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Photon-assisted shot noise through a quantum dot coupled to Luttinger liquid
Physics Letters A 375 (2011) 747–755
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Physics Letters A www.elsevier.com/locate/pla
Photon-assisted shot noise through a quantum dot coupled to Luttinger liquid Kai-Hua Yang ∗ , Yan-Ju Wu, Yu-Peng Wu, Ya-Liang Zhao College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 2 July 2010 Received in revised form 2 December 2010 Accepted 10 December 2010 Available online 17 December 2010 Communicated by R. Wu
a b s t r a c t Photon-assisted shot noise through a quantum dot coupled to Luttinger liquid leads is considered using nonequilibrium-Green-function-method. We find that the effect of ac field on the differential shot noise is different for different intralead electron interaction. The inelastic channels associated with photonassisted-tunneling can dominate electron transport for some ac parameters. © 2010 Elsevier B.V. All rights reserved.
Keywords: Luttinger liquid Quantum dot Electronic transport
1. Introduction The study of shot noise, nonequilibrium current fluctuation with time due to the discreteness of charge carriers, has become one of the most active topics in condensed matter physics. Unlike thermal noise, which is related to the linear conductance by the Fluctuation– Dissipation theorem, shot noise provides valuable information not available only in an average current measurement. For example, shotnoise experiments can determine the kinetics of electrons and reveal information about the correlation of electronic wave functions. Therefore, shot noise has been extensively studied in a wide variety of systems [1–8]. For uncorrelated carriers described by a Poissonian distribution, the well-known Schottky’s formula S P = 2e I is referred to Poissonian shot noise, where e is the charge of the carrier and I is the average current. In mesoscopic systems, the nonequilibrium shot noise can be suppressed or enhanced either by the Pauli exclusion principle or by Coulomb interactions between electrons with respect to the Poissonian value. Recently, the advances in nanofabrication techniques have opened a new path in studying electric correlation effects. One of special interests in this respect is the shot noise of one-dimensional interacting electron systems described by the Luttinger liquid (LL) model [9]. Shot noise in LL system is a powerful tool to extract important information on the charge and the mutual correlations of the particles. For example, the nonequilibrium shot noise of the edge states has been used to measure the effective charge of quasiparticles in fractional quantum Hall system [10,11]. In single walled carbon nanotubes, shot noise is expected to be a valuable tool for studying the physics of charged elementary excitations [12–19]. For this reason, shot noise has been a subject of a great number of theoretical and experimental works in one-dimensional correlated system. Experimentally, shot noise has been studied by Kim et al. [17] who found power law dependence at low bias voltages as well as oscillations as a function of bias voltage. Similar investigation was also carried by Onac [16]. There are also several theoretical investigations on the shot noise of the LL system with different method [18,20–25]. So far, these studies on shot noise in LL system have concentrated on steady-state transport phenomena. Shot noise involving time-dependent transport phenomena has received less attention. As far as strongly correlated LL systems are concerned, one can imagine that more rich physics could be exploited if the device is subject to a microwave irradiation field. It has been reported that microwave spectroscopy is a possible tool to probe the energy spectrum of small quantum systems and to measure the decoherence time of the quantum states [26]. The essential effect of photonassisted tunneling (PAT) on transport properties is that the electrons tunneling through the system can exchange energy with microwave field, opening new inelastic tunneling channels and introducing many effects such as the sideband effect, the turnstile effect, photonelectron pumping, etc., [27,28]. Therefore the PAT could provide a new way of understanding more essence of the transport properties. In particular, the photon-assisted shot noise [29,30] allows one to extract information on the finite frequency noise, which was measured
*
Corresponding author. E-mail address: [email protected] (K.-H. Yang).
