Nonequilibrium plasmons and transport properties of a double-junction quantum wire
We study theoretically the current-voltage characteristics, shot noise, and full counting statistics of a quantum wire double barrier structure. We model each wire segment by a spinless Luttinger liquid. Within the sequential tunneling approach, we describe the system’s dynamics using a master eq...
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| Cite this: | Nonequilibrium plasmons and transport properties of a double-junction quantum wire / J.U. Kim, Mahn-Soo Choi, I.V. Krive, J.M. Kinaret // Физика низких температур. — 2006. — Т. 32, № 12. — С. 1522–1544. — Бібліогр.: 78 назв. — англ. |
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Kim, J.U. Mahn-Soo Choi Krive, I.V. Kinaret, J.M. 2017-06-13T06:01:02Z 2017-06-13T06:01:02Z 2006 Nonequilibrium plasmons and transport properties of a double-junction quantum wire / J.U. Kim, Mahn-Soo Choi, I.V. Krive, J.M. Kinaret // Физика низких температур. — 2006. — Т. 32, № 12. — С. 1522–1544. — Бібліогр.: 78 назв. — англ. 0132-6414 PACS: 71.10.Pm, 72.70.+m, 73.23.Hk, 73.63.–b https://nasplib.isofts.kiev.ua/handle/123456789/120865 We study theoretically the current-voltage characteristics, shot noise, and full counting statistics of a quantum wire double barrier structure. We model each wire segment by a spinless Luttinger liquid. Within the sequential tunneling approach, we describe the system’s dynamics using a master equation. We show that at finite bias the nonequilibrium distribution of plasmons in the central wire segment leads to increased average current, enhanced shot noise, and full counting statistics corresponding to a super-Poissonian process. These effects are particularly pronounced in the strong interaction regime, while in the noninteracting case we recover results obtained earlier using detailed balance arguments. This work has been supported by the Swedish Foundation for Strategic Research through the CARAMEL consortium, STINT, the SKORE-A program, the eSSC at Postech, and the SK-Fund. J.U. Kim acknowledges partial financial support from Stiftelsen Fru Mary von Sydows, fodd Wijk, donationsfond. I.V. Krive gratefully acknowledges the hospitality of the Department of Applied Physies at Chalmers University of Technology. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкоразмерные и неупорядоченные системы Nonequilibrium plasmons and transport properties of a double-junction quantum wire Article published earlier |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire Kim, J.U. Mahn-Soo Choi Krive, I.V. Kinaret, J.M. Низкоразмерные и неупорядоченные системы |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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Nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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nonequilibrium plasmons and transport properties of a double-junction quantum wire |
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Kim, J.U. Mahn-Soo Choi Krive, I.V. Kinaret, J.M. |
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Kim, J.U. Mahn-Soo Choi Krive, I.V. Kinaret, J.M. |
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Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы |
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2006 |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study theoretically the current-voltage characteristics, shot noise, and full counting statistics
of a quantum wire double barrier structure. We model each wire segment by a spinless
Luttinger liquid. Within the sequential tunneling approach, we describe the system’s dynamics
using a master equation. We show that at finite bias the nonequilibrium distribution of plasmons
in the central wire segment leads to increased average current, enhanced shot noise, and full counting
statistics corresponding to a super-Poissonian process. These effects are particularly pronounced
in the strong interaction regime, while in the noninteracting case we recover results obtained
earlier using detailed balance arguments.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120865 |
| citation_txt |
Nonequilibrium plasmons and transport properties of a double-junction quantum wire / J.U. Kim, Mahn-Soo Choi, I.V. Krive, J.M. Kinaret // Физика низких температур. — 2006. — Т. 32, № 12. — С. 1522–1544. — Бібліогр.: 78 назв. — англ. |
| work_keys_str_mv |
AT kimju nonequilibriumplasmonsandtransportpropertiesofadoublejunctionquantumwire AT mahnsoochoi nonequilibriumplasmonsandtransportpropertiesofadoublejunctionquantumwire AT kriveiv nonequilibriumplasmonsandtransportpropertiesofadoublejunctionquantumwire AT kinaretjm nonequilibriumplasmonsandtransportpropertiesofadoublejunctionquantumwire |
| first_indexed |
2025-11-26T23:38:51Z |
| last_indexed |
2025-11-26T23:38:51Z |
| _version_ |
1850781668269359104 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12, p. 1522–1544
Nonequilibrium plasmons and transport properties of a
double-junction quantum wire
Jaeuk U. Kim1, Mahn-Soo Choi2, Ilya V. Krive3,4, and Jari M. Kinaret3
1Department of Physics, G�teborg University, SE-412 96 G�teborg, Sweden
2Department of Physics, Korea University, Seoul 136-701, Korea
3Department of Applied Physics, Chalmers University of Technology, SE-412 96 G�teborg, Sweden
4B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkov 61103, Ukraine
Received January 3, 2006, revised January 22, 2006
We study theoretically the current-voltage characteristics, shot noise, and full counting statis-
tics of a quantum wire double barrier structure. We model each wire segment by a spinless
Luttinger liquid. Within the sequential tunneling approach, we describe the system’s dynamics
using a master equation. We show that at finite bias the nonequilibrium distribution of plasmons
in the central wire segment leads to increased average current, enhanced shot noise, and full count-
ing statistics corresponding to a super-Poissonian process. These effects are particularly pro-
nounced in the strong interaction regime, while in the noninteracting case we recover results ob-
tained earlier using detailed balance arguments.
PACS: 71.10.Pm, 72.70.+m, 73.23.Hk, 73.63.–b
Keywords: quantum wire, Luttinger liquid, shot noise, full counting statistics.
1. Introduction
The recent discovery of novel one-dimensional
(1D) conductors with non-Fermi liquid behaviors has
inspired extensive research activities both in theory
and experiment. The generic behavior of electrons in
1D conductors is well described by the Luttinger
liquid (LL) theory, a generalization of the Tomo-
naga–Luttinger (TL) model [1–3]. Luttinger liquids
are clearly distinguished from Fermi liquids by many
interesting characteristics. The most important exam-
ples among others would be (a) the bosonic nature of
the elementary excitations (i.e., the collective density
fluctuations) [4], (b) the power-law behavior of cor-
relations with interaction dependent exponents [5–8],
and (c) the spin-charge separation [9]. All these cha-
racteristics cast direct impacts on transport properties
of an 1D system of interacting electrons, including the
shot noise and the full counting statistics as we discuss
in this paper.
In this paper we will consider a particular struc-
ture, namely, the single-electron transistor (SET),
made of 1D quantum wires (QWs). A conventional
SET with noninteracting electrodes itself is one of the
most extensively studied devices in recent years in
various contexts [10]. The SET of interacting 1D
QWs has attracted renewed interest as a tunable de-
vice to test the LL theory and thereby improve the un-
derstanding of the 1D interacting systems. In parti-
cular, the recent experimental reports on the
temperature dependence of the resonance level width
in a SET of semiconductor QWs [11] and on the tem-
perature dependence in a SWNT SET [12] stirred a
controversy, with the former consistent with the con-
ventional sequential tunneling picture [13,14] and the
latter supporting the correlated sequential tunneling
picture [12,15]. The issue motivated the more recent
theoretical works based on a dynamical quantum
Monte Carlo method [16] and on a function renor-
malization group method [17–20], and still remain
controversial.
Another interesting issue on the SET structure of
1D QWs is the effects of the plasmon modes in the
central QW [21–24]. It was suggested that the
plasmon excitations in the central QW leads to a
© Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret, 2006
power-law behavior of the differential conductance
with sharp peaks at resonances to the plasmon modes
[21]. The plasmon excitations were also ascribed to
the shot noise characteristics reflecting the strong LL
correlations in the device [22]. Note that in these both
works, they assumed fast relaxation of plasmon exci-
tations. We hereafter refer to this approach as «equi-
librium plasmons». However, it is not clear, espe-
cially, in the presence of strong bias (eV k TB�� ) how
well the assumption of equilibrium plasmons can be
justified. This question is important since the plasmon
modes excited by electron tunneling events can influ-
ence the subsequent tunnelings of electrons and hence
act as an additional source of fluctuations in current.
In other words, while the average conductance may
not be affected significantly, the effects of the
«nonequilibrium plasmons» can be substantial on shot
noise characteristics [25–31]. Indeed, a recent work
suggest that the nonlinear distribution of the plasmon
excitations itself is of considerable interest [23] and it
can affect the transport properties through the de-
vices, especially the shot noise, significantly [23,24].
Moreover, even if the precise mechanism of plasmon
relaxation in nanoscale structures is not well known
and it is difficult to estimate its rate, recent computer
simulations on carbon nanotubes indicate that the
plasmon life time could be of the order of a picosec-
ond, much longer than those in three-dimensional
structures [32].
It is therefore valuable to investigate systemati-
cally the effects of nonequilibrium distribution of
plasmon excitations with finite relaxation rate on the
transport properties through a SET of 1D QWs. In
this work we will focus on the shot noise (SN) charac-
teristics and full counting statistics (FCS), which are
more sensitive to the nonequilibrium plasmon excita-
tions than average current-voltage characteristics. We
found that the SN characteristics and the FCS both
indicate that the fluctuations in the current through
the devices is highly super-Poissonian. We ascribe this
effects to the additional conduction channels via exci-
tation of the plasmon modes. To demonstrate this rig-
orously, we present both analytic expression of simpli-
fied approximate models and numerical results for the
full model system. Interestingly, the enhancement of
the noise and hence the super-Poissonian character of
the FCS is more severe in the strong interaction limit.
Further more, the sensitive dependence of the super-
Poissonian shot noise on the nonequilibrium plasmon
excitations and their relaxation may provide a useful
tool to investigate the plasmon relaxation phenomena
in 1D QWs.
The paper is organized as following: In Sec. 2 we
introduce our model for a 1D QW SET and briefly ex-
amine the basic properties of the tunneling rates
within the golden rule approximation. We also intro-
duce the master equation approach to be used through-
out the paper, and discuss the possible experimental
realizations. Before going to the main parts of the pa-
per, in Sec. 3, we first review the results of the previ-
ous work [23], namely, the nonequilibrium distribu-
tion of the plasmons in the central QW in the limit of
vanishing plasmon relaxation. This property will be
useful to understand the results in the subsequent sec-
tions. We then proceed to investigate the consequence
of the nonequilibrium plasmons in the context of the
transport properties. We first consider average current
in Sec. 4, and then discuss shot noise in Sec. 5.
Finally, we investigate full counting statistics in
Sec. 6. Section 7 concludes the paper.
2. Formalism
The electric transport of a double barrier structure
in the (incoherent) sequential tunneling regime can be
described by the master equation [14,33,34]
�
�
� � � � � � � �
t
P N n t N n N n P N n t N( , { }, ) [ ( , { } , { }) ( , { }, ) ( , {� � n N n P N n t
nN
� �
��
�� } , { }) ( , { }, )]
{ }
, (1)
where P N n t( , { }, ) is the probability that at time t there
are N (excess) electrons and { } ( , , ..., , ...)n n n nm� 1 2
plasmon excitations (i.e., collective charge excitations),
that is, nm plasmons in the mode m on the quantum
dot. The transitions occur via single-electron tunneling
through the left (L) or right (R) junctions (see Fig. 1).
The total transition rates � in master equation (1) are
sums of the two transition rates �L and �R where
�L/R N n N n( , { } , { })� � � is the transition rate from a
quantum state ( , { })N n� � to another quantum state
( , { })N n via electron tunneling through L/R-junction.
Master equation (1) implies that, with known transi-
tion rates, the occupation probabilities P N n t( , { }, ) can
be obtained by solving a set of linear first order differen-
tial equations with the probability conservation
P N n t
N n
( , { }, )
,{ }
� � 1.
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1523
In the long time limit the system converges to a
steady-state with probability distribution
lim ( , { }, )
t
P N n t
��
�
P N nst ( , { }),
irrespective of the initial preparation of the system.
