time dependent potential barrier. (2)
The coupling Tl
qnbetween a state with momentum qof
the leadland the state nof the isolated sample, with
wave function n. The coupling coecients Tl
qndepend
on the energies of the coupled states and, maybe more
important, on the amplitude of the wave functions in the
contact region. As we have shown in our previous work7,8
this construction allows us to capture geometrical eects
in the electronic transfer. A precise denition of the cou-
pling coecients is however model specic, and will be
mentioned in the next section.
The evolution of our system is completely determined
by the statistical operator W(t) associated to the total
Hamiltonian H(t) dened in Eq.(1). W(t) is the solution
of the quantum Liouville equation with a known initial
value, prior to the coupling of the sample and leads:
i~_W(t) = [H(t);W(t)]; W (tt0) =LRS;(6)
The isolated leads are described by equilibrium distribu-
tions,
l=e(HllNl)
Trlfe(HllNl)g; l=L;R; (7)
and the isolated sample by the density operator S. Af-
ter the coupling moment the dynamics of the sample is
conveniently described by the RDO which is dened by
averaging the total statistical operator over those degrees
of freedom belonging to the leads:
(t) = TrLTrRW(t); (t0) =S: (8)
In the absence of the electron-electron interaction the
MES eigenvectors of HSare bit-strings of the form ji=
ji
1;i
2;::;i
n:::i, wherei
n= 0;1 is the occupation number
of then-th SES. The set fgis a basis in the Fock space
of the isolated sample and the RDO can be seen as a
matrix in this basis. From Eqs. (6)-(8) we obtain in the
lowest (2-nd) order in the coupling parameters Tl
qnthe
GME (see Ref. 7 for details):
_(t) =i
~[HS;(t)]
1
~2X
l=L;RZ
dql(t)([Tql;
ql(t)] +h:c:);(9)
where the coupling operator Tqlhas matrix elements
(Tql)=X
nTl
qnhjdyji: (10)
The operators 
 qland qlare dened as

ql(t) =eitHSZt
t0dsl(s)ql(s)ei(st)"qleitHS;
ql(s) =eisHS
Ty
ql(s)(1fl)(s)Ty
qlfl
eisHSandflis the Fermi function of the lead l.
In the presence of the electron-electron interaction in
the sample the MES which are eigenstates of HSare lin-
ear combinations of bit-strings: HSj) =Ej), where
j) =P
Cji,Cbeing the mixing coecients
which can be found together with the energies Eby
diagonalizing HS. (To distinguish better between the
noninteracting and the interacting MES we use the right
angular bracket for the former and the regular curved
bracket for the later.) Using now the set fgas a basis,
i. e.theinteracting MES, the GME has the same form
as Eq. (9), where the matrix elements of all operators
are now dened in the interacting basis and the matrix
elements of the coupling operators are
(Tql)=X
nTl
qn(jdyj): (11)
Because the sample is open the number of electrons N
contained in the sample is not xed. The Hamiltonian
HSgiven in Eq. (3) commutes with the total \number"
operatorP
ndy
ndn. ThusNis a \good quantum number"
such that any state j) has a xed number of electrons.
So the MES can also be labeled as j) =jN;i) with
i= 0;1;2;:::an index for the ground and excited states
of the MES subset with Nelectrons. The many-body
energies can also be written as E=E(i)
N. In the practical
calculations Nvaries between 0 (the vacuum state) and
Nmaxwhich is the total number of SES considered in the
numerical diagonalization of HS. The total number of
MES is thus 2Nmax.
If the coupling between the leads and the sample is
not too strong we expect that only a limited number of
MES participate eectively to the electronic transport.