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experimentally in diffusive metallic wires [31] and theoretically investigated for ballistic systems [32]. In practice, it is challenging on finite frequency measurements in experiment. However, information on the finite frequency noise spectrum can be extracted from zerofrequency noise, provided that the system has an added, external finite frequency perturbation. Experimentally, photon-assisted shot noise has been measured in diffusive metallic conductor [33], diffusive junctions and quantum point contacts [34]. Photon-assisted shot noise was also studied theoretically by many authors [6,28–30,35–38]. The transport properties for a LL wire in the presence of a time-dependent impurity were studied only at zero temperature by Fang Cheng and Guanghui Zhou [38]. For normal metals, the differential shot noise as the function of bias voltage exhibits a series of steps, whose width is Ω and heights are proportional to Bessel function, where Ω is the microwave field frequency [28,35]. One naturally expects that this step behavior is modified by the electron–electron interacting in the leads, i.e. a smoothing of the steps due to electronic correlations. There have been many studies on the photon-assisted shot noise of a quantum dot (QD) system coupled to Fermi liquid (FL) leads irradiated by microwave fields [6,28,37]. However, the photon-assisted shot noise is not still investigated through a QD coupled to LL leads. QD have often been used to study various transport properties of strongly correlated systems, such as most notably the Kondo effect [39]. The experimental control over the properties of the QD allows a more detailed investigation of the strongly correlated states than that accessible in bulk systems. An experimental study of LL behavior in QD has been suggested by Kinaret et al. [40] who theoretically investigated tunneling into a QD in a strong magnetic field. In this Letter, we will study photon-assisted shot noise through a QD coupled to LL. A general photon-assisted shot noise formula is firstly derived by applying the nonequilibrium Green function approach [41]. We in detail study the effect of the intralead electron interaction and ac parameters on the transport properties. We find that, for strong intensity of the microwave field, the inelastic channels associated with photon-assisted electron tunneling can control over electron transport. For a weak electron interaction, the differential conductance and differential shot noise display resonant-like behavior as a function of bias and gate voltages. In the limit of strong interaction, resonant behavior disappears due to strong electronic correlations and the differential conductance and differential shot noise scale as a power law in bias voltage. In strong coupling between a QD and LL, the differential shot noise as a function of voltage exhibits steps at integer values of the ac frequency Ω when no electron–electron interaction in the leads. Besides, the numerical results show the width of the dip in Fano factor becomes broader with the increase of the intensities of the ac field. The steps are smoothed out due to the fact that electrons tunnel in a strongly correlated one-dimensional system. The Letter is organized as follows: in Section 2, we introduce the bosonized representation Hamiltonians of the LL leads, which are diagonal to obtain the shot noise formula of the LL-QD-LL system by nonequilibrium Green function technique. Section 3 discusses our numerical results for differential conductance and differential shot noise. Finally, we conclude in Section 4. 2. Model and formulation We start out by introducing a model for a QD coupled to the LL leads under a time-dependent gate voltage. The Hamiltonian of the system can be described as follows
H = H leads + H D + H T
(1)
where H leads = H L + H R represents the Hamiltonians of the left and right LL leads and its standard form [42] is H leads = h¯ v c
∞ 0
†
ak ak k dk,
H D = ε (t )d d is the Hamiltonian of the QD, with {d , d} the creation/annihilation operators of the electron in the QD, ε (t ) = ε + cos Ω t, ε is the time-independent single electron energy of the QD without MW fields, and Ω are the amplitude and frequency of the ac gate †
†
voltage, respectively. It causes an alternating current through the dot. H T is the tunneling Hamiltonian and can be written as
HT =
t α d† ψα + h.c.
(2)
α †
where t α is the electron tunneling constant and {ψα , ψα }(α = L / R ) are the Fermi operators at the end points of the left/right lead. The operator ψα could be written in a “bosonized” form [42]
ψα =
∞
2
πα
exp(−α k/2) † dk akα − akα 2kρ k
exp 0
(3)
where α is a short-distance cutoff of the order of the reciprocal of the Fermi wave number k F , and kρ = (2/ g − 1)−1 is the interaction parameter in the “fermionic” form of the LL Hamiltonian, it defines the LL parameter g which is varied between 0 and 1: the case g = 1 describes the noninteracting (FL) leads, while in the case g → 0 the interaction in the leads goes to infinity. Firstly, we apply a time-dependent canonical transformation [43] (hereafter h¯ = 1) to Hamiltonian H D
t U (t ) = exp i
†
dt cos Ω t d d .