To calculate the transition rates, we start from the
Hamiltonian of the system. The reservoir temperature
is assumed zero (T � 0), unless it is stated explicitly.
2.1. Model and Hamiltonian
The system we consider is a 1D quantum wire SET.
Schematic description of the system is that a finite
wire segment, which we call a quantum dot, is weakly
coupled to two long wires as depicted in Fig. 1. The
chemical potential of the quantum dot is controlled by
the gate voltage (VG) via a capacitively coupled gate
electrode. In the low-energy regime, physical proper-
ties of the metallic conductors are well described by
linearized dispersion relations near the Fermi points,
which allows us to adopt the Tomonaga–Luttinger
Hamiltonian for each wire segment. We model the
system with two semi-infinite LL leads and a finite LL
for the central segment. The leads are adiabatically
connected to reservoirs which keep them in internal
equilibria. The chemical potentials of the leads are
controlled by source-drain voltage (V), and the wires
are weakly coupled so that the single-electron tunnel-
ing is the dominant charge transport mechanism, i.e.,
we are interested in the sequential tunneling regime.
Rigorously speaking, the voltage drop between the
two leads (V) deviates from the voltage drop between
the left and right reservoirs (sayU) if electron trans-
port is activated [35,36]. However, as long as the tun-
neling amplitudes through the junctions (barriers) are
weak so that the Fermi golden rule approach is appro-
priate, we estimate V U� .
The total Hamiltonian of the system is then given
by the sum of the bosonized LL Hamiltonian
� � � �H H H HL D R0 � accounting for three isolated
wire segments labeled by � � ( , , )L D R , and the tunnel-
ing Hamiltonian �HT accounting for single-electron
hops through the junctions L and R at XL and XR, re-
spectively:
� � � .H H HT� 0 (2)
Using standard bosonization technique, the Hamil-
tonian describing each wire segment can be expressed
in terms of creation and annihilation operators for col-
lective excitations ( �†b and �b) [4,37]. For the semi-in-
finite leads, it reads
� � �
,
†
,H mb b L R
M
m
m
m�
� ��
� �
�
� ���
�
� �
1 1
for = , , (3)
where the index
labels the M transport sectors of the
conductor and m the wave-like collective excitations on
each transport sector. The effects of the Coulomb inter-
action in 1D wire are characterized by the Luttinger pa-
rameter g�: g � 1 for noninteracting Fermi gas and
0 1� �g for the repulsive interactions (g �� 1 in the
strong interaction limit). Accordingly, the velocities of
the collective excitations are also renormalized as
v v /gF� �� . The energy of an elementary excitation in
sector
is given by � �� �� �v /L where L is the length
of the wire and � the Planck constant. For instance, if
the wire has a single transport channel (usually referred
to as spinless electrons), e.g., a wire with one transport
channel under a strong magnetic field, the system’s dy-
namics is determined by collective charge excitations
(plasmons) alone (
�� and M � 1). If, however, the
spin degrees of freedom survive, the wire has two trans-
port sectors (M � 2); plasmons (
�� ) and spin-waves
(
�� ) [37]. If the system has two transport channels
with electrons carrying spin (M � 4), as is the case with
SWNTs, the transport sectors are total-charge-plasmons
(
�� ), relative-charge-plasmons (
�� � ), total-spin-
waves (
��� , and relative-spin-waves (
�� � ) [38,39].
For the short central segment, the zero-mode need
to be accounted for as well, which yields
� � �
,
†
,H mb bD
m
m m� � �
�
�
��
�
� �
1
� �
�
�� ��
�
�
�
�� �
�2 2
2 2
Mg
N N
Mg
N EG r( � ) � . (4)
In the second line of Eq. (4), which represents the
zero-mode energy of the quantum dot, the operator �N�
measures the ground-state charge, i.e., with no excita-
tions, in the
-sector. The zero-mode energy systemati-
cally incorporates Coulomb interaction in terms of the
1524 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
3 quantum wires
connected to reservoirs
Fig. 1. Model system. Two long wires are adiabatically
connected to reservoirs and a short wire is weakly coupled
to the two leads. Tunneling resistances at junction points
XL and XR are RL/R, and the junction capacitances are
considered equal C CL R� . Quantum dot is capacitively
coupled to the gate electrode.
Luttinger parameter g� in the QD. To refer to the
zero-mode energy later in this paper, we define the
«charging energy» EC as the minimum energy cost to
add an excess electron to the QD in the off-Coulomb
blockade regime, i.e.,
E E E /gC D D N /G
� � ��[ ( , ) ( , )] .2 0 1 0 1 2 �� � (5)
Note that this is twice the conventional definition,
and includes the effects of finite level spacing. Hence,
the charging energy is smallest in the noninteracting
limit (g� � 1) and becomes the governing energy scale
in the strong interaction limit (g� �� 1). The origin of
the charging energy in conventional quantum dots is
the long range nature of the Coulomb interaction. In
the theory of Luttinger liquid, the long range inter-
action can easily be incorporated microscopically
through the interaction strength g� . For the effect of
the finite-range interaction across a tunneling junc-
tion, for instance see Refs. 40, 41.
Note that charge and spin are decoupled in Lut-
tinger liquids, which implies the electric forces affect
the (total) charge sector only; due to intrinsic e e� in-
teractions, g� � 1 but g� �� � 1, and the gate voltage
shifts the band bottom of the (total) charge sector as
seen by the dimensionless gate voltage parameter NG
in Eq. (4). As will be shown shortly, the transport
properties of the L/R-leads are determined by the LL
interaction parameter g� and the number of the trans-
port sector M. In this work, we consider each wire
segment has the same interaction strength for the (to-
tal) charge sector, g g L� ��
( ) g gR D
� �
( ) ( )� . Accord-
ingly, the energy scales in the quantum are written by
� � �� � �� �( )D
F Dv /g L� and � � �� �0 � �� �v /LF D.
We consider the ground-state energy in the QD is
the same as those in the leads, by choosing the refer-
ence energy Er in Eq. (4) equals the minimum value
of the zero-mode energy,
E
N
Mg
N
Mg
M
Mr
G G�
�
��
�
�
�
�
�
�
�
min ,
( ) ( )� � ��
�
�
�
2 2
0
2
1
2
1
2
, (6)
where min( , )x x� denotes the smaller of x and �x , and
the gate charge NG is in the range N MG � [ , ]0 .
The zero-mode energy in the QD,
E
Mg
N N
M
N EG r0
2 0 2
2 2
� � �
�
�� ��
�
�
� �
( ) (7)
yields degenerate ground states for N � 0 and N � 1
excess electrons when N M g /G � � [( ) ]1 1 22
� . Here
we replaced N� by the number of the total excess
electrons N since N N� � , and N� are all either even
or odd integers, simultaneously; in the case of the
SWNTs with N N
i s i s� �
, , excess electrons, where
i � 1 2, is the channel index and s � � �, is the spin
index of conduction electrons (M = 4), N N� � ,
N N Ni
i
i� � �
�( ), , , N N Ns
s
s�� � ��( ), ,1 2 ,
N N Ni
i
i i�� � � ��
( ) ( ), ,1 .
From now on we consider only one spin-polarized
(or spinless) channel unless otherwise stated — our
focus is on the role of Coulomb interactions, and the
additional channels only lead to more complicated ex-
citation spectra without any qualitative change in the
physics we address below. A physical realization of
the single-channel case may be obtained, e.g., by ex-
posing the quantum wire to a large magnetic field.
For the system with high tunneling barriers, the
electron transport is determined by the bare electron
hops at the tunneling barriers. The tunneling events in
the DB structure are described by the Hamiltonian
� [ � ( ) � ( ) .]†
,
H t X XT D
L R
�
�
� � � � �
�
� � h.c , (8)
where � ( )†�
� �X and � ( )�� �X are the electron creation
and annihilation operators at the edges of the wires
near the junctions at XL and XR. As mentioned ear-
lier, the electron field operators �� and � †� are related
to the plasmon creation and annihilation operators �b
and �†b by the standard bosonization formulae. Differ-
ent boundary conditions yield different relations be-
tween electron field operators and plasmon operators.
Exact solutions for the periodic boundary condition
have been known for decades [3,37] but the open
boundary conditions which are apt for our system of
consideration has been investigated only recently (see,
for example, Refs. 38, 42–44).
The dc bias voltage V V VL R� between L and R
leads is incorporated into the phase factor of the
tunneling matrix elements t t ieV t/� � �� �� | | exp( ) by a
time-dependent unitary transformation [10]. Here
VL/R � VC/CR/L is voltage drop across the L/R-
junction where C C C / C CL R L R� ( ) is the effective
total capacitance of the double junction, and the bare
tunneling matrix amplitudes | |tL/R are assumed to be
energy independent. Experimentally, the tunneling
matrix amplitude is sensitive to the junction properties
while the capacitance is not. For simplicity, the capaci-
tances are thus assumed to be symmetric C CL R�
throughout this work. By junction asymmetry we mean
the asymmetry in (bare) squared tunneling amplitudes
| |tL/R
2. The parameter R t / tL R� | | | |2 2 is used to de-
scribe junction asymmetry; R � 1 for symmetric junc-
tions and R �� 1 for highly asymmetric junctions.
It is known that, at low energy scales in the
quantum wires with the electron density away from
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1525
half-filling, the processes of backward and Umklapp
scattering, whose processes generate momentum trans-
fer across the Fermi sea (� 2kF), can be safely ignored
in the middle of ideal 1D conductors [37], including
armchair SWNTs [38]. The Hamiltonian (2) does not
include the backward and Umklapp scattering (except
at the tunneling barriers) and therefore it is valid
away from half-electron-filling.
We find that, in the regime where electron spin
does not play a role, the addition of a transport chan-
nel does not change essential physics present in a sin-
gle transport channel. Therefore, we primarily focus
attention to a QW of single transport channel with
spinless electrons and will comment on the effects due
to multiple channel generalization, if needed.
2.2. Electron transition rates
The occupation probability of the quantum states
in the SET system changes via electron tunneling
events across L/R-junctions. In the single-electron
tunneling regime, the bare tunneling amplitudes
| |tL/R are small compared to the characteristic energy
scales of the system and the electron tunneling is the
source of small perturbation of three isolated LLs. In
this regime, we calculate transition rates �L/R be-
tween eigenstates of the unperturbed Hamiltonian H0
to the lowest nonvanishing order in the tunneling am-
plitudes | |tL/R . In this golden rule approximation, we
integrate out lead degrees of freedom since the leads
are in internal equilibria, and the transition rates are
given as a function of the state variables and the ener-
gies of the QD only [21,23],
�L/R N n N n( , { } , { })� � � �
� �2 2�
� �
�
| | ( ) ({ }, { })t W n nL/R L/R D .
(9)
In Eq. (9)WL/R is the change in the Gibbs free energy
associated with the tunneling across the L/R-junction,
W E N n E N n N N eVL/R D D L/R� � � � ��( , { }) ( , { }) ( )� .
(10)
Here E N n N n H N nD D( , { }) , { }| � | , { }� ! is the energy of
the eigenstate | , { }N n ! of the dot and L/R correspond
to � / . For the QD with only one transport channel
with spinless electrons only,
E N n mn
N N
g
ED p
m
m
G
D
r( , { })
( )
,�
�"
#
$
$
%
&
'
'
�
�
�
��
1
2
2 (11)
where � ��p � accounting that we consider only
charge plasmons and n n b b nm m m m m� !| � � |† is the num-
ber of plasmons in the mode m. Note that the excita-
tion spectrum (11) in the QD consists of charge
excitations which are formed by changing electron
number N in the zero-mode, and plasmon excitations
which are neutral excitations. In this work, we
sharply distinguish those two different excitations.