These states are naturally selected by the bias window
[R;L]. In the following examples, by selecting suitable
values of the chemical potentials in the leads, we will
truncate the basis of interacting MES to a reasonably
small subset such that we can solve numerically Eq. (9)
with our available computing resources. To relate the
bias window with the eective MES we need to consider
the chemical potential of the isolated sample containing
Nelectrons,
(i)
N=E(i)
NE(0)
N1; (12)
which is the energy required to add the N-th electron on
top of the ground state with N1 to obtain the i-th MES
withNparticles.23We expect the current associated to
the MESjN;i) to depend on the location of the chemical
potential(i)
Nrelatively to the bias window. In particular
it is clear that if at the coupling moment t0the sample is
empty all MES with (i)
NLwill remain empty both
during the transient and the steady states, so they can
be safely ignored when solving the GME.4
III. MODELS AND RESULTS
We have numerically implemented the GME method
both for lattice and continuous models. The sample mod-
els are: a short 1D chain with 5 sites, a 2D rectangular
lattice with 1210 = 120 sites, and a short quantum
wire withe parabolic lateral connement. In all cases the
coupling functions have the form
l(t) = 12
e
t+ 1(13)
with
a constant parameter, such that at the initial mo-
ment, which is t0= 0, we have l(0) = 0 (no coupling),
and in the steady state, for t!1 ,l= 1 (full coupling).
A. A toy model: short 1D chain
In this model the two semi-innite leads are attached
to the ends of a 1D chain with 5 sites. The coupling
between a lead state with wave function  qland a sam-
ple state with wave function nis given by the product
between the wave functions at the contact site:
Tl
qn=Vl 
ql(0)n(il); (14)
where 0 is the contact site of the lead l=L;R, the end
sites of the sample being iL= 1 andiR= 5.
FIG. 1: (Color online) The equilibrium chemical potentials
(0)
Nfor 1N5 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 gure,
i. e.L= 5:25 andR= 4:75.
The reason to call this a toy model is that we can ob-
tain the complete set of 25= 32 MES, i. e.we do not
need to cut the basis of the 5 SES. We also do not need
to cut the MES basis, all matrix elements of the statisti-
cal operator can be numerically calculated, even if not all
of them might be important for the currents. In addition
we will consider the strength of the Coulomb interaction
as a free parameter U, whereas in a realistic systems this
is xed by the electron charge and the dielectric con-
stant of the material. Our goal is to have a qualitativeunderstanding of the underlying physics, and in particu-
lar to show the presence of the Coulomb blocking eects
at certain values of Uor of the chemical potentials of the
leads. The Coulomb matrix elements dened in Eq. (4)
are calculated as
Vnm;n0m0=X
i6=i0
n(i)
m(i0)U
jii0jn0(i)m0(i0):(15)
In Fig.1 we show the equilibrium chemical potentials
(0)
Ncorresponding to ground states with 1 N5 par-
ticles against the interaction strength U. One observes
a linear dependence of (0)
NonU, with slope increasing
withN. Obviously the total Coulomb energy increases
both withUandN.
Let us now brie
y review the Coulomb blockade
scenario.24Suppose the isolated sample contains Nelec-
trons and the chemical potentials of the leads are cho-
sen such that (0)
N< R< L< (0)
N+1. Then the bias
window [R;L] may include one or more of the excited
congurations with Nparticles. In general some states
withNelectrons may have excitation energies exceed-
ingLor even(0)
N+1. This situation corresponds to the
Coulomb blockade phenomenon. Indeed, the addition of
the (N+ 1)-th electron is energetically forbidden. Con-
sequently the current in the steady state should vanish.
However, shorter or longer transient currents are gener-
ated by all many-body congurations in the vicinity of
the bias window.
Fig. 2(a) and 2(b) show the total currents in the left
lead and the total charge residing in the sample for sev-
eral values of the interaction strength. Uis measured
in units of tS, the hopping parameter in the sample,7
and the time is expressed in units of ~=tSwhile the cur-
rent is in units of etS=~. The coupling constant in Eq.
(13) is
= 1. The system is initially empty and thus
(0) =j00000ih00000j.
The chemical potentials of the leads, L= 5:25 and
R= 4:75, are chosen such that in the absence of
Coulomb interaction, i. e. forU= 0,(0)
4is located
within the bias window. In this case we obtain in the
steady state the mean number of electrons about 3.6 and
a non-vanishing current in the leads. This is understand-
able, since (0)
4=E4= 5, which is the 4-th level of
the isolated sample. The occupation of this level in the
steady state is about 0.6, the other states being either full
or empty. Also in this case, the excited states have small
contributions to the steady state current as the system
tends to be in the ground state with N= 3 electrons.