(4)
−∞
Under this transformation, we obtain H D (t ) = U (t ) H (t )U −1 (t ) − iU (t )∂t U −1 (t ) = εd† d, the time-dependence of the gate voltage removed. Instead, the electron tunnel coupling
t
t α (t ) = t α exp −i −∞
is now time-dependent.
dt cos Ω t
ε is
(5)
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The current operator which describes tunneling from the L lead into the QD at time t is found to be: (in units of h¯ = 1)
ˆI L (t ) = ie t L (t )ψ † d − t ∗L (t )d† ψ L . L
(6)
Using nonequilibrium-Green-function technique and Langreth theorem of analytic continuation, the current can then be expressed as:
I L (t ) = −2e Re
∗ < a dt 1 t L (t ) G r (t , t 1 ) g < L (t 1 , t ) + G (t , t 1 ) g L (t 1 , t ) t L (t 1 )
(7)
where G r (t , t 1 ) and G < (t , t 1 ) are the Green’s function of the QD. We have defined G r (t , t 1 ) = −i θ(t − t 1 ){d(t ), d† (t 1 )} H and all the other <,a Green functions with the standard forms [41]. g L are the lesser (advanced) Green functions of the left leads without coupling to the QD. The retarded Green function G r (t , t 1 ) and lesser Green function G < (t , t 1 ) can be calculated from the following Dyson equation:
dτ dτ g r (t , τ )Σ r
G r (t , t 1 ) = g r (t , t 1 ) +
τ , τ G r τ , t1
(8)
and the Keldysh equation
dτ dτ G r (t , τ )Σ <
G < (t , t 1 ) =
τ , τ G a τ , t1
(9)
where g r (t , t 1 ) is the free retarded Green function of isolated dot which depends only on the time difference t − t 1 . Σ r /a,< (τ , τ ) = r /a,< ∗ (τ , τ )t α (τ ) is the self-energy. We now make Fourier transformation over the two times t and t which switches from α = L , R t α (τ ) g α the time-domain into energy representation through a double-time Fourier-transform defined as [44,45]
F (ω, ω1 ) = and
dt dt 1 F (t , t 1 )e i ωt e −i ω1 t1
dω dω1
F (t , t 1 ) =
2π 2π
(10)
F (ω, ω1 )e −i ωt e i ω1 t1 .
(11)
Then from Eq. (8), one can derive the Dyson equation
dω dω G r
ω, ω e iωt e−iω t1 =
dω g r (ω)e i ω(t −t1 ) +
1
(2π
)2
Next we take the time average over the above equation with respect to
F (t , t 1 ) = lim
T →∞
T
1
F
2T
dω dω1 dω g r (ω)Σ r (ω, ω1 )G r
ω1 , ω e iωt e−iω t1 .
(12)
τ = (t + t 1 )/2 and let t = t − t 1 .
τ + t /2 , τ − t /2 d τ .
(13)
−T
With the help of the time average Eq. (13), we finally obtain following expression for Dyson equation
dω G r (ω)e i ω(t −t1 ) =
dω g r (ω)e i ω(t −t1 ) +
1
(2π )2
dω dω1 g r (ω)Σ r (ω, ω1 )G r (ω1 , ω)e i ω(t −t1 ) .
(14)
Since we are interested in the time-averaged tunneling current, we only need to calculate the Green functions related to the Fourier transformed versions in diagonal variable forms by using the resonant approximation described in Refs. [6,46] to find
G r (ω, ω1 ) = 2π G r (ω)δ(ω − ω1 ).