The function �( )x in Eq. (9) is responsible for the
plasmon excitations on the leads, and given by (see,
e.g., Ref. 14)
� �
�
�( ) ( , ) ( , )†� ! �
�
�
(1
2
0
�
� � � �dt X X ti te � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
1
2
1 2 1
2 2
1 1
� �
�
)
* )�
�
�
� �v
g
v
i
F
/ g
F
+
�
( ) ,
,
, ,
,
,
2 2
1
e ��
*
/
�( )
,
(12)
where ) � 1/k TB is the inverse temperature in the
leads, + is a short wavelength cutoff, and �( )z is the
gamma function. The exponent * � �
( )g /M1 1 is a
characteristic power law exponent indicating interac-
tion strength of the leads with M transport sectors
(hence, in our case, M � 1). At g � 1 (noninteracting
case), the exponent * � 0 and it grows as g - 0
(strong interaction). The decrease of the exponent *
with increasing M implies that the effective interac-
tion decreases due to multi-channel effect, and the
Luttinger liquid eventually crosses over to a Fermi
liquid in the many transport channel limit [45,46].
For the noninteracting electron gas, the spectral density
� �( ) is the TDOS multiplied by the Fermi–Dirac
distribution function fFD( )� � [ exp( )]1 1
)� ; � �( ) �
� f / vFD F( ) ( )� �� for g � 1. At zero temperature, � �( ) is
proportional to a power of energy,
lim ( ) ( )
(| | )
( )T Fv
/
�
� �
0
1
1
� � �
�
� �
*
�
.
�
�
�
, (13)
where � � +�
�v / gF
/ g1 1( ) is a high energy cut-off.
At zero temperature � �( ) is the TDOS for the negative
energies and zero otherwise (as it should be), imposed
by the unit step function .( )�� .
The function � D in Eq. (9) accounts for the plasmon
transition amplitudes in the QD, and is given by
� /D N N Dn n N n X N n({ }, { }) | , { }| � ( )| , { } |,
†� � � � ! � �1
2� �
� � !�
/N N DN n X N n, | , { }| � ( )| , { } |1
2� � , (14)
where we used that the zero-mode overlap is unity for
N N� � 0 1 and vanishes otherwise. The overlap
integrals between plasmon modes are, although
straightforward, quite tedious to calculate, and we
refer to Appendix A for the details. The resulting
overlap of the plasmon states can be written as a
function of the mode occupations nm ,
1526 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
| { }| � ( )| { } | | { }| � ( )| { } |† � ! � � ! �n X n n X nD D� �� �
2 2
�
�
�
��
�
�
�� �
1
L L
n n
D D
D�
1
�
+
({ }, { }) (15)
with
1({ }, { })
!
!
| |
( )
( )
n n
gm
n
n
L
m
m
m
n
nm nm
� �
�
�
��
�
�
��
�
� �
�2
� �
1
1 m
m mn n
gm( )
| | ,
�
�
�
�
��
�
�
��
"
#
$
%
&
'
1
2
(16)
where n n nm m m
( ) ( , )� � �min and n n nm m m
( ) ( , )� � �max ,
* D Dg� �
( )1 1 for the QD with one transport sector,
and L xa
b ( ) are Laguerre polynomials. Additional tran-
sport sectors would appear as multiplicative factors of
the same form as 1({ }, { })n n� and result in a reduction
of the exponent * D. Notice that in the low energy
scale only the first few occupations nm and n m� in the
product differ from zero, participating to the transi-
tion rate (9) with nontrivial contributions.
2.3. Plasmon relaxation process in the quantum dot
In general, plasmons on the dot are excited by tun-
neling events and have a highly nonequilibrium distri-
bution. The coupling of the system to the environment
such as external circuit or background charge in the
substrate leads to relaxation towards the equilibrium.
While the precise form of the relaxation rate, �p , de-
pends on the details of the relaxation mechanism, the
physical properties of our concern do not depend on
the details. Here we take a phenomenological model
where the plasmons are coupled to a bath of harmonic
oscillators by
H g b b amn m n
m n
plasmon bath h. c.
� �� ( )†
,
�
�
�
. (17)
In Eq. (17) a� and a�
† are bosonic operators describ-
ing the oscillator bath and gmn
� is the coupling con-
stants. We will assume an Ohmic form of the bath
spectral density function
J gmn mn p( ) | | ( ) ,3 / 3 3 � 3�
�
�� � �� 2
0� (18)
where 3� is the frequency of the oscillator cor-
responding to a� , � p is a dimensionless constant
characterizing the bath spectral density, and �0
1
�
�
�
2 2 2v L t tF D L R(| | | | ) is the natural time scale of
the system. Within the rotating-wave approximation,
the plasmon transition rate due to the harmonic oscil-
lator bath is given by
� �p p
p p
W
n n
W /
p
({ } { })� � �
�
�
�
�0
1e
(19)
with
W n np p m
m
m� � ��� ( ) ,
where � ��p � is the plasmon energy. Note that these
phenomenological rates obey detailed balance and,
therefore, at low temperatures only processes that
reduce the total plasmon energy occur with apprecia-
ble rates.
2.4. Matrix formulation
For later convenience, we introduce a matrix nota-
tion for the transition rates �, with the matrix ele-
ments defined by
[ � ( )] ( , { } , { }) ,{ },{ }� � �
�
� � 0 � �N N n N nn n � 1 (20)
i.e, the element ({ }, { })�n n of the matrix block � ( )� �
� N
is the transition rate ��( , { } , { })N n N n0 � �1 . Simi-
larly,
[ � ] [ � � ]{ },{ } { },{ } { },{ }
{ }
� � �� � �
0
n n n n n n
n
� �
�
�
��
� �/ , (21)
and
[ � ( )] ({ }, { }){ },{ }�p n n pN n n� � � � �
���
��
�/{ },{ }
{ }
({ }, { })n n p
n
n n� . (22)
Master equation (1) can now be conveniently ex-
pressed as
d
dt
P t P t| ( ) � | ( )! � � !� (23)
with
� � ( � � � )
,
� � � � �� � ��
�
�p
L R
� � �
�
0 ,
where | ( )P t ! is the column vector (not to be confused
with the «ket» in quantum mechanics) with elements
given by ! �N n P t P N n t, { }| ( ) ( , { }, ). Therefore, the
time evolution of the probability vector satisfies
| ( ) exp( � )| ( )P t t P! � � !� 0 . (24)
In the long-time limit, the system reaches a steady
state | ( )P 4 !.
The ensemble averages of the matrices �� �
� can then
be defined by
! � !� ��� ( ) , { }| � | ( )
,{ }
� �� �t N n P t
N n
. (25)
We will construct other statistical quantities such as av-
erage current and noise power density based on Eq. (25).
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1527
3. Steady-state probability distribution
of nonequilibrium plasmons
By solving the master equation (23) numerically
(without plasmon relaxation), in Ref. 23, we obtained
the occupation probabilities of the plasmonic many-
body excitations as a function of the bias voltage and
the interaction strength. We found that in the weak to
noninteracting regime, * � 0 or g /� �1 1 1( )* for
the wire with one transport sector, the nonequilibrium
probability of plasmon excitations is a complicated
function of the detailed configuration of state occupa-
tions { } ( , , ..., , ...)n n n nm� 1 2 .
In contrast, the nonequilibrium occupation proba-
bility in the strong interaction regime with (nearly)
symmetric tunneling barriers depends only on the to-
tal energy of the states, and follows a universal form
irrespective of electron charge N in the QD. In the
leading order approximation, it is given by
P
Z eV p
( )( ) exp
( )
log0 1 3 1
2
�
* � �
�
� �
�
�
�
�
�
�
�
� , (26)
where Z is a normalization constant. Notice that � is
the total energy, including zero-mode and plasmon
contributions. The distribution has a universal form
which depends on the bias voltage and the interaction
strength. The detailed derivation is in Appendix B.
This analytic form is valid for the not too low ener-
gies ( , )� eV � 3�p and in the strongly interacting re-
gime * � 1. More accurate approximation formula
(Eq. (B.13)) is derived in Appendix B.
For symmetric junctions, the occupation probabili-
ties fall on a single curve, well approximated by the an-
alytic formulas Eqs. (26) and (B.13), as seen in the in-
sets in Fig. 2, where P( )� is depicted as a function of
the state energies for g � 0 2. (a) and g � 0 5. (b), with
parameters R � 1 for the inset and R � 100 for the main
figures (eV p� 6� , N /G � 1 2). For the asymmetric
junctions, the line splits into several branches, one for
each electric charge N, see the figure. However, as seen
in Fig. 2,a, if the interaction is strong enough (g � 0.3
for R � 100 and eV p� 6� ), each branch is, independ-
ently, well described by Eq. (26) or (B.13). For
weaker interactions, g � 0.3 for R � 100 and eV p� 6� ,
the analytic approximation is considerably less accurate
as shown in Fig. 2,b. Even in the case of weaker inter-
actions, however, the logarithms of the plasmon occu-
pation probabilities continue to be nearly linear in � but
with a slope that deviates from that seen for symmetric
junctions.
4. Average current
In terms of the tunneling current matrices across
the junction L/R
� ( � � )IL/R L/R L/Re� ��
� � � , (27)
the average current I t tL/R L/R( ) � ( )� !I through
L/R-junction is
I t N n P tL/R
N n
L/R( ) , { }| � | ( )
,{ }
� !� I . (28)
The total external current I t t( ) �( )� !I , which in-
cludes the displacement currents associated with
charging and discharging the capacitors at the left
and right tunnel junctions, is then conveniently writ-
ten as
1528 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
a
100
10–5
10–10
10–15
10–20
10–25
P( )� 010
–1010
–20
10
0 2 4 6 8 10
0 5 10
� �/ p
N = 0
N = 1
N = 2
N = 3
P(0)
P (1)
100
10–5
10–10
10–15
P( )�
0 2 4 6 8 10 � �/ p
N = 0
N = 1
N = 2
N = 3
P(0)
P (1)
100
10–5
10
–10
0 5 10
Fig. 2. The occupation occupation probability P( )� as a
function of the mode energy � �/ p. The energy � is abbrevia-
tion for E N nD( ,{ }), the bias eV p� 6� , the asymmetry pa-
rameter R � 100, and N /G � 1 2 (T � 0). Two analytic ap-
proximations, Eq. (26) (blue curve) and Eq. (B.13) (cyan
curve), are fitted to the probability distribution of the
charge mode N � 1 (blue circle). The interaction parameter
is g � 02. (a) and 05. (b). In the inset the case of symmetric
junctions (R � 1) is plotted with the same conditions.
I t C/C I t
L R
( ) ( ) ( ),
,
�
�
�
�
� � (29)
where C C CL R
� 1 1 1. As the system reaches
steady-state in the long-time limit, the charge current is
conserved throughout the system, I I� 4 �( ) IL( )4 �
� 4IR( ).
One consequence of nonequilibrium plasmons is the
increase in current as shown in Fig. 3, where the aver-
age current is shown as a function of the bias for dif-
ferent interaction strengths. The currents are normal-
ized by I I eV Ec C� �( )2 with no plasmon relaxation
(� p � 0) for each interaction strength g, and we see
that the current enhancement is substantial in the
strong interaction regime (g � 0.5), while there is ef-
fectively no enhancement in noninteracting limit g � 1
(the two black lines are indistinguishable in the fig-
ure). In the weak interaction limit the current in-
creases in discrete steps as new transport channels
become energetically allowed, while at stronger inter-
actions the steps are smeared to power laws with expo-
nents that depend on the number of the plasmon states
involved in the transport processes.
Including the spin sector results in additional peaks
in the average current voltage characteristic that can
be controlled by the transverse magnetic field [47,48].
The current-voltage characteristics show that, in
the noninteraction limit, the nonequilibrium approach
predicts similar behavior for the average current as the
detailed balance approach which assumes thermal
equilibrium in the QD. In contrast, in the strong
interaction regime, nonequilibrium effects give rise to
an enhancement of the particle current.