Those contributions may also depend on the coupling
strength of individual states with the leads, but in gen-
eral remain small.25
The situation may change for U6= 0. For the inter-
acting system, e. g. forU= 0:3, the system settles down
in the Coulomb blockade regime, the total current be-
ing almost suppressed in the steady state. This happens
because the interaction pushes the chemical potentials
upwards such that for U= 0:3 both ground states with5
FIG. 2: (Color online) The total current entering the 5 1
sample from the left lead as a function of time for the dierent
values of the interaction strength U. The chemical potentials
of the leads L= 5:25 andR= 4:75.
N= 3 andN= 4 electrons are outside the bias win-
dow and cannot produce steady currents. When the in-
teraction strength is further increased to U= 0:5 and
U= 1 the steady state currents are gradually restored.
This could look surprising, but one can see in Fig.1 that
by increasing Uthe ground state conguration with 3
electrons approaches and enters the bias window. Con-
sequently the transport becomes again possible. Note
that while the steady state currents are not monotonous
w.r.t.Uthe charge absorbed in the system continuously
decreases, Fig. 2(b).
In transport experiments the strength of the electron-
electron interaction is indeed xed. The usual way to
obtain the Coulomb blockade is to vary the chemical po-
tentials of the leads relatively to the energy levels of the
sample, or vice versa. In Fig. 3 we show the currents
in both leads for dierent values of the chemical poten-
tialR, while keeping xed L= 7. The strength of
the Coulomb interaction is U= 1 and(0)
4almost equals
L. The steady state value of the current decreases as
Rincreases, because fewer states are included in the
bias window. The Coulomb blockade onset occurs for
R>5, when(0)
3drops below R. We observe that the
FIG. 3: The time-dependent total currents in the left and
right leads at dierent values of the chemical potential R.
The current in the right lead starts at negative values. Other
parameters: VL=VR= 0:750,U= 1:0.
maximum value of the total current in the left lead does
not change much when Rvaries. In contrast, the tran-
sient current in the right lead is negative and increases
in magnitude as Rincreases. This means that the right
lead feeds the many-body congurations that fall below
R.
The contribution of the excited states to the transient
and steady state currents depends strongly on the bias
window. In Fig. 4 we show the currents entering the sam-
ple from the left lead, carried by the states with N= 2
andN= 3 electrons, for R= 3;4;5 (the cases with non-
vanishing current in the steady state). We also show sep-
arately the contribution to the currents associated to the
ground state congurations, related to (0)
2and(0)
3, and
the complementary contribution of all the excited states
with 2 and 3 particles. In this case the wave vectors of
the ground states are mostly given by the non-interacting
wave vectors:j11000iwith weight 97% and j11100iwith
98% forN= 2 andN= 3 respectively.
ForR= 3 the steady state current of the ground
state conguration is vanishingly small and so the total
negative current associated to two-particle states comes
mostly from the excited states. In the many-body energy
spectrum of the isolated sample we obtain 5 excited con-
gurations with (i)
22[R;L] = [3;7]. AsRmoves up
the steady state current of the ground state with N= 2
becomes also negative. The combined contributions of
the excited states vanishes at R= 5. As can be seen
from Fig. 1 R= 5 is well above (0)
2, but very close to
(0)
3. Consequently, the ground conguration with N= 2
is heavily populated in the steady state, whereas the ex-
cited states have low probability and thus weak current.
Actually, as we have checked, all the currents associated
to each excited state with N= 2 vanish individually. In
the transient regime however the N= 2 currents in all
three cases are dominated by the excites states.
The currents of the excited states having N= 3 elec-6
FIG. 4: The separate contributions to the current of the
ground state with Nparticles and of allexcited states with
Nparticles, for dierent values of R. For completeness we
also include the total currents JLfor the same congura-
tions. The discussion is made in the text. Other parameters:
VL=VR= 0:750,U= 1:0.
trons are positive at R= 3, but change sign at R= 4.