(15)
This signifies that the electrons tunneling through the QD possess resonant structure. The similar approximation method has been also adopted by the other authors [47,48]. Now we can obtain the Dyson equation for the QD as a single time function: G r (ω) = g r (ω) + g r (ω)Σ r (ω)G r (ω). Then the retarded Green’s function is related to
G r (a) (ω) =
1
(16)
ω − ε − Σ r (a) (ω)
where the retarded self-energy Σ r (ω) can be obtained by Σ r (ω) = (Σ > (ω) − Σ < (ω))/2. The greater (lesser) self-energy Σ > (ω) (Σ < (ω)) can be obtained by the time-averaged double-time self energy as
Σ
>,<
(t 1 , t 2 ) =
∗
>,<
t α (t 1 ) g α
α=L/ R
(t 1 , t 2 )t α (t 2 ) =
α = L / R ,nm
Jn
Ω
Jm
|t α |2 g α>,< (t 1 − t 2 )e inΩ t1 e −imΩ t2 Ω
(17)
∞
where J m ( z) is the mth order Bessel function. In deriving this equation, we used identity exp(iz sin θ) = m=−∞ J m ( z)e imθ , which is a signature of photon-assisted processes [49]. With above the same method, the time-averaged self-energy can be obtained Σ >,< (ω) =
2 2 >,< α = L / R ,n J n ( Ω )|t α | g α (ω + nΩ). Similarly, using above the same procedure to the lesser Green function, after Fourier transform and time average, one obtains the following expression for the lesser Green function from Eq. (9)
G < (ω) = G r (ω)Σ < (ω)G a (ω).
(18)
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Once the G r ,a is calculated, the lesser Green’s function is obtained from the Keldysh equation. After using Langreth theorem of analytic T /2 continuation, and taking Fourier transformation over the current equation (7) and the time average, F (t ) = lim T →∞ T1 − T /2 F (t ) dt, the time-averaged tunneling current can be expressed as
IL =
e 2π
2 2 Jm J n |t L |2 |t R |2
> > < dω G dr G ad g < Ln (ω) g Rm (ω) − g Ln (ω) g Rm (ω)
(19)
m,n
with g Lm/ Rm (ω) = g L / R (ω + mΩ). This formula describes the time-averaged current through the LL-QD-LL system in the presence of ac >,< >,< fields. g L and g R are the greater (lesser) Green functions of the left and right leads without the couplings to the dot, and can be directly quoted from [42]
T
<,>
g L / R (ω) = ±i
exp ∓(ω − μ L / R )/2T
|t L / R |2
where T is the temperature. μ L / R = μ F + eV L / R , system. γ L / R (ω) is defined as
γL / R (ω) =
ΓL/ R 2π T
πT Λ
1 / g −1
γL / R (ω − μL / R )
(20)
μ F is the chemical potential of LL leads, V L − V R = V is the bias voltage imposed on the
|(1/2g + i ω/2π T )|2 (1/ g )
(21)
here (x) is the Gamma function, g is the interaction parameter characterizing the left/right LL leads, which should not be confused with the Green functions with the same symbols, Γ L / R = 2π ct 2L / R /α Λ describe the effective level broadening of the dot, c is a dimensionless constant of order 1, and Λ is the high energy cutoff or a band width [42,50]. If we introduce the electron occupation number for interacting |[1/2g +i (ω−μ L / R )/2π T ]|2 which is analogous to the Fermi distribution f (ω) for noninteracting (1/ g ) 2 |[ 1 / 2g + i ( ω − μ )/ 2 π T ]| L/ R 1 (ω−μ L / R )/2T π T 1/ g −1 e (Λ) is analogous to 1 − f (ω) for FL leads, then we finally obtains the 2π (1/ g )
electrons F L/ R = PAT current
I =e
2 2 Jm Jn
m,n
dω 2π
> < < > Γ L Γ R G r G a F Rm (ω) F Ln (ω) − F Rm (ω) F Ln (ω)
(22)
with F Lm/ Rm (ω) = F L / R (ω + mΩ). The spectral density of shot noise is defined by the Fourier transformation of the current correlation
S αα t , t = ˆI α (t ) ˆI α t + ˆI α t ˆI α (t )
(23)
where ˆI α (t ) = ˆI α (t ) − ˆI α (t ). By substituting the current operator Eq. (6) into Eq. (23), the shot noise can be expressed in terms of †
the four-operator Green functions which is related to the electrons operators in the leads ψα (ψα ) and in the QD d† (d). The Hamiltonian H Eq. (1) is noninteracting because the bosonized form H L / R have been mapped to a noninteracting boson-type model with renormalized parameters. We can factorize the four-operator Green functions into a product of the two-operator Green functions with an annihilation electron operator and a creation electron operator by using the Wick theorem [51]. There appear the off-diagonal functions < † G d<α (t , t ) ≡ i ψα (t )d(t ), G < αd (t , t ) ≡ i d (t )ψα (t ) and G α ,β (t , t ) ≡ i ψβ (t )ψα (t ) in the shot noise expression. In order to obtain these Green functions, as the starting point, we construct the contour-ordered Green function and derive functions equation of motion (EOM). And then, via analytical continuation rules, we obtain these off-diagonal Green functions which are expressed in terms of both the Green functions of the QD and the bare Green functions of the leads. With the help of the Keldysh Green functions [52], after a straightforward calculation and derivation via EOM, we obtain the double-time shot noise S (t , t ) under the dc bias and time-dependent voltage field. The more details of derivation process on shot noise can be found in Ref. [24]. Usually, we are interested in the spectral density of current noise in the pseudoequilibrium state, which is defined as the Fourier transformed current correlation of Eq. (23) by the expression: †
†
†
†
∞ ∞ S (ω1 , ω2 ) =
dt 1 dt 2 e i ω1 t1 e i ω2 t2 S (t 1 , t 2 ).