Experimentally, however, the current enhancement
may be difficult to attribute to plasmon distribution
as the current levels depend on barrier transparencies
and plasmon relaxation rates, and neither of them can
be easily tuned. We now turn to another experimental
probe, the shot noise, which is more sensitive to
nonequilibrium effects.
5. Current noise
Noise, defined by
S d t t t
t
i( ) lim [ �( )�( ) �( ) ]3
��� ! � !
��
�
�
(2 2e I I I , (30)
describes the fluctuations in the current through a
conductor [49,50]. Thermal fluctuations and the
discreteness of the electron charge are two fundamen-
tal sources of the noise among others which are sys-
tem specific [30,31]. Thermal (equilibrium) noise is
not very informative since it does not provide more
information than the equilibrium conductance of the
system. In contrast, shot noise is a consequence of the
discreteness of charge and the stochastic nature of
transport. It thus can provide further insight beyond
average current since it is a sensitive function of the
correlation mechanism [29], internal excitations
[51–53], and the statistics of the charge carriers
[54–57]. Here we will particularly focus on the roles
of the nonequilibrium plasmon excitations in the
SETs of 1D QWs [21–24].
Within the master equation approach, the correlation
functions K t t
t�� � �� �� �� !( ) lim � ( ) � ( )
I I in Eq. (30)
can be deduced from the master equation (23). In the
matrix notation they can be written as [27,28]
K e N n
N n
�� � �� �� � �( ) , { }| ( )� exp( � �
,{ }
[
�2 . I I�
� 4 !� �
�
.( )� exp( � )� ( ) ( � � ) | ( ) ,]
/
/I I� � �� � �� � � P
(31)
where .( )x is the unit step function. In principle,
there is no well-known justification of the master
equation approach, which ignores the quantum coher-
ence effects, for the shot noise. Actually, influence of
quantum coherence on shot noise is an intriguing is-
sue [31]. However, we note that both quantum me-
chanical approaches [58,59] and semiclassical deriva-
tions based on a master equation approach predict
identical shot noise results [26,60,61], implying that
the shot noise is not sensitive to the quantum coher-
ence in double-barrier structures. Master equation ap-
proach was used by many authors for shot noise in
SET devices as well [26–28,60,62,63], and some pre-
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1529
0 0.5 1.0
eV/2E c
4
3
2
1
cI/I
g = 0.3
g = 0.3, �p
g = 0.5
g = 0.5, �p
g = 0.7
g = 0.7, �p
g = 1.0
g = 1.0, �p
Fig. 3. Average current I I/ c as a function of the bias volt-
age eV/ EC2 and LL interaction parameter g for R � 100
(highly asymmetric junctions) with no plasmon relaxation
(�p � 0, solid lines) or with fast plasmon relaxation
(�0
410� , dashed lines). The bias voltage is normalized by
the charging energy 2EC and current is normalized by the
current at eV EC� 2 with no plasmon relaxation for each
g. Other parameters are NG � 1/2, T � 0.
dictions of them [27] have been experimentally con-
firmed [64]. On this ground, here we adopt the mas-
ter equation approach and leave the precise test of the
justification open to either experimental or further
theoretical test.
To investigate the correlation effects, the noise
power customarily compared to the Poisson value
S eIPoisson � 2 . The Fano factor is defined as the ratio
of the actual noise power and the Poisson value,
F
S
eI
�
( )0
2
. (32)
Since thermal noise ( ( ))S k TG VB� �4 0 is not par-
ticularly interesting, we focus on the zero frequency
shot noise, in the low bias voltage regime where the
Coulomb blockade governs the electric transport;
T � 0 and eV � 2EC.
5.1. Qualitative discussions
As we discuss below in detail, we have found in the
absence of substantial plasmon relaxation a giant en-
hancement of the shot noise beyond Poissonian limit
(F � 1) over wide ranges of bias and gate voltages;
i.e., the statistics of the charge transport through the
device is highly super-Poissonian. In the opposite
limit of fast plasmon relaxation rate, the shot noise is
reduced below the Poissonian limit (still exhibiting
features specific to LL correlations) in accordance
with the previous work [22]. Before going directly
into details, it will be useful to provide a possible
physical interpretation of the result.
We ascribe the giant super-Poissonian noise to the
opening of additional conduction channels via the
plasmon modes. In the parameter ranges where the gi-
ant super-Poissonian noise is observed, there are a con-
siderable amount of plasmon excitations [23]. It
means that the charges can have more than one possi-
ble paths (or, equivalently, more than two local states
in the central island involved in the transport) from
left to right leads, making use of different plasmon
modes. Similar effects have been reported in conven-
tional SET devices [29] and single-electron shuttles
[65]. It is a rather general feature as long as the multi-
ple transport channels are incoherent and have differ-
ent tunneling rates. An interesting difference between
our results and the previous results [29,65] is that the
super-Poissonian noise is observed even in the sequen-
tial tunneling regime whereas in Ref. 29 it was ob-
served in the incoherent cotunneling regime and in
[65] due to the mechanical instability of the elec-
tron-mediating shuttle.
Notice that the fast plasmon relaxation prevents
the additional conduction channel opening, and in the
presence of fast plasmon relaxation, the device is
qualitatively the same as the conventional SET. The
noise is thus reduced to the Poissonian or weakly
sub-Poissonian noise.
To justify our interpretation, in the following two
subsections we compare two parameter regimes with
only a few states involved in the transport, which are
analytically tractable. When only two charge states
(with no plasmon excitations) are involved (Sec. 5.2),
the transport mechanism is qualitatively the same as the
usual sequential tunneling in a conventional SET.
Therefore, one cannot expect an enhancement of noise
beyond the Poissonian limit. As the bias voltage in-
creases, there can be one plasmon excitation associated
with the lowest charging level (Sec. 5.3). In this case,
through the three-state approximation, we will expli-
citly demonstrate that the super-Poissonian noise arises
due to fluctuations induced by additional conduction
channels. Detailed analysis of the full model system is
provided in subsequent subsections.
5.2. Two-state model (eV p� � )
The electron transport involving only two lowest
energy states in the quantum dot are well studied by
many authors (see, for instance, Ref. 31). Neverthe-
less, for later reference we begin the discussion of shot
noise with two-state process, which provides a reason-
able approximation for eV p� � . At biases such that
eV p� � and sufficiently low temperatures, the two
lowest states | , | ,N n1 0 0! � ! and | ,10! dominate the
transport process and the rate matrix is given by
� ,� �
�
�
"
#
$
$
%
&
'
'
�
�
� �
� �
(33)
where the matrix elements are � � � ��L( , , )10 0 0 and
�
� ��R( , , )0 0 10 .
With the current matrices defined by Eq. (27)
� ,IL �
"
#
$
%
&
'�
0 0
0�
� ,IR �
"
#
$
%
&
'
0
0 0
�
the noise power is obtained straightforwardly by
Eqs. (30) and (31) using the steady state probability
| ( )P
P
P
4 ! �
"
#
$
%
&
' �
"
#
$
$
%
&
'
'�
�
00
10
1
� �
�
�
. (34)
The Fano factor (32) takes a simple form
F P P
/
/
2 00
2
10
2
2
2
1
1
� �
�
�
( )
( )
� �
� �
, (35)
where
1530 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
�
�
/
/
�
�
�
�
�
�
��
�
�
��
1 2
2
0
0R
eV/ E
eV/ E
and /E0 is the shift of the bottom of the zero mode
energy induced by the gate voltage,
/ / � /E N /g N N /G p G G0 1 2� � �( ) , . (36)
Note that Eq. (36) is valid for | |/E eV/0 25 , other-
wise I � 0 and S � 0 due to Coulomb blockade. We
see from Eq. (35) that the Fano factor is minimized
for � �
� �/ 1 and maximized for ( )� �
� � �/ 1 0, with
the bounds 1 2 12/ F5 � . At the gate charge
N /G � 1 2, it is determined only by the junction
asymmetry parameter R: F R / R2
2 21 1� ( ) ( ) .
Note that when only two states are involved in the
current carrying process (ground state to ground state
transitions), the Fano factor cannot exceed the Pois-
son value F � 1.
As a consequence of the power law dependence of
the transition rates on the transfer energy (13), the
Fano factor is a function of the bias voltage, and the
interaction strengts, it varies between the minimum
and maximum values
F
E
R
R
eV
E
eV
/
/
2
0
1
1
0
1
2
1
1 2
1
2
�
�
�
� 0
6
7
88
9
8
8
at
at
/
/
�
�
(37)
(cf. Eq. (5) in Ref. 22). Figure 4 depicts the Fano
factor as
N / g eV/
R
R
G p
/
/
dip �
�
1 2 2
1
1
1
1
( )�
�
�
. (38)
The Fano factor independently of the interaction
strength crosses F R / R2
2 21 1� ( ) ( ) at N /G � 1 2,
and it approaches maximum F2 1� at N /G � 01 2
0 g eV/ p( )2� .
5.3. Three-state model (eV � 2�p)
The two-state model is applicable for bias voltages
below eV Epth � �2 0( | | )� / . Above this threshold
voltage, three or more states are involved in the trans-
port. For electron transport involving three lowest en-
ergy states in the quantum dot | , | , , | ,N n1 0 0 10! � ! !, and
| ,11! with nm � 0 for m : 2, the noise power can be cal-
culated exactly if the the contribution from the (back-
ward) transitions against the bias is negligible, as is
typically the case at zero temperature. In practice,
however, the backward transitions are not completely
blocked for the bias above the threshold voltage of the
plasmon excitations, even at zero temperature: once
the bias voltage reaches the threshold to initiate
plasmon excitations, the high-energy plasmons in the
QD above the Fermi energies of the leads are also par-
tially populated, opening the possibility of backward
transitions.
A qualitatively new feature that can be studied in
the three-state model as compared to the two-state
model is plasmon relaxation: the system with a con-
stant total charge may undergo transitions between
different plasmon configurations.
We will show in this subsection that the analytic
solution of the Fano factor of the three-state process
yields an excellent agreement with the low bias nu-
merical results in the strong interaction regime, while
it shows small discrepancy in the weak interaction re-
gime (due to non-negligible contribution from the
high-energy plasmons). We will also show that within
the three state model the Fano factor may exceed the
Poisson value.
Analytic results
By allowing plasmon relaxation, the rate matrix in-
volving three lowest energy states | , | , ,| , ,N n1 0 0 10! � ! !
and | ,11! is given by
�� �
� � � �
� � �
� � �
0 1 0 1
0 0
1 10
� �
�
�
� �
� �
�
"
#
$
$
$
$
%
&
'
'
'
p
p '
, (39)
with the matrix elements � i L i� � �� ( , , ),1 0 0 � i
�
� � ��R i i( , , ), ,0 0 1 01 and � p introduced in Eq. (19).
Current matrices defined by Eq. (27) are
[� ] , [� ]{ , } { , }I IL L21 0 31 1� �� �� � ,
[� ] , [� ] ,{ , } { , }I IR R12 0 13 1� �
� �
with [� ]{ , }I� i j � 0 for other set of i j, , ,� 1 2 3, where
� � L R, . The noise power is obtained straightfor-
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1531
0 0.2 0.4 0.6 0.8
NG
1.5
1.0
0.5
F
g = 0.2
g = 0.3
g = 0.5
g = 0.7
g = 1.0
1 –
2 –
3 –
5 –
4 –
1234
5
Fig. 4. Fano factor F S / eI� ( )0 2 as a function of the gate
charge NG and LL interaction parameter g for R = 100
(highly asymmetric junctions) at eV p� � (T � 0).
wardly by Eqs. (30) and (31) using the steady state
probability
| ( )
( )
(P
P
P
P
Z
p
p4 ! �
"
#
$
$
$
%
&
'
'
'
�
�
00
10
11
0 1
0 1
1
� � �
� � � )
"
#
$
$
$
$
%
&
'
'
'
'
�
�
� �
� �
1
1 0
p , (40)
with normalization constant Z � �
�
� � � � � �0 1 1 0 0 1
�
�( )� � � �0 0 1 p . Using the average current I �
�
e P P( )10 0 11 1� � , the Fano factor F S / eI� ( )0 2 is
given by
F P P P P
Z3 00
2
10
2
11
2
11
0
0
2
1
� ;
�
�
�
; �
"
#
$
$
%
&
'
'
� �
�
( ) ( ) .� � � �
� �
�
� �0
2
1
2
0 1
0 1
0
1 p (41)
Compared to the Fano factor (35) in the two-state
process, complication arises already in the three-state
process due to the last term in Eq. (41) which results
from the coupling of P11 and the rates which cannot be
expressed by the components of the probability vector.