ForR= 5 their magnitude exceeds the contribution of
the ground state which is always positive. A more de-
tailed analysis of the currents carried by specic excited
states will be given for the 2D model.
Finally, both in the transient and in the steady states
the currents have small periodic 
uctuations determined
by the permanent transitions of electrons between the
states in the sample and the states in the leads and
back.25They are best seen in Fig. 2(a). Such 
uctuations
have also been obtained very recently by Kurth et al. us-
ing combination of the non-equilibrium Green's functions
and the time dependent density-functional theory of the
Coulomb interaction.26
B. 2D lattice
We show now results for a 2D rectangular lattice with
1210 sites. For a lattice constant of a= 5 nm this
sample can be seen as a discrete version of a quantum
dot of 60 nm50 nm. We used the lowest 10 SES of the
non-interacting sample in the numerical diagonalization
of the interacting Hamiltonian. This number is sucient
to produce convergent results for the rst 50 MES for
an interaction strength U= 0:8. The Coulomb matrix
elements are calculated in the same way as for the 1Dcase, Eq. (15), except that now the site indices are two-
dimensional, i. e.i= (ix;iy) andi0= (i0
x;i0
y).
The two contact sites are chosen at diagonally opposite
corners of the sample. The coupling coecients are cal-
culated with Eq. (14), like for the 1D chain, and depend
on the wave function of the particular SES at the con-
tact sites. These coecients are illustrated in Fig. 5(a).
The reduced density matrix is calculated using the rst
50 MES. This allowed us to take into account many-body
congurations with up to 3 electrons.
In Fig. 5(b) we show the chemical potentials (i)
Nfor
the ground and excited states with N= 1;2;and 3 par-
ticles. At the initial moment t0= 0 the system is empty.
Based on the previous example, the main contribution to
the currents in the steady states is expected from those
MES with ground state chemical potentials located inside
the bias window [ R;L]. One also observes excited con-
gurations with Nparticles having chemical potentials
larger than (0)
N+1.
FIG. 5: (Color online) (a) The coupling amplitudes jTqnj2
forn= 1;::;5 between single-particle states in the leads with
momentum qand the lowest 5 single-particle states of the
isolated dot. (b) The generalized chemical potentials for N-
particle interacting congurations. The red crosses mark (0)
N
while the other ones correspond to generalized potentials (i)
N
related to the i-th excited state of the Nparticle system.
In the following we discuss the currents carried by the7
various many-body states involved in transport. In a rst
series of calculations we selected the chemical potential
R= 0:2 and used two values of the chemical potential
of the left lead L= 0:4 andL= 0:6. ForR= 0:2 and
L= 0:4 the bias window contains only the 1-st and the
2-nd excited congurations with N= 1, Fig. 5(b). The
ground states for N= 1 andN= 2 are instead located
below and above the bias window, respectively. Conse-
quently the steady state current is very small. When
Lincreases to 0.6 the ground state conguration with
N= 2 enters the bias window and the current increases,
Fig. 6(a).
To analyze the transient regime we split the current
into contributions given by the ground state and excited
states with 1 electron (see Fig. 6(b)). When L= 0:4 the
1-st and 2-nd excited state carry currents exceeding the
current associated to the ground state, which survive all
the way to the steady state. The current corresponding
to the 2-nd excited state is smaller than the current of the
1-st excited state, but comparable to that of the ground
state. This is explained by the strength of the coupling
coecients shown in Fig. 5(a), the 2-nd single-particle
state being stronger coupled to the leads. The remaining
higher excited states give oscillating and fast decaying
transient currents. In Fig. 6(c) L= 0:6 and therefore
higher excited states enter the bias window; their tran-
sient currents are still decaying but at a smaller rate.
Comparing with Fig. 6(a) it in clear that the transient
regime is dominated by excited states.