(24)
−∞ −∞
In this Letter, we consider the spectral density of shot noise in the zero-frequency limit due to the fact that the presence of the timedependent field perturbation mimics a finite frequency shot noise measurement. The shot noise spectral density contains the effect of photon absorption and emission induced by the microwave fields. In experiment, in order to obtain the observed shot noise spectral density, the absorption and emission of photon numbers must be equal, which comes from the requirement of Fourier transformation of the current correlation function in the pseudoequilibrium state. In fact, it is the selection rule for the absorption and emission of photons associated with the energy conservation, i.e., the total absorbed photon energy equals the total emitted photon energy in the system. In this Letter, we are interested in the time-averaged zero-frequency shot noise for the balanced absorption case where the absorption and emission of photon numbers are equal in the same lead, and there exist no correlations between the sidebands of photon between different terminal currents. Since the shot noise spectrum possesses the property S LL (ω) = S R R (ω), we present only the noise on the left terminal, and denote it as S = S LL (0). Therefore, following standard mathematical procedure, making the Fourier transformation over the two times t and t and the time average over τ = (t + t )/2, we finally obtain the time-averaged spectral density of shot noise in the zero-frequency limit for the balanced absorption case:
K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
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Fig. 1. Differential conductance (a,b,c) and differential shot noise (d,e,f) as a function of bias voltage for several different microwave intensities. Here we put Γ = 0.02Ω , Ω = 1.0, ε = 2.0, here = 0.7 (a,d), = 1.0 (b,e) and = 1.7 (c,f). Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from bottom to top. For 1/2 < g 1, the main resonance peak is accompanied by a set of satellite resonant peaks and when > 1 the first satellite peak is higher than the main resonant peak. In the limit of the strong interaction, resonance behavior disappears and shows bias-voltage-dependent power law scalings.
S = e2
2 2 Jm Jn
m,n
− e2
dω 2π
> < < > Γ L Γ R G r G a F Rm (ω) F Ln (ω) + F Rm (ω) F Ln (ω)
2 2 2 2 Jm Jn J p Jq
m,n, p ,q
dω 2π
ΓL Γ R G r G a
2
> < > < > < > < F Rq (ω) F Lm (ω) − F Lm (ω) F Rq (ω) F Rp (ω) F Ln (ω) − F Ln (ω) F Rp (ω) .