In order to have | , | , , | , ,N n! � ! !0 0 10 and | ,11! as the
relevant states, we assume � �0 0
�
� or more explicitly
N NG G: dip which is introduced in Eq. (38). In the
opposite situation ( )� �0 0
�
� , the relevant states are
| , | , , | , ,N n1 0 0 01! � ! ! and | ,10!, and the above description
is still valid with the exchange of electron number
N � <0 1 and the corresponding notations � �i i
�
< ,
i � 0 1, .
To see the implications of Eq. (41), we plot the
Fano factor in Fig. 5, with respect to the gate charge
NG for symmetric junctions at eV p� 2� (T � 0), with
no plasmon relaxation (� p � 0).
Two main features are seen in Fig. 5. Firstly, the
shot noise is enhanced over the Poisson limit (F � 1)
in the strong interaction regime, g � 0.5, for a range
of parameters with gate charges away from N /G � 1 2.
As discussed above, in the low bias regime
eV Ep� �2 0( | | )� / at zero temperature, no plasmons
are excited and the electric charges are transported via
only the two-state process following the Fano factor
(35) which results in the sub-Poissonian shot noise
( )1 2 1/ F5 5 . Once the bias reaches the threshold
eV Epth � �2 0( | | )� / , it initiates plasmon excitations
which enhance the shot noise over the Poisson limit.
This feature is discussed in more detail below.
Secondly, in the weak interaction regime (g � 0.5)
a small discrepancy between the analytic result (41)
(dashed line) and the numerical result (solid line) is
found. It results from the partially populated states of
the high-energy plasmons over the bias due to
nonvanishing transition rates. On the other hand, a
simple three-state approximation shows excellent
agreement in the strong interaction regime (g � 0 3. in
the Fig. 5), indicating negligible contribution of the
high-energy plasmons (E N n eV/D( , { }) � 2) to the
charge transport mechanism. This is due to the power
law suppression of the transition rates (13) as a func-
tion of the transfer energy (10).
Limiting cases
To verify the role of nonequilibrium plasmons as
the cause of the shot noise enhancement, we consider
two limiting cases of Eq. (41): � p � 0 and � �p i�� � .
In the limit of no plasmon relaxation (� p � 0), the
Fano factor (41) of the three-state process is simpli-
fied as
F P P P P3
0
10 10 11 111 2 1 1( ) [( ) ( ) ]� � � �
2 10 11
0
2
1
2
0 1
P P
( ) ( )
,
� �
� � (42)
with steady-state probability
| ( )P
P
P
P
Z
4 ! �
"
#
$
$
$
%
&
'
'
'
�
"
#
$
�
�
00
10
11
0 1
0 1
1 0
1
� �
� �
� �
$
$
$
%
&
'
'
'
'
, (43)
where the new normalization constant is Z � �
� �0 1
�
� � � �1 0 0 1 .
The three-state approximation is most accurate in
the low bias regime 2 0( | | )� /p E� � eV EC�� 2 (/E0
is defined in Eq. (36)) and for gate voltages away from
N /G � 1 2, i.e., for 1 2/ NG� � 1 (or 0 � N /G � 1 2
with the exchange of indices regarding particle number
N � <0 1) . In this regime, �0
� and/or �1
dominate
1532 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
2.0
1.5
1.0
0.5
0
F
NG0.2 0.5 0.8
g = 0.3, N
g = 0.3, A
g = 0.5, N
g = 0.5, A
g = 1.0, N
g = 1.0, A
g = 0.3
0.5
1.0
Fig. 5. Fano factor F S / eI� ( )0 2 as a function of the gate
charge NG for symmetric junctions (R � 1) at voltage
eV p� 2� (T � 0) with no plasmon relaxation. Numerical
results (solid lines) versus analytic results with three
states, Eq. (41) (dashed lines) for g � 03. , 05. , and 10. .
over the other rates, ( , ) ( , )� � � �0 1 0 1
�
��� , which results
in 1 � P P P10 0 11 0�� �( , ) . The Fano factor (42) now
is reduced to
F P P3
0
10 11
0
2
1
2
0 1
1
0
1 2 1 2( ) ( ) ( )
.�
�
�
�
� �
� �
�
�
(44)
Notice that while Eq. (42) is an exact solution for the
three-state process with no plasmon relaxation, Eq.
(44) is a good approximation only sufficiently far
from N /G � 1 2. In this range, Eq. (44) explicitly
shows that the opening of new charge transport chan-
nels accompanied by the plasmon excitations causes
the enhancement of the shot noise (over the Poisson
limit).
Recently, super-Poissonian shot noises, i.e., the Fano
factor F � 1, were found in several different situations in
quantum systems. Sukhorukov et al. [29] studied the
noise of the co-tunneling current through one or several
quantum dots coupled by tunneling junctions, in the
Coulomb blockade regime, and showed that strong in-
elastic co-tunneling could induce super-Poissonian shot
noise due to switching between quantum states carrying
currents of different strengths. Thielmann et al. [66]
showed similar super-Poissonian effect in a single-level
quantum dot due to spin-flip co-tunneling processes,
with a sensitive dependence on the coupling strength.
The electron spin in a quantum dot in the Coulomb
blockade regime can generate super-Poissonian shot
noise also at high frequencies [67]. The shot noise en-
hancement over the Poisson limit can be observed by
studying the resonant tunneling through localized states
in a tunnel-barrier, resulting from Coulomb interaction
between the localized states [68]. In nano-electro-me-
chanical systems, in the semiclassical limit, the Fano
factor exceeds the Poisson limit at the shuttle threshold
[65,69]. Commonly, the super-Poissonian shot noise is
accompanied by internal instability or a multi-channel
process in the course of electrical transport.
In the limit of fast plasmon relaxation, on the other
hand, � �p i�� � and effectively no plasmon is excited,
| ( )P
P
P
P
4 ! �
"
#
$
$
$
%
&
'
'
'
�
� �
� �
00
10
11
0 1 0
0
0 1
1
0� � �
�
� �
"
#
$
$
$
$
%
&
'
'
'
'
. (45)
Consequently, the Fano factor (41) is given by
F P3 00
21
2
2
1
2
1
2
1( ) ,� � ��
�
�
�
�
� � "
#$
%
&'
. (46)
The maximum Fano factor F � 1 is reached if one of
the rates �0
� or �0
dominates, while the minimum
value F /� 1 2 requires that � � �0 0 1
� �� , i.e., that
the total tunneling-in and tunneling-out rates are
equal; more explicitly,
1
1
2
2
1
1
2
0
0 0R
E
eV/ E g eV/ E
p�
�
�
��
�
�
�� � �
�
�
�
�
�
�
�
� �
/
/
�
/
� �
1.
(47)
5.4. Numerical results
The limiting cases of no plasmon relaxation
(� p � 0) and a fast plasmon relaxation (� p � 104) are
summarized in Fig. 6,a and b, respectively. In the fig-
ure the Fano factor is plotted as a function of gate
voltage and interaction parameter g in the strong in-
teraction regime 0 3 0 6. .5 5g at eV p� 2� for R � 100
(strongly asymmetric junctions).
As shown in Fig. 6,a, the Fano factor is enhanced
above the Poisson limit (F � 1) for a range of gate
charges not very close to N /G � 1 2, especially in the
strong interaction regime, as expected from the
three-state model. The shot noise enhancement is lost
in the presence of a fast plasmon relaxation process, in
agreement with analytic arguments, as seen in Fig. 6,b
when F is bounded by 1 2 1/ F5 5 . Hence, slowly re-
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1533
F
g
NG
2.5
2.0
1.5
1.0
0.5
0.7 0.6 0.5 0.4 0.3 0.2
0.3
0.4
0.5
0.6
F
1.0
0.9
0.8
0.7
0.6
0.5
NG
g
0.7 0.6 0.5 0.4 0.3 0.2 0.6
0.5
0.4
0.3
b
Fig. 6. Fano factor F S / eI� ( )0 2 as a function of the gate
charge NG and LL interaction parameter g for R � 100
(highly asymmetric junctions) at eV p� 2� (T � 0), with
no plasmon relaxation (�p � 0) (a) and with fast plasmon
relaxation (�p � 104) (b).
laxing plasmon excitations enhance shot noise, and
this enhancement is most pronounced in the strongly
interacting regime.
In the limit of fast plasmon relaxation F exhibits a
minimum value F /� 1 2 at positions consistent with
predictions of the three-state model: the voltage polar-
ity and ratio of tunneling matrix elements at the two
junctions is such that total tunneling-in and tunnel-
ing-out rates are roughly equal for small values of NG .
If plasmon relaxation is slow, F still has minima at ap-
proximately same values of NG but the minimal value
of the Fano factor is considerably larger due to the
presence of several transport channels.
5.5. Interplay between several charge states and
plasmon excitations near eV EC� 2
So far, we have investigated the role of nonequi-
librium plasmons as the cause of the shot noise en-
hancement and focused on a voltage range when only
two charge states are significantly involved in trans-
port. The question naturally follows what is the conse-
quence of the involvement of several charge states. Do
they enhance shot noise, too?
To answer this question, we first consider a toy
model in which the plasmon excitations are absent dur-
ing the single-charge transport. In the three-N-state re-
gime where the relevant states are | | , |N! � � ! !1 0 and |1!
with no plasmon excitations at all. At zero tempera-
ture, the rate matrix in this regime is given by
�� �
�
� �
�
"
#
$
$
$
$
%
&
'
'
'
'
�
�
�
�
� �
� � � �
� �
1 0
1 0 0 1
0 1
0
0
, (48)
where the matrix elements are
�i
� � �L i i( , { } , { }), �1 0 0
� �i R i i
� � �( , { } , ),1 0 0 i � �1 0 1, , .
Repeating the procedure in Sec. 5.3, we arrive at a
Fano factor that has a similar form as Eq. (42),
F P P P PN3 1 1 1 11 2 1 1� � � �
[( ) ( ) ]
�
�
�
�
�
�
�
�
�
�
2 1 1
1
1
1
1
P P
�
�
�
�
, (49)
with the steady-state probability vector
| ( )P
P
P
P
Z
4 ! �
"
#
$
$
$
%
&
'
'
'
�
"
#
$
$
�
� �
1
0
0 1
1 1
1 0
1
� �
� �
� �
$
$
%
&
'
'
'
'
, (50)
where Z �
�
� �� � � � � �0 1 1 1 1 0 .
Despite the formal similarity of Eq. (49) with
Eq. (42), its implication is quite different. In terms of
the transition rates, F N3 reads
F
Z
N3 2 1 0 1 0 1 0 11
2
� � �
� �
� �
[ ( ( ) )� � � � � � �
�
� �
� � � � � � �1 0 0 1 1 0 1( ( ) )] . (51)
Since � �
� ��1 0 and � �1 0
� in the three-N-state re-
gime, the Fano factor F N3 is sub-Poissonian, i.e.,
F N3 1� , consistent with the conventional equilibrium
descriptions [22,27].
We conclude that while plasmon excitations may
enhance the shot noise over the Poisson limit, the
involvement of several charge states and the ensuing
correlations, in contrast, do not alter the sub-Pois-
sonian nature of the Fano factor in the low-energy re-
gime. This qualitative difference is due to the fact that
certain transition rates between different charge states
vanish identically (in the absence of co-tunneling): it
is impossible for the system to move directly from a
state with N � �1 to N � 1 or vice versa.