Next we discuss currents associated with states having
2 and 3 electrons. We keep now xed R= 0:35 and
again increase Lstarting with 0.6. Fig. 7(a) shows the
total currents in the left lead for N= 2 andN= 3. As
the bias increases the transient currents are enhanced,
but they become comparable as the system approaches
the steady state. In Fig. 7(b) the total current on three
particle states shows a dierent behavior: the steady
states value increases drastically when Lmoves up. To
explain this one can look again at the diagram of the
chemical potentials, Fig. 5(b). At L= 0:6 the 3-particle
congurations are above the bias window and as such
they contribute less to the current. In contrast, as L
increases the ground state conguration with N= 3 en-
ters the bias window, the window is closer to the excited
states, and thus the total current increases. Actually, for
L= 0:8 and 0:9 the current for N= 3 does not reach
the steady state in the time interval considered.
Now we look at the contribution of the excited states
withN= 2 for two cases, L= 0:6 andL= 0:9. Again,
the inspection of the diagram in Fig. 5(b) predicts the
results of Fig. 8. When L= 0:6 there is just one excited
conguration within the bias window, in addition to the
ground state. In Fig. 8(a) we see that in the steady state
these two congurations give signicant contributions to
the current, whereas the higher excited states play a role
only in the transient regime. Fig. 8(b) shows that at
L= 0:9 the currents of the excited states and of the
ground state are decreasing, some of them reaching even
FIG. 6: (a) The total currents in the left and right leads for
L= 0:6 andL= 0:4, while keeping R= 0:2. (b) The
partial currents in the left lead for single-particle states when
L= 0:4 andR= 0:2. (c) The partial currents in the left
lead for single-particle states when L= 0:6 andR= 0:2
negative values towards the steady state. This happens
because the bias window includes now the ground state
withN= 3 and the excited states with N= 2 deplete in
the favor of the ground state.
The sign of the current carried by states with Npar-
ticles depends on the placement of the corresponding
ground state chemical potential relatively to the bias win-
dow. For example if we x L= 1:5 andR= 0:65 we8
FIG. 7: (a) The total current in the left lead carried by all
many-body congurations with N= 2, for increasing values
ofL(i. e.0.6,0.7,0.8 and 0.9) and R= 0:2. (b) The same
forN= 3.
obtain(0)
2< L. Fig. 9(a) shows the N-particle cur-
rents when the sample initially contains two electrons in
the ground state. This initial state evolves faster to the
steady state than the empty system. While for N= 3
the current in the left lead is positive, for both N= 2
andN= 1 the currents are negative. The charge re-
siding on each N-particle state and the total charge are
shown in Fig. 9(b). Since single-particle congurations
are unlikely their occupation vanishes. The total charge
accumulated on the N= 3 states increases up to 2, while
the total charge on the N= 2 states decreases from 2 to
0.75. The sign of the current for N= 2 becomes pos-
itive when Ris lowered to 0.2, Fig. 9(c), and exceeds
the current carried by the states with N= 3. This is
because the 1-st and the 2-nd SES practically determine
the ground state with two electrons and thus (0)
2, and
also because the 1-st SES is strongly coupled to the leads.
However, the current with N= 1 is still negative.
FIG. 8: (a) The total current in the left lead carried by all
many-body congurations with N= 2 atL= 0:6. (b) The
same forL= 0:9. Other parameters R= 0:35.
C. Parabolic quantum wire
In this subsection we apply the GME with Coulomb
interaction to describe the transport through a short
quantum wire of length Lx= 300 nm with a parabolic
connement in the y-direction perpendicular to the di-
rection of transport. The contact ends of the isolated
wire atLx=2 are described by hard walls. This is
now a continuous model, where a large functional ba-
sis is used to expand the eigenfunctions of the system
in. In a similar manner we use a functional basis with
complete truncated sets of continuous and discrete func-
tions to expand the eigenfunctions of the semi-innite
parabolic leads in. To show that we can describe the com-
bined geometrical eects imposed on the system by it's
geometry and an external perpendicular magnetic eld
we place the quantum wire is in an external magnetic
eld of strength 1 :0 T. The characteristic connement
energy is given by ~
0= 1:0 meV. We assume GaAs
parameters with m= 0:067me,= 12:4 meV. The
magnetic length modied by the parabolic connement
isaw=p
~=(m
w), with 
2
w= 
2
0+!2
c. and the
cyclotron frequency !c=eB=(mc). AtB= 1:0 T,
aw= 23:87 nm. The semi-innite leads having the same9
FIG. 9: (a) The total current in the left lead carried by N-
particle states and the total charge. for L= 1:5 and for
R= 0:65. (b) The occupation number of the N-particle
states. (c) The total current in the left lead carried by N-
particle states for L= 1:2 andR= 0:2. (d) The occupation
number of the N-particle states and the total charge.parabolic connement and being subject to the same ex-
ternal perpendicular magnetic eld have a continuous en-
ergy spectrum with discrete Landau sub-bands.