(25)
Eq. (25) describes the photon-assisted shot noise through the QD coupled to the LL leads under the alternating field, which contains the thermal noise and shot noise. Eq. (25) is a basic and general formula for the time-averaged zero-frequency shot noise through a QD. It is the central result of this work, and is different from Eq. (30) in Ref. [38] which was obtained only at zero temperature. From Eq. (25) we can find that the effect of photon absorption and emission induced by the microwave fields is included in the shot noise spectral density. The expression for the noise becomes more complicated due to the opening of multiple inelastic tunneling channels. In the absence of the ac gate voltage (i.e., = 0), Eq. (25) can be expressed in a standard form as Eq. (21) without the ac field [37]. More details of the shot noise formula can be found in the literature [37]. When g = 1, Eq. (25) is reduced into the formula (23) in Ref. [24]. In order to observe qualitative effect of the intralead electron interaction on the transport properties, we consider the weakly coupling between QD and LL. After taking a contour integral, Eq. (22) reduces to the
I =e
> (ε ) F < (ε ) − F < (ε ) F > (ε )) ( F Rm Ln Rm Ln 2 2 Jm J n ΓL Γ R 2 > (ε ) + F < (ε )] J [ F m α , m α , m α ,m m,n
(26)
and Eq. (25) becomes
S = e2
> < < > 2 2 ( F Rm (ε ) F Ln (ε ) + F Rm (ε ) F Ln (ε )) Jm Jn 2 > < α ,m J m [ F α ,m (ε ) + F α ,m (ε )] m,n
− e2
m,n, p ,q
2 2 2 2 Jm Jn J p Jq
> (ε ) F < (ε ) − F > (ε ) F < (ε ))( F > (ε ) F < (ε ) − F > (ε ) F < (ε )) ( F Rq Lm Lm Rq Rp Ln Ln Rp .
2 > (ε ) + F < (ε )]3 J [ F α ,m α ,m m α ,m
(27)
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K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
Fig. 2. Differential shot noise as a function of bias voltage. The parameters are Γ L = 40.0, Γ R = 20.0, Ω = 1.0, = 1.7,
ε = 2.0, g = 1.0.
Fig. 3. Differential conductance (a,b,c) and differential shot noise (d,e,f) as a function of the level energy ε . V = 5.0, all the other parameters are the same as Fig. 1. Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from bottom to top. For g 1, the differential shot noise exhibits a double peak, however, only a single peak shows up in differential conductance.
The validity of the approximation has been given by the authors [50]. This evaluation validates only when T = 0. Here we no longer restate the approximation. 3. Numerical results and discussion In this section, we present the numerical results of the photon-assisted zero frequency differential shot noise and Fano factor. For simplicity, we only consider the symmetric systems Γ L = Γ R and symmetric bias V L = − V R = V /2. The Fermi energy E F = 0 and temperature T = 0.1. We take the microwave field frequency Ω as the energy unit throughout the Letter. Moreover, we set h¯ = e = 1.
K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
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Fig. 4. Fano factor as a function of the level energy ε with V = 5.0 (a,b,c) and the bias voltage with ε = 5.0 (d,e,f). Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from top to bottom. The width of dip becomes broader with the increase of /Ω in (a,b,c). Strong intralead electron interaction smears out the steps in Fano factor.
First in Fig. 1, we plot the differential shot noise as a function of the bias voltage for different strengths on the basis of Eqs. (26), (27). We can find the following features in this figure: firstly, the main resonant peak is suppressed while the satellite peak connected with the photon-absorption processes grows up gradually with the increase of ac intensity for 1/2 < g 1. And the main resonance peak is now accompanied by a set of satellites spaced by δ = h¯ Ω . Secondly, when the amplitude becomes larger than one, = 1.7, the first satellite peak related to the photon-absorption processes is higher than the main resonant peak. It implies that increasing intensity of the ac field will enhance the photon-assisted processes and the inelastic channels associated with photon-assisted electron tunneling can dominate electron transport. This is the most interesting phenomenon in this Letter. For moderate interaction, the sideband peaks become indistinct, and finally all the resonance peaks disappear for LL leads with strong enough electron interactions. It is due to the fact that the tunneling density of states on the LL is modified by the Coulomb interactions in the LL. They scale as a power law in higher bias voltage, dI /dV ∝ n J n2 ( Ω )| V + nΩ|1/ g −2 and dS /dV ∝ n J n2 ( Ω )| V + nΩ|1/ g −2 , respectively, in accordance with the LL prediction in the absence of an ac gate voltage. These features are related directly to the power law behavior of the density of state. In order to further check our model and theory, we plot