Therefore, we expect that for bias voltages near
eV EC p� �2 2� , when both plasmon excitations and
several charge excitations are relevant, the Fano fac-
tor will exhibit complicated nonmonotonic behavior.
Exact solution is not tractable in this regime since it
involves too many states. Instead, we calculate the
shot noise numerically, with results depicted in
Fig. 7, where the zero temperature Fano factor is
shown as a function of the bias eV and LL interaction
parameter g for R � 100 at N /G � 1 2, (a) with no
plasmon relaxation ( )� p � 0 and (b) with fast
plasmon relaxation (� p � 104).
In the bias regime up to the charging energy
eV EC5 2 , the Fano factor increases monotonically
due to nonequilibrium plasmons. On the other hand,
additional charge states contribute at eV EC: 2 which
tends to suppress the Fano factor. As a consequence of
this competition, the Fano factor reaches its peak at
eV EC� 2 and is followed by a steep decrease as
shown in Fig. 7,a. Note the significant enhancement
of the Fano factor in the strong interaction regime,
which is due in part to the power law dependence of
the transition rates with exponent * � �( )1 1/g as dis-
cussed earlier, and in part to more plasmon states be-
ing involved for smaller g since E /gC p� � . The latter
reason also accounts for the fact that the Fano factor
begins to rise at a lower apparent bias for smaller g:
the bias voltage is normalized by EC so that plasmon
excitation is possible for lower values of eV/EC for
stronger interactions.
In the case of fast plasmon relaxation, the rich
structure of the Fano factor due to nonequilibrium
1534 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
plasmons is absent as shown in Fig. 7,b, in agreement
with the discussions in previous subsections. The only
remaining structure is a sharp dip around eV EC� 2
that can be attributed to the involvement of addi-
tional charge states at eV EC: 2 . Not only the mini-
mum value of the Fano factor is a function of the in-
teraction strength but also the bias voltage at which it
occurs depends on g. The minimum Fano factor occurs
at higher bias voltage, and the dip tends to be deeper
with increasing interaction strength. Note for g � 0.5,
the Fano factor did not reach its minimum still at larg-
est voltages plotted (eV/ EC2 12� . ).
The Fano factor at very low voltages eV p� 2� for
N /G � 1 2 is F R / R( ) ( ) ( )0 2 21 1� (see Eq. (35)),
regardless of the plasmon relaxation mechanism. As
shown in Fig. 7,b, the Fano factor is bounded above by
this value in the case of fast plasmon relaxation. Notice
that in the case of no plasmon relaxation (Fig. 7,a), the
dips in the Fano factor at eV EC� 2 reach below F ( )0 .
See Fig. 1,a in Ref. 24 for more detail.
Since both the minimum value of F and the voltage
at which it occurs are determined by a competition be-
tween charge excitations and plasmonic excitations,
they cannot be accurately predicted by any of the sim-
ple analytic models discussed above.
6. Full counting statistics
Since shot noise, that is a current-current correla-
tion, is more informative than the average current, we
expect even more information with higher-order cur-
rents or charge correlations. The method of counting
statistics, which was introduced to mesoscopic physics
by Levitov and Lesovik [70] followed by Muzy-
kantskii and Khmelnitskii [71] and Lee et al. [72],
shows that all orders of charge correlation functions
can be obtained as a function related to the probabil-
ity distribution of transported electrons for a given
time interval. This powerful approach is known as full
counting statistics (FCS). The first experimental
study of the third cumulant of the voltage fluctuations
in a tunnel junction was carried out by Reulet et al.
[73]. The experiment indicates that the higher
cumulants are more sensitive to the coupling of the
system to the electromagnetic environment. See also
Refs. 74, 75 for the theoretical discussions on the third
cumulant in a tunnel-barrier.
We will now carry out a FCS analysis of transport
through a double barrier quantum wire system. The
analysis will provide a more complete characterization
of the transport properties of the system than either
average current or shot noise, and shed further light
on the role of nonequilibrium versus equilibrium
plasmon distribution in this structure.
Let P M( , )
be the probability that M electrons
have tunneled across the right junction to the right
lead during the time
. We note that
P M( , )
�
� � �
��
��lim ( , , { }, ; , , {
, ,{ },{ }
t
P M M N n t M N n
M N nN n
0 0 0 0
0 0 0
� }, ),t
where P M M N n t M N n t( , , { }, ; , , { }, )0 0 0 0
is cal-
led joint probability since it is the probability that,
up to time t, M0 electrons have passed across the
right junction and N0 electrons are confined in the
QD with { }n0 plasmon excitations, and that M M0
electrons have passed R-junction with ( , { })N n excita-
tions in the QD up to time t
. The master equation
for the joint probability can easily be constructed
from Eq. (23) by noting that M M- 0 1 as
N N- � 1 via only R-junction hopping
To obtain P M( , )
, it is convenient to define the
characteristic function conjugate to the joint proba-
bility as
g N n( , , { }, )=
�
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1535
F
7
5
3
1
0
1
2eV/E C 0.4 0.6 0.8 1.0
g
a
F
2.0
2.2
2.4eV/E C g
0.4 0.6 0.8 1.0
1.0
0.9
0.8
0.7
0.6
0.5
b
Fig. 7. Fano factor F S / eI� ( )0 2 as a function of the bias
eV and LL interaction parameter g for R � 100 (highly
asymmetric junctions) at N /G � 1 2 (T � 0): with no plas-
mon relaxation (�p � 0) (a) and with fast plasmon relax-
ation (�p � 104) (b).
� � �
��
� �lim ( , , { }, ; , , {
, ,{ }
t
i M
M M N n
P M M N n t M N ne � �0 0 0
0 0 0
0 }, ).t
The characteristic function satisfies the master
equation
�
�
! � � !
=
= =
| ( , ) �( )| ( , )g g�
with the initial condition | ( , ) | ( )g P=
� ! � 4 !0 . The
=-dependent �� in Eq. (52) is related to the previously
defined transition rate matrices through
�( ) � ( � � � )� � � �= � � � �
�p L L L
0
� ��
( � � � ) .� � �R R
i
R
i0 e e� �
(53)
The characteristic function G( , )=
conjugate to
P M( , )
is now given by G N n g
N n
( , ) , { }| ( , )
,{ }
=
=
� !� ,
or
G N n P
N n
( , ) , { }| exp [ �( ) ]| ( ) .
,{ }
=
=
� � 4 !� � (54)
Finally, the probability P M( , )
is obtained by
P M
d
G
dz
i
G z
z
i M
M
( , ) ( , )
( , )
=
�
=
�
�
�� �( ( �2 2
0
2
2 1
e � (55)
with z i /�
e � 2, where the contour runs counterclock-
wise along the unit circle and we have used the symme-
try property G z G z( , ) ( , )
� � for the second equality.
Taylor expansion of the logarithm of the character-
istic function in i= defines the cumulants or irreducible
correlators >
k( ):
ln ( , )
( )
!
( )G
i
k
k
k
k=
=
>
�
�
�
�
1
. (56)
The cumulants have a direct polynomial relation with
the moments n n P nk k
n
( ) ( , )
� � . The first two
cumulants are the mean and the variance, and the
third cumulant characterizes the asymmetry (or skew-
ness) of the P M( , )
distribution and is given by
>
/
3
3 3( ) ( ) ( ( ) ( ))� � �n n n . (57)
In this section, we investigate FCS mainly in the
context of the probability P M( , )
that M electrons
have passed through the right junction during the
time
. Since the average current and the shot noise
are proportional to the average number of the tunnel-
ing electrons !M and the width of the distribution of
P M( , )
, respectively, we focus on the new aspects
that are not covered by the study of the average cur-
rent or shot noise.
In order to get FCS in general cases we integrate
the master equation (52) numerically (see Sec. 6.3).
In the low-bias regime, however, some analytic calcu-
lations can be made. We will show through the fol-
lowing subsections that for symmetric junctions in the
low-bias regime (2 2�p CeV E� � ), and irrespective of
the junction symmetry in the very low bias regime
(eV p� 2� ), P M( , )
is given by the residue at z � 0
alone,
P M
M
d
dz
G z
M
M
z
( , )
( )!
( , ) .
�
,
,
,
�
1
2
2
2
0
(58)
Through this section we assume that the gate charge
is N /G � 1 2, unless stated explicitly not so.
We will now follow the outline of the previous sec-
tion and start by considering two analytically tracta-
ble cases before proceeding with the full numerical re-
sults.
6.1. Two-state process; eV p� �
For the very low bias eV p� 2� at zero temperature,
no plasmons are excited and electrons are carried by
transitions between two states ( , ) ( , ) ( , )N n � <0 0 10 .
In this simplest case, the rate matrix �( )� = in Eq. (52)
is determined by only two participating transition
rates � � � ��L( , , )10 0 0 and �
� ��R( , , )0 0 10 ,
�( )
( )
( )
� =
/ /
/ /
�
�
� �
� �
"
#
$
$
%
&
'
'
� �
� �
0 0
0 0
e i
(59)
with �0 2 2� � ��
�
( ) , ( )� � / � �/ / .
Substituting the steady-state probability Eq. (34)
and the transition rate matrix (59) to Eq. (54), one
finds
G z
f z
f z f z
2
2
2
20
0 2
4
1
1( , )
( )
[( ( )) ( )
�
�� �
e
e
�
�
� �
( ( )) ],( )1 2
2 0 2f z f ze � �
(60)
where f z / z2 0
2 2( ) ( )� �� � �� , with � �� /� � �0
2 2
� �
� � and � �2 2 2� / �/ .
Now, it is straightforward to calculate the cu-
mulants. In the long time limit
��� �
( ) 1, for
instance, in terms of the average current I2 �
� �
�
e /� � � �( ) and the Fano factor F2 in (35),
the three lowest cumulants are given by
>
1 2( ) ,� I >
2 2 2( ) ,� eI F
>
3
2
2 2
23 1 2 1 4( ) [ ( ) ]� � e I F / / , (61)
where the electron charge (�e) is revived. These are in
agreement with the phase-coherent quantum-mechani-
cal results [76]. It is convenient to discuss the asym-
1536 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
metry (skewness) by the ratio A /e� > >3
2
1 ( )
- 4 ,
noticing the Fano factor F /e�
��
lim ( ( )
�
>
� >
2 1 . The
factor A F / /2 2
23 1 2 1 4� � ( ) is positive definite
(positive skewness) and bounded by 1 4 12/ A5 5 . It
is interesting to notice that A2 is a monotonic func-
tion of F2 and has its minimum A /2 1 4� for the mini-
mum F /2 1 2� and maximum A2 1� for the maximum
F2 1� . Notice A � 1 for a Poissonian and A � 0 for a
Gaussian distribution. Therefore, the dependence of
A2 on the gate charge NG is similar to that of the
Fano factor F2 (see Fig. 4) with dips at NG
dip in Eq.
(38). Together with Eq. (37), it implies that the ef-
fective shot noise and the asymmetry of the probabil-
ity distribution per unit charge transfer have their re-
spective minimum values at the gate charge NG
dip
which depends on the tunnel-junction asymmetry and
the interaction strength of the leads.
The integral in Eq. (55) is along the contour de-
picted in Fig. 8. Notice that the contributions from the
part along the branch cuts are zero and we are left with
the multiple poles at z � 0. By residue theorem, the
two-state probability P M2( , )
is given by Eq. (58).
The exact expression of P M2( , )
is cumbersome.
For symmetric tunneling barriers with N /G � 1 2
( , )/ � �� � ��
0 0� , however, Eq. (60) is reduced to
G z
z
z zs z z
2
2 20
0 0
4
1 1( )( , ) [( ) ( ) ]
�
� �� � �
e
e e
�
� � .
(62)
Accordingly, P Ms
2
( )( , )
is concisely given by
P M
M M
s
M M
2
0
2 1
0
2
0 1
2 2 1 2
( )( , )
( )
( )!
( )
( )!