The Coulomb potential in Eq. (4) in the 2D wire is
described by
u(rr0) =e2
p
(xx0)2+ (yy0)2+2; (16)
with the small convergence parameter ( =aw) = 0:01
to facilitate the two-dimensional numerical integration
needed for the matrix elements (4).
After the GME (9) has been transformed to the in-
teracting many-electron basis by the unitary transfor-
mation obtained by the diagonalization of HS(3) we
truncate the RDO (8) to 32 MES. For the bias range
0:0
=LR1:7 meV used here 10 SES are
sucient to obtain these lowest 32 states with good accu-
racy. We will be omitting singly occupied states of high
energy that should not be relavant for the parameters
here. The natural strength of the Coulomb interaction
will only give us MES that are occupied by one or two
electrons in the energy range 0 to 6 meV covered by the
32 MES.
Since in the partitioning approach [ HS;HL] = 0 we
have to construct Tl
qnas a non-local overlap ofnand
 L;R
qon the contact regions Cl; l=L;R:8
Tl
qn=Z
Cldrdr0
 
ql(r)n(r)gl
qn(r;r0) +h:c:

:(17)
gl
qn(r;r0) =gl
0exp
l
1(xx0)2l
2(yy0)2
expjEn"qlj

l
E
: (18)
As before"qlis the energy spectrum of lead l, andEn
is the energy of the SES numbered by nin the quan-
tum wire. The quantum number qfor the states in leads
represents both the discrete Landau band number and a
continuous quantum number that can be related to the
momentum of a particular state. Here we use the pa-
rameters1a2
w=2a2
w= 0:25, 
LR
E= 0:25 meV, and
gLR
0= 40 meV for B= 1:0 T. The domain of the over-
lap integral for the leads is 2awinto the lead or the
system forxandx0from each end of the wire at Lx=2
and between4awforyandy0, see Ref. (8) for an exact
denition. All the SES will be coupled to the leads, but
the coupling strength will depend on the character of the
SES, whether it is an edge- or bulk state and other ner
geometrical details that is brought about by the magnetic
eld.
The right chemical potential Ris held at 1 :4 meV
and the transport properties are calculated for dierent
values of the bias 
 by varyingL. Figure 10 compares
the total occupation of all one- electron and two-electron
MES for the interacting system at two dierent values of
the bias. At, 
 = 0:2 meV we see that almost solely10
FIG. 10: (Color online) The total charge residing in one- and
two-electron states as a function of time for two dierent val-
ues of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
0= 1:0
meV.
one-electron states are occupied, while for 
 = 1:2 meV
initially it is likely to have one-electron states occupied,
but very soon the occupation of the two-electron states
becomes as probable with the likelihood of the occupa-
tion of the one-electron states fast reducing with time.
We also have to admit here that even though the steady
state value of the total current through the system can
be deduced by the values of the current at 270 ps, the
charging of the system takes much longer time, since we
are using here a very weak coupling to the leads that
mimics a tunneling regime.
If we now use the average value of the current in the left
and right leads at t= 270 ps as a measure of the steady
state current we get the information displayed in Fig.
11, where the steady state value of the current is shown
for the interacting system as a function of the bias and
compared to the charge in the system. We have a clear
Coulomb blocking in the interacting system. In the case
of a non-interacting system the lack of a gap between the
one- and two electron MES and a strong mixing of the
energy regimes of two- and three-electron states the two-
electron plateau only appears as a small shoulder. The
FIG. 11: The total steady state current for interacting 10
SES, and the total charge at t= 270 ps. for dierent values
of the bias 
 .B= 1:0 T,Lx= 300 nm, ~
0= 1:0 meV.
