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Physics Letters A 378 (2014) 3136–3143

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

The shot noise of a strongly correlated quantum dot coupled to

the Luttinger liquid leads

Kai-Hua Yang

a,∗

, Xian He

, Huai-Yu Wang

, Kai-Di Liu

, Bei-Yun Liu

College of Applied Sciences, Beijing University of Technology, Beijing 100122, China

Department of Physics, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o a b s t r a c t

Article history:

Received

11 June 2014

Received

in revised form 24 August 2014

Accepted

4 September 2014

Available

online 8 September 2014

Communicated

by R. Wu

Keywords:

Kondo

effect

Luttinger liquid

Shot

noise

We study the shot noise of a strongly correlated quantum dot weakly coupled to Luttinger liquid leads in

the Kondo regime by means of the extended equation of motion method. A general zero-frequency shot

noise formula with good convergence is derived. The shot noise exhibits a non-monotonic dependence

on voltage for weak intralead interaction. There is a peak around the Kondo temperature at low voltage

when the interaction is very weak, and its height decreases rapidly with the intralead interaction

increasing. When the interaction is moderately strong the peak disappears and the shot noise scales

as a power law in bias voltage, indicating that the intralead electron interaction suppresses the shot

noise. It is possible that the measurements of the shot noise spectrum can extract the information of the

intralead interaction.

1. Introduction

As techniques in fabricating devices of nanometer scale develop,

the research on shot noise has become a very active ﬁeld in meso-

scope

physics [1–7]. The shot noise is caused by the discreteness of

charge carriers and is unavoidable even at zero temperature. It is

sensitive to the second-order correlations of the current and thus

provides additional information about conduction process. Shot-

noise

measurements have proven to be a valuable tool to investi-

gate

many-body effects in mesoscopic transport. For example, shot

noise experiments can determine the kinetics of electrons and re-

veal

information on the correlations of electronic wave function.

Therefore, it can provide some valuable information unavailable by

the usual conductance measurements [1].

number of experiments [2–7] tried to exploit the control-

lable

physical characteristics of a quantum dot (QD) in order to

acquire important information of Kondo systems unavailable from

bulk systems. The study of shot noise through strongly corre-

lated

QD devices in the Kondo regime has received increasing

attention [8–15]. These theoretical studies focused on the sys-

tems

with a QD coupled to the three-dimensional interacting elec-

tron

leads which could be described by Landau’s Fermi liquid (FL)

theory. However, one-dimensional electron systems, where inter-

actions

between electrons are of great importance in transport,

Corresponding author.

E-mail

address: khy@bjut.edu.cn (K.-H. Yang).

cannot be described by the FL model. No matter how weak the

fermion interactions are, they cannot be taken into account per-

turbatively

[16–19]. The low-energy properties of one-dimensional

metals have been investigated successfully within the framework

of the Luttinger liquid (LL) theory [16–19]. It is expected that in

such systems the shot noise will behave differently from those

with FL leads.

For

a classical conductor the zero-frequency shot noise S(0)

is characterized by the Poisson value S(0) = 2eI, where I is the

average current. In a mesoscopic conductor, the shot noise is in-

ﬂuenced

by quantum statistics and interactions between charged

carriers. In the shot-noise measurements of single-wall-nanotubes

(SWNT), the power law dependence at low bias voltages was

found, which exhibited the LL behavior [20–22]. Researches deal-

ing

with the shot noise in LL systems have been few [23–33],

especially in the Kondo regime. For example, with the help of mas-

ter

equation approach, the shot noise in the sequential tunneling

regime was studied [27–29]. The idea of using the Bethe ansatz

to describe transport through a quantum impurity was pursued

in works [30,31] leading to the shot noise [30] and third cumu-

lants

of the current [31]. However, the properties of the shot noise

in the Kondo regime has not been touched in previous works, al-

though

the Kondo problems have been investigated in LL systems

[32,33]. This paper intends to do the theoretical work. The method

we use is the equation of motion (EOM) approach of the many-

body

Green’s function technique. The EOM treatment is a self-

consistent

procedure. The decoupling scheme used in determining

http://dx.doi.org/10.1016/j.physleta.2014.09.007

0375-9601/

K.-H. Yang et al. / Physics Letters A 378 (2014) 3136–3143 3137

the dot retarded Green function contains the same order of corre-

lations [34,35].

The

numerical results show that intralead electron interac-

tion

suppresses the shot noise. For weak intralead interaction, the

height of the peak around the Kondo temperature T

is decreased

by the intralead interaction, while the peak position remains al-

most

unchanged, which manifests the one-channel Kondo (1CK)

effect. This can help to obtain reliable estimate of the Kondo tem-

perature

by shot-noise measurements. With the increase of the

intralead interaction, the peak will disappear, indicating the tran-

sition

from the 1CK effect to two-channel Kondo (2CK) effect. In

the limit of the strong intralead interaction, the shot noise scales

as a power law in bias voltage. We also calculate the Fano factor

F = S/2eI, the ratio of the noise power spectral density S to the

full shot noise value. The intralead interaction enhances the Fano

factor, implying that the reduction rate of the current is greater

than that of its noise. This result also reveals that the Fano factor

can be controlled by tuning the LL parameters. In the high voltage

regime, the Fano factor manifests bias-voltage-dependent power-

law

scaling. The present work enriches the existing studies of the

noise in the strongly correlated Kondo regime. We believe that it

will be informative for future experimental effort.