��
�
"
#
$
e � � �
$
%
&
'
'
:
�1
2 2 1
10
2 1( )
( )!
for
�
M
M
M , (63)
with P /s
2 00 1 20( )( , ) ( )
��
e � � , in agreement with
Eq. (24) of Ref. 76. While this distribution resembles
a sum of three Poisson distributions, it is not exactly
Poissonian.
For a highly asymmetric junctions R �� 1
( )� ��
�� , the first term in Eq. (60) dominates the
dynamics of G z2( , )
and its derivatives, and the char-
acteristic function is approximated by
G z za
2 0
2 2( )( , ) exp ( ( ) )
� � /� � �
� . (64)
Now, the solution of P Ma
2
( )( , )
is calculated by this
equation and Eq. (58). The leading order approxima-
tion in � �
�/ leads to the Poisson distribution,
P M
M
a
M
2
( )( , )
( )
!
.
�
� ��
�
e (65)
For a single tunneling-barrier, the charges are trans-
ported by the Poisson process (F � 1) [31]. Therefore,
we recover the Poisson distribution in the limit of
strongly asymmetric junctions and in the regime of
the two-state process, in which electrons see effec-
tively single tunnel-barrier.
For the intermediate barrier asymmetry the proba-
bility P M2( , )
of a two-state process is given by a dis-
tribution between Eq. (63) (for symmetric junctions)
and the Poissonian (65) (for the most asymmetric
junctions).
6.2. Four-state process; 2�p � eV � 2EC
Since we focus the FCS analysis on the case
N /G � 1 2, the next simplest case to study is a four-
state-model rather than the three-state-model dis-
cussed in the connection of the shot noise in the previ-
ous section.
In the bias regime where the transport is governed
by Coulomb blockade (eV EC� 2 ) and yet the plas-
mons play an important role (eV p: 2� ), it is a fairly
good approximation to include only the four states
with N � 0 1, and n1 0 1� , (nm � 0 for m : 2). For
general asymmetric cases, the rate matrix �( )� = in
Eq. (52) is determined by ten participating transition
rates (four rates from each junction, and two relax-
ation rates). The resulting eigenvalues of | ( , )g =
! are
the solutions of a quartic equation, which is in general
very laborious to present analytically.
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1537
C2 C1
– /2 + /2
+ /2 – /2
C0
C 3 C4
Fig. 8. Contour of Eq. (55). The arguments of f z2( ) �
� �( )� � ��/ z0
2 2 are / / / /2 2 2 2, , ,
along the
branch cuts C C C1 2 3, , , and C4, respectively.
For symmetric junctions (R � 1) with no plasmon
relaxation, however, the rate matrix is simplified to
�( )� =
� � � �
� � � �
�
�
� �
� �
�
00 10 00
2
01
2
01 11 10
2
11
2
00
0
0
z z
z z
�
� �
"
#
$
$
$
$
$
%
&
'
'
'
'
'
� � �
� � � �
01 00 10
10 11 01 11
0
0
(66)
with the matrix elements � ij L i j� � �� ( , , )1 0
� ��R i j( , , )0 1 . The steady-state probability is then
given by
| ( ) | ( , )
( )
( )P g zs
4
01 10
01
10
10
0
1
2
4 ! � � ! �
"
#
$
$
$
� �
�
�
�
�
��
$
%
&
'
'
'
'
. (67)
Solving Eq. (54) with this probability and the rate
matrix (66) is laborious but straightforward and one
finds
G z /( , ) ( )
� � � � �� ;
� � �e 00 10 01 11 2
;
6
7
8
98
?
@
8
A8
- �
"
#
$
$
%
&
'
'
G z
z
z
z zI ( , )
( )
{ }
1
8
2
, (68)
with
G z
A z
f zI
z z f z /( , )
( )
( )
( ( ))
� � ��
�
�
��
�
�
��
6
7
� �e 00 11 4 2
4
1
9
;
; �
�
�
�
��
�
�
��
?
@
A
- �1
2
4
01 10
4 4
A z f z
z
f z f z
( ) ( )
( )
{ ( ) ( )}
� �
, (69)
where A z( ) and f z4( ) are given by
A z z( ) ( )� � � � � �� � � � � � �00 10 01 11 00 11 012 ,
f z z a b4 00 11
2
10 01
2 24( ) ( ) ( )� � � � � � � � (70)
with dimensionless parameters a b, given by
a �
� � �
� �
( )( )
( )
,
� � � � � �
� � � �
00 11 00 10 01 11
00 11
2
10 014
b �
� �
� �
4
4
10 01 00 10 01 11
00 11
2
10 01
� � � � � �
� � � �
( )
( )
.
Integral of G zI ( , )
along the contour | |z � 1 contains
two branch points at z a ibc � 0 , however, the inte-
gral along the branch cuts cancel out due to the sym-
metry under [ ( ) ( )]f z f z4 4- � . Therefore, the contri-
bution from the branch cuts due to G zI ( , )
and
G zI ( , )�
is zero to the probability P Ms
4
( )( , )
, and it
is given by the residues only at z � 0, i.e., by
Eq. (58). The explicit expression of P Ms
4
( )( , )
is
cumbersome.
The probability distribution P M2( , )
for the two-
state process deviates from Eq. (63) as a function of
the asymmetry parameter R and reaches Poissonian in
the case of strongly asymmetric junctions. In a similar
manner, P Ms
4
( )( , )
deviates from P Ms
2
( )( , )
as a
function of ratio of the transition rates � ij .
6.3. Numerical results
It is worth mentioning that for strongly asymmetric
junctions P M( , )
is Poissonian in the very low bias
regime (eV p� 2� ), as seen from Eq. (65). It exhibits
a crossover at eV p� 2� : P M( , )
deviates from Pois-
son distribution for 2 2�p CeV E� � while it is
Poissonian for eV p� 2� (at T � 0), as shown by the
shot noise calculation.
Voltage dependence
The analytic results presented above are useful in
interpreting the numerical results in Fig. 9, where
probability P M( , )
for symmetric junctions (R � 1)
and R � 100, in the case of LL parameter g � 0 5. with
no plasmon relaxation (� p � 0), is shown as a function
of eV and M, that is the number of transported elec-
trons to the right lead during
such that during this
time ! � �M Ic
10 electrons have passed to the right
lead at eV EC� 2 .
The peak position of the distribution of P M( , )
is
roughly linearly proportional to the average particle
flow !M , and the width is proportional to the shot
noise but in a nonlinear manner. In a rough estimate,
therefore, the ratio of the peak width to the peak posi-
tion is proportional to the Fano factor. Two features
are shown in the Fig. 9. First, the average particle flow
(the peak position) runs with different slope when the
bias voltage crosses new energy levels, i.e., at eV p� 2�
and eV EC� 2 , that is consistent with the I V� study
(compare Fig. 9,b with Fig. 3). Notice E /gC p� ��
� 2�p for g � 0 5. . Second, the width of the distribution
increases with increasing voltage, with different char-
acteristics categorized by eV p� 2� and eV EC� 2 . Es-
pecially in the bias regime eV EC� 2 in which several
charge states participate to the charge transport, for the
highly asymmetric case, the peak runs very fast while
its width does not show noticeable increase. It causes
the dramatic peak structure in the Fano factor around
eV EC� 2 as discussed in Sec. 5.
The deviation of the distribution of probability
P M( , )
due to the nonequilibrium plasmons from its
low voltage (equilibrium) counterpart is shown in the
insets. Notice in the low-bias regime eV p� 2� , it fol-
lows Eq. (63) for the symmetric case (Fig. 9,a, inset
I), and the Poissonian distribution (65) for the highly
1538 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
asymmetric case (Fig. 9,b, inset I). The deviation is
already noticeable at eV Ep C� �3 15� . for R � 1 (in-
set II in Fig. 9,a), while it deviates strongly around
eV EC� 2 for R � 100 (inset III in Fig. 9,a).
Interaction strength dependence
We have concluded in Sec. 5 that shot noise shows
most dramatic behavior around eV EC� 2 due to inter-
play between the nonequilibrium plasmons and the
charge excitations. To see its consequence in FCS, we
plot in Fig. 10 the probability P M( , )
as a function of
the particle number M and the interaction parameter g
for
�� � �10 2 0e/I eV EC p( , ) with no plasmon re-
laxation and with fast plasmon relaxation.
The main message of Fig. 10,a is that the shot noise
enhancement, i.e., the broadening of the distribution
curve, is significant in the strong interaction regime
with gradual increase with decreasing g. Fast plasmon
relaxation consequently suppresses the average current
and the shot noise dramatically as shown in Fig. 10,b
implying the Fano factor enhancement is lost. Effec-
tively, the probability distribution of P M( , )
for dif-
ferent interaction parameters maps on each other al-
most identically if the time duration is chosen such that
!M equals for all g.
7. Conclusions
We have studied different transport properties of a
Luttinger-liquid single-electron transistor including
average current, shot noise, and full counting statis-
tics, within the conventional sequential tunneling
approach.
At finite bias voltages, the occupation probabilities
of the many-body states on the central segment is
found to follow a highly nonequilibrium distribution.
The energy is transferred between the leads and the
quantum dot by the tunneling electrons, and the elec-
tronic energy is dispersed into the plasmonic collective
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1539
40
30
20
10
0
M
0.5 1.0 1.5 2.0
eV/E C
0.8
0.6
0.4
0.2
0
a
1.0
0 0.5 1.0 1.5 2.0
eV/E C
40
30
20
10
M
0.8
0.6
0.4
0.2
0
b
1.0
Fig. 9. Probability P M( , )� with no plasmon relaxation
��p � 0) during the time (� � 10/Ic, where Ic is the particle
current at eV EC� 2 with no plasmon relaxation (�p � 0):
for symmetric junctions (R � 1) (a) and for a highly asym-
metric junctions (R � 100) (b). Here g � 0 5. , N /G � 1 2,
and T � 0. Inset shows cross-sectional image of P M( , )�
(blue solid line) as a function of M and the reference dis-
tribution function (a) Eq. (63) (magenta dashed line) and
(b) the Poisson distribution Eq. (65) (red dashed), at
eV EC� 05. (I), eV EC� 15. (II), and eV EC� 2 (III).
0
5 10 15 20
M
P(M, )�
0.20
0.15
0.10
0.05
0.4
0.6
0.8
1.0
g
a
0
5 10 15 20
M
0.4
0.6
0.8
1.0
g
P(M, )�
0.20
0.15
0.10
0.05
b
Fig. 10. The probability P M( , )� that M electrons have
passed through the right junction during the time
� � 10 0/I , where I0 is the particle current with no
plasmon relaxation (�p � 0); with no plasmon relaxation
��p � 0) (a) and with fast plasmon relaxation (�p � 104)
(b). Here eV E RC� �2 100, , N /G � 1 2 and T � 0.
excitations after the tunneling event. In the case of
nearly symmetric barriers, the distribution of the oc-
cupation probabilities of the nonequilibrium plasmons
shows impressive contrast depending on the interac-
tion strength: In the weakly interacting regime, it is a
complicated function of the many-body occupation
configuration, while in the strongly interacting re-
gime, the occupation probabilities are determined al-
most entirely by the state energies and the bias volt-
age, and follow a universal distribution resembling
Gibbs (equilibrium) distribution. This feature in the
strong interaction regime fades out with the increasing
asymmetry of the tunnel-barriers.
We have studied the consequences of these non-
equilibrium plasmons on the average current, shot
noise, and counting statistics. Most importantly, we
find that the average current is increased, shot noise is
enhanced beyond the Poisson limit, and full counting
statistics deviates strongly from the Poisson distribu-
tion. These nonequilibrium effects are pronounced es-
pecially in the strong interaction regime, i.e. g � 0.5.
The overall transport properties are determined by a
balance between phenomena associated with nonequi-
librium plasmon distribution that tend to increase
noise, and involvement of several charge states and
the ensuing correlations that tend to decrease noise.