32 MES selected here include no three-electron or MES
with higher number of electrons. It should be mentioned
here that a dierent choice of the right bias Rcan result
in the system charging faster and thus at the same time
the total current through it being smaller. This comes
from the fact that the states have a dierent coupling to
the leads and the time range shown here is very much in
the transient- or it's long exponential decay regime.
Figure 12 displaying the current in the right lead gives
an idea how the Coulomb blocking plateau appears after
the transition regime. The transition regime where the
FIG. 12: The total current in the right lead for interacting
and non-interacting 10 SES as a function of the bias 
 and
time.B= 1:0 T,Lx= 300 nm, ~
0= 1:0 meV.
right current goes negative, i. e.where it supplies charge
to the system is partially truncated from the gure.11
IV. SUMMARY AND CONCLUSIONS
We calculated time-dependent currents in open meso-
scopic systems composed by a sample attached to two
semi-innite leads, by solving the generalized master
equation for the reduced density operator acting in the
Fock space of the sample. This is the natural frame-
work for including the Coulomb electron-electron inter-
action in the sample, which is the main achievement of
this work. The Coulomb interaction is treated in the
spirit of the exact diagonalization method, i. e.in a pure
many-body manner. The interacting many-body states
of the sample are expanded in the basis of non-interacting
\bit-string" states with unspecied number of electrons.
We believe our method is a viable alternative to a recent
approach based on a time-dependent density-functional
model.9,10,26We used three sample models, a short 1D
wire with 5 sites, but also a larger 2D lattice with 120
sites and a continuous model, whereas the cited group
used much smaller samples even with no structure.26
Indeed, due to computational limitations we could use
only a restricted, eective number of many-body states in
the GME, between 30-50 depending on the model, from
the bottom of the energy spectrum. We chose the bias
window [R;L] and the strength of the sample-leads
coupling parameters VR;Lsuch that only the eective
states contribute to the transport of electrons through
the sample, whereas the states with higher energy are
unreachable by the electrons. Consequently the number
of electrons in the sample can be only up to 3 or 4.
We could calculate the contribution to the charge and
currents in the sample and in the leads respectively, cor-
responding to any particular many-body state. We use
the 1D chain as a toy model to emphasize the dominant
role of the excited states in the transient regime and the
onset of the Coulomb blockade in the steady state. A
similar 1D model with 4 sites 1D has been considered
recently by My oh anen et al.10
As shown also in our previous works on time-dependenttransport in non-interacting systems the GME method
includes information on the energy structure of the sam-
ple, but also on the geometrical properties re
ected in
the wave functions and sample-lead contacts.7,8,25Here
we illustrate these aspects, in the interacting case, for two
nanosystems: a two-dimensional quantum dot described
by a lattice Hamiltonian and a short parabolic quantum
wire. The time-dependent occupation of specic many-
body states was thoroughly analyzed, for dierent values
of the chemical potentials of the leads. It turned out
that the excited states with Nelectrons contribute to
the steady state currents if the ground state conguration
withN+ 1 particles is not available for transport. How-
ever, if(0)
N<Rand at the same time (0)
N+1lies within
the bias window the excited states with Nparticles
are active only in the transient regime and become de-
populated in the steady state regime. This behavior is of
interest in the excited-state spectroscopy experiments.11
To our knowledge the time-dependent currents associated
to excited states have not been discussed theoretically so
far.
Acknowledgments
The authors acknowledge nancial support from the
Development Fund of the Reykjavik University Grant
No. T09001, the Research and Instruments Funds of the
Icelandic State, the Research Fund of the University of
Iceland, the Icelandic Science and Technology Research
Programme for Postgenomic Biomedicine, Nanoscience
and Nanotechnology, the National Science Council of Tai-
wan under contract No. NSC97-2112-M-239-003-MY3.
V.M. also acknowledges the hospitality of the Reykjavik
University, Science Institute and the partial nancial sup-
port from PNCDI2 program (grant No. 515/2009) and
grant No. 45N/2009.
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