The

structure of the paper is as the following. In Section 2,

we present the model Hamiltonian of the system. Then, a general

formula of the shot noise is derived by using the EOM method.

Numerical results are analyzed in Section 3. Section 4 gives our

summary.

2. Hamiltonian and formulation

2.1. Model Hamiltonian

The device under consideration consists of a QD coupled to two

semi-ﬁnite LL leads. Its Hamiltonian is

H = H

leads

+ H

. (1)

The ﬁrst-term describes the left (L) and right (R) electrodes,

leads

= H

+ H

. To derive it we follow the procedure in Ref. [36].

First, we write down the kinetic energy of the electrons with open

boundary conditions. Then, the electron operators are bosonized.

An outline is presented in Appendix A. Finally, the lead Hamilto-

nian

can be written in the bosonized form [37] as:

∞





cα

†

kα

+ v

sα

†

kα



dk (α = L, R), (2)

where the operator a

k,α

) is an annihilation operator of bosons

describing charge (spin) density ﬂuctuations propagating with ve-

locity

c(s)α

. In this paper, we set

h = 1.

The

second-term H

in Eq. (1) describes the QD and has the

form



†

+ Ud

†

↑

†

↓

, (3)

where ε

denotes the discrete energy-levels of the QD which can

be tuned by a gate voltage, and U is the on-dot Coulomb inter-

action.

Whereas d

†

) is the creation (annihilation) operator of

electrons with spin σ , and the corresponding occupation number

is n

= d

†

. The third-term in Eq. (1) describes the tunneling

processes between the QD and electrodes with the tunneling ma-

trix

components t



ασ



†

ασ

+H.c.



(4)

The operator ψ

at the boundary could be written in a bosonized

form [37]:

ασ

(t) =



πα



exp



∞



dke

−α



k/2



kα

−iω

cαk

−a

†

iω

cαk

√

+σ

kα

−iω

sαk

−c

†

iω

sαk

√

2kK





(5)

where α



is a short-distance cutoff with the same order of magni-

tude

of the reciprocal of the Fermi wave number k

, ω

cαk

= v

cα

and

sαk

= v

sα

k. The intralead electron interaction parameter for

the α lead is K

for the charge sector and K

for the spin sector.

Due to the repulsive electron–electron interactions, K

is less than

1 while K

is ﬁxed as 1because of the SU(2) spin symmetry.

2.2. The shot noise formula

The cur rent operator ﬂowing from the α lead to the QD can be

evaluated from the time evolution of the occupied number opera-

tor

of the lead as

ασ



ασ





†

ασ



−





†

ασ



=−

2e Re





dασ

(t, t)



, (6)

where G

dασ

(t, t



) = iψ

†

ασ



(t) is the lesser Green’s function

(GF) b etween the dot and α lead. To calculate it we construct the

complex time GF,

dασ



τ , τ





=−



(τ )ψ

†

ασ







(7)

where T

is the contour time-ordering operator and τ and τ



are

complex times running along a complex contour. Then we ﬁnd

dασ

(τ , τ



) =t





dτ

dσ

(τ , τ

ασ

(τ

, τ



), where

dσ

(τ , τ

) =−i



(τ )d

†

(τ

)



(8)

is the full Green’s function of the QD, and

ασ



, τ





=−



ασ

(τ

)ψ

†

ασ







(9)

is the local GF at the end point of the isolated α lead. After using

Langreth theorem of analytic continuation [38,39], the lesser GF

appearing in Eq. (6) is expressed as

dασ



t, t









(t, t

ασ



, t





dσ

(t, t

ασ



, t





(10)

where the “retarded” and “lesser” GFs G

r,<

dσ

for the QD are de-

ﬁned

as G

≡ iθ(t − t



){d

(t), d

†



)} and G

dσ

≡ id

†



(t).

Besides, in order to describe the nonequilibrium state of elec-

trons,

some other lesser GFs are also deﬁned for later use,

such as G

αdσ

(t, t



) ≡ id

†



)ψ

ασ

(t), G

dασ

(t, t



) ≡ iψ

†

ασ



(t),

αβσ

(t, t



) ≡ iψ

†

βσ



)ψ

ασ

(t), (α, β = L, R), as well as their cor-

responding

greater, advanced, and retarded GFs, e.g., G

ασ

≡

iθ(t −t



){d

(t), ψ

†

ασ



)}.

The

Langreth theorem helps us to get

Ldσ



t, t









Lσ

(t, t



, t





(t, t

dσ



, t





(11)

and

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