The result of this competition is, for instance, a
nonmonotonic voltage dependence of the Fano factor.
At the lowest voltages when charge can be trans-
ported through the system, the plasmon excitations are
suppressed, and the Fano factor is determined by
charge oscillations between the two lowest zero-modes.
Charge correlations are maximized when the tunnel-
ing-in and tunneling-out rates are equal, which for
symmetric junctions occurs at gate charge N /G � 1 2.
At these gate charges the Fano factor acquires its low-
est value which at low voltages is given by a half of the
Poisson value, known as 1/2 suppression, as only two
states are involved in the transport, at somewhat larger
voltages increases beyond the Poisson limit as plasmon
excitations are allowed, and at even higher voltages ex-
hibits a local minimum when additional charge states
are important. If the nonequilibrium plasmon effects
are suppressed, e.g., by fast plasmon relaxation, only
the charge excitations and the ensuing correlation ef-
fects survive, and the Fano factor is reduced below its
low-voltage value. The nonequilibrium plasmon effects
are also suppressed in the noninteracting limit.
Acknowledgments
This work has been supported by the Swedish Founda-
tion for Strategic Research through the CARAMEL con-
sortium, STINT, the SKORE-A program, the eSSC at
Postech, and the SK-Fund. J.U. Kim acknowledges par-
tial financial support from Stiftelsen Fru Mary von
Sydows, f�dd Wijk, donationsfond. I.V. Krive gratefully
acknowledges the hospitality of the Department of Ap-
plied Physies at Chalmers University of Technology.
Appendix A: The transition amplitudes
in the quantum dot
In this appendix, we derive the transition ampli-
tudes, Eqs. (15) and (16). As shown in Eq. (14), the
zero-mode overlap of the QD transition amplitude is
either unity or zero. Therefore, we focus on the over-
lap of the plasmon states. It is enough to consider
| { }| ( )| { } | , ,† � ! �n X n X X X
D L R� � �
2
due to the symmetry between matrix elements of tun-
neling-in and tunneling-out transitions, as in Eq. (15),
| { }| ( )| { } | [| { }| ( )| { } |† † � ! � � !
��
�n X n n X nr
r
� � �
2 2B
� ! � ! �
{ }| ( )| { } { }| ( )| { } ]†n X n n X nr rB B� �
� � !2 2| { }| ( )| { } |†n X nrB � ,
(A.1)
where r � �( ) denotes the right (left)-moving com-
ponent, and the cross terms of oppositely moving
components cancel out due to fermionic anti-commu-
tation relations.
The transition amplitudes at XL is identical to that
at XR. For simplicity, we consider the case at XL � 0
only. The overlap elements of the many-body
occupations � { }| , , ..., ,... |n n n nm1 2 and | { }n� ! �
� � � � !| , ,..., ,...n n nm1 2 are
| { }| ( )| { } | | | | |† � � ! � � !
�
�
2n x n n nr
m
m m mB
�
C0
1
2
2
1
2
+
,
(A.2)
where C Dm m m mb b� exp[ ( )]† with Dm i/ gmM� �
is the bosonized field operator at an edge of the wire
with open boundary conditions (see for instance Ref.
44). Here g is the interaction parameter, m is the mode
index (and the integer momentum of it), and M is the
number of transport sectors; if M � 1, the contribu-
tions of the different sectors must be multiplied. The
operators bm and bm
† denote plasmon annihilation and
creation and + is a high energy cut-off.
Using the Baker–Haussdorf formula
C D
�
� �
m m m m
a aa a m m m m� �
exp[ ( )]† †
e e e
2
2 , (A.3)
and the harmonic oscillator states
|
( )
!
| , | |
!
,
†
n
a
n
n
a
n
n n
! � ! � 0 0 (A.4)
1540 Fizika Nizkikh Temperatur, 2006, v. 32, No. 12
Jaeuk U. Kim, Mahn-Soo Choi, Ilya V. Krive, and Jari M. Kinaret
one can show that, if n n5 �,
| | | |
| |
! !
!
( )!
| |
( )
� ! �
� � �
�
�
�
�
�
� ;
�
n n
n n
n
n n
n n
C
D�2
2 22
e
; � � �E( , ; | | )n n n1 1 2 2D , (A.5)
where E( , ; )x x z� is a degenerate hypergeometric func-
tion defined by [77]
E( , ; )
!
( )!
( )!
( )!
( )!
x x z
z x
x
x
x
� �
�
�
� �
� �
"
�
�
�
�
�
�
�
�
0
1
1
1
1#
$
$
%
&
'
'
. (A.6)
If n n�5 , the indices n and n� are exchanged in Eq.
(A.5). The function E( , ; )x x z� is a solution of the
equation
z x z xz z� � � � � �2 0E E E( ) .
By solving this differential equation with the proper
normalization constant, one obtains
E( , ; | | )
!( )!
!
(| || |n n n
n n n
n
Ln
n n � � � �
� � �
�
�1 1 2 22
D D�e ),
where L ya
b ( ) is the Laguerre polynomials [77]. In
terms of the Laguerre polynomials, therefore, the
transition amplitude (A.5) is written by
| | | |
( )
!
!
/
| |
( )
( )
� ! � ;
�
�
�
n n
gmM
n
n
m m m
gmM
n n
m
mm m
C 2
1e
;
�
�
��
�
�
��
"
#
$
%
&
'�
�
L
gmMn
n n
m
m m
( )
| | 1
2
, (A.7)
where n n n( ) ( , )� � �min and n n n( ) ( , )� � �max .
We introduce a high frequency cut-off m k L /c F D~ �
to cure the vanishing contribution due to e
1/gmM ,
1
2
1
2
1 4
1
�
�
�
+
+
e
�
2 -
�
�
��
�
�
��
/ mg
m
m
D
c
D
c
L L
, (A.8)
where the exponent is * � �
( )g /M1 1 . We arrive at
the desired form of the on-dot transition matrix
elements,
| { }| ( { } |†
)|� ! �
�
�
��
�
�
�� ;n X n
L La
D D
�
+
�
2 1 �
�
;
�
�
��
�
�
��
�
�
� �
�
�22 �
1
1
g mM
n
n
L
n n
m
m
m
n
n
m m
m��
| | ( )
( )
|!
!
( )
m mn
g mM
�
�
��
�
�
��
"
#
$
%
&
'
| ,
1
2
�
(A.9)
Appendix B: Universal occupation probability
In this appendix, we derive the universal distribu-
tion of the occupation probability, which becomes
Eq. (26) in the leading order approximation.
Since the occupation probability of the plasmon
many-body states is a function of the state energy in
the strong interaction regime, we introduce the
dimensionless energy n mn
m m� � of the state with
{ } ( , ,..., ,...)n n n nm� 1 2 plasmon occupations. Exclud-
ing the zero mode energy, therefore, the energy of the
state { }n is given by E n E n nD D p({ }) ( )� � � with the
state degeneracy D n( ), i.e., the number of many-body
states n satisfying n mn
m m� � , asymptotically follow-
ing the Hardy–Ramanujan formula [78]
D n / nn/( ) ( ).� e � 2 3 4 3 (B.1)
We denote n sd by the corresponding dimensionless
bias voltage eV n sd p� � .
We obtain an analytic approximation to the occu-
pation probability P n( ) at zero temperature by setting
the on-dot transition elements in (15) to unity and
considering the scattering-in and scattering-out pro-
cesses for a particular many-body state { }n .
The total scattering rates at zero temperature are
given by a simple power-law Eq. (13),
� .( ) ( )( )n m n m n / n m n /sd sd� � � � � � 2 2 �
� � �.( )( )n m n / n m n /sd sd2 2 �
� � � � �.( )( )n m n / n m n /sd sd2 2 � , (B.2)
where the constant factor in Eq. (13) is set to unity.
The master equation now reads
�
�
�
�t
P n P m D m m n P n D m n m
m
( ) [ ( ) ( ) ( ) ( ) ( ) ( )]� � .
(B.3)
To solve this master equation, we assume an ansatz
of a power-law
P m P n qn
m n( ) ( ) .�
(B.4)
In the steady-state, master equation (B.3) in terms of
this ansatz becomes
( ) ( )m n n / D m q
m m
sd n
m n
i
� �
�
�
� 2 �
� � � �
� �
�
�( ) ( ),n m n / D m
m
n n /
sd
sd
0
2
2 � (B.5)
in which the sum in the LHS runs from m ni � �max( ,0
� n /sd 2), where max( , )x x� gives larger of x and x�.
Nonequilibrium plasmons and transport properties of a double-junction quantum wire
Fizika Nizkikh Temperatur, 2006, v. 32, No. 12 1541
Using a saddle point integral approximation
e e ef x f x f x x x
dx dx( ) ( ) ( )( )
,( (�
��
0
0 0
21
2
if ( ) , ( )� � �� �f x f x0 00 0 ,
(B.6)
we solve Eq. (B.5) to obtain an equation for lnqn
and find that for a large n,
exp( ) exp( ( )),z /z n F n� (B.7)
where
z
q
C
C
n /sd
� �
| |
, ,
ln
�
�
* 1
2
and F n( ) is a slowly varying function of n for n �� 1,
F n
C n
C
n
n
sd( ) ln .�
"
#
$
%
&
' �
�
�
�
�
�
�
2
3
6 1
4
1
1
�
� *�
�
(B.8)
We assume an ansatz for the solution of z in Eq. (B.7),
z n / F n n / F n K� � ( ) ( ) ( ) ( )ln ln2 2 F
(B.9)
and find a constant K which minimizes the correction
term F. Putting this ansatz into Eq. (B.7) and solving
the equation for F, we find at K � 0 8. , that the correc-
tion term F is negligibly small (F � 0 01. ).
Noting f n n / F n K( ) ( ) ( )� �ln 2 is almost li-
near function in the regime of our interest (3 � n � 15),
we linearize it around a valuen n� 0 (for instance,n0 9� ),
f n f n n n f n( ) ( )( ) ( ),� � � 0 0 0 and solve Eq. (B.7) by
ansatz (B.9) with above linearized form;
� � � � �
ln
ln ( ) ( ) ( ) ( ) .
q
C
n F n f n n f n f n nn
�
1
2 0 0 0 0
(B.10)
Apply � �m nP m qln ln[ ( )] [ ] to Eq. (B.10), and
solve the integral equation for ln [ ( )]P n ,
ln ln[ ( )] [( ) ( )P n C dn n / F n
n
� � (� 2
� � �f n n f n f n n( ) ( ) ( ) ]0 0 0 0 . (B.11)
The leading order approximation P n( )( )0 to the
probability P n( ) of the average occupation from this
integral results in Eq. (26)
P n Z n Zn
n
n
n n
sd sd( )( ) ,0 1
3
2
1
1
3
2
1
� �
�
�� �
e
log
(B.12)
where Z is the partition function.
A more accurate approximation P n( )( )1 can be de-
rived by solving the integral Eq. (B.11) to a higher
degree of precision, which yields
P Z n n
n
Csd
C n/ C
( )1 1
2
2
6
� �
�
�
�
�
�
�
�
��
�
�
��
�
�
��
�
�
��
� �
� �
n
C n F n Kexp ( )
ln
� � �"
#$
%
&'
�
�
�
�
�
� ;� 0
3
2
3
4
; �
�
�
�
�
�
�
�
�
�
��
�
�
��
exp
4
6
2
2 2
2
�
n
n n /
n n /
n
nsd
sd
n/n
sd
sd �
�
�
�
�
� � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�( )� 1 2
2
2
2
/
sdn
n
, (B.13)
where n0 9� is used and f n f n( ) ( )0
2
0� with minor
correction is utilized for formal simplicity. Note the
first term with normalization constant Z approaches
P n( )( )0 in Eq. (B.12) as n/n sd - 0, noticing
C /nsd� *� 2 1( ) . The integral equation (B.11) can
be solved even without approximating on f n( ), with
the expense of more cumbersome appearance of P n( ).
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|