dS /dV vs the bias voltage in the case when Γ Ω in the Fig. 2 based on Eq. (25). At g = 1, the differential shot noise exhibits steps whose spacing is an integer value of Ω and the step heights are proportional to J n2 (n = ±1, ±2, . . .). This is in complete agreement with the theoretical results obtained by Lesovik and Levitov [53] and experiment result obtained by Schoelkopf et al. [54] for a FL, respectively. Discussion of the strongly coupling between QD and Luttinger leads is beyond the scope of this Letter, and the effects of strongly coupling on transport under ac field will be investigated in our future publication. In Fig. 3, to further understand the PAT, the differential shot noise and differential conductance are plotted as a function of level energy ε , or, equivalently, as a function of gate voltage V g . The bias voltage is V = 5.0. For g = 1, the main resonance peaks correspond to the level positions at ε = ± V /2. Comparing Fig. 1 and Fig. 2, one can see that the some similarities are preserved for different the intensity of ac field. For example, when = 1.7, the first satellite peak connected is higher than the main resonant peak. On the other hand, with the intralead electron interaction increasing, all the peaks become blurring and then merge into a single central peak for g = 0.2 in differential conductance. These features are due to the power law behavior of the density of states and the suppression of the electron tunneling near the Fermi energies of LL leads. Correspondingly, the differential shot noise power exhibits two peaks located symmetrically around the conductance peak position. This central peak in differential conductance and differential shot noise entirely narrows for g = 0.2. They come from the contribution of the direct tunneling process and many PAT processes. In Fig. 4, we study the variation of the Fano factor F as the function of both the bias voltage and the level energy of the QD at a given bias V = 5.0 with different strengths and frequencies of the external fields, respectively. From the Fig. 4 we find that, the steps appear for noninteraction in the leads. For a weak interaction, the sideband steps become blurring and finally disappear for strong interaction
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K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
and exhibit a simple behavior. In particular, we observe that, in Fig. 4a, the width of the dip becomes broader with the increase of /Ω , originated from the contributions of the multi-photon-assisted tunneling processes and the summation of the Bessel function. 4. Summary In conclusion, we have investigated the photon-assisted transport properties of QD coupled to LL by applying nonequilibrium Green function method. A general expression for the photon-assisted shot noise and current is obtained, which includes the arbitrary ac field and the Coulomb effects in the leads. Numerical results have shown that the resonant-like behavior of the photon-assisted shot noise for weak electron interaction in leads cannot survive in the strong Coulomb interaction leads. For strong interaction, photon-assisted shot noise has a power-law dependence with the voltage. This conclusion is qualitatively in agreement with a recent study [36] where the step structure of the ac noise of LL in carbon nanotube is found to be smoothed out by the Coulomb interaction. These are due to the fact that power law behavior of the density of states of the electrons in a strongly correlated one-dimensional system. We reproduce the result of the step-like behavior of the photon-assisted noise in noninteracting leads for the case of Γ Ω . In different parameter region, the effect of the electron interaction and ac field on the transport properties is different. These results may shed light on the understanding of the behavior of LL. We expect that the work can be tested in experiment. In addition, photon-assisted transport in this work is a first step for studying the behavior of mesoscopic systems in nonstationary situations, which is relevant to the understanding of the finite-frequency shot noise simulated by a zero-frequency noise measurement operated on a nanotube in presence of PAT processes. Acknowledgements This work was supported by the Beijing Novel Program No. 2005B11 and Fund of Beijing Municipal Education Commission. References [1] A. Hewson, D. Newns, J. Phys. C 13 (1980) 4477. [2] Y.M. Blanter, M. Buttiker, Phys. Rep. 336 (2000) 1. [3] K.E. Nagaev, Phys. Rev. B 57 (1998) 4628; Y. Wei, B. Wang, J. Wang, H. Guo, Phys. Rev. B 60 (1999) 16900. [4] B. Dong, X.L. Lei, J. Phys.: Condens. Matter 14 (2002) 4963. [5] N.Y. Kim, P. Recher, W.D. Oliver, Y. Yamamoto, J. Kong, H.-J. Dai, Phys. Rev. Lett. 99 (2007) 036802. [6] H.-K. Zhao, J. Wang, Phys. Rev. B 74 (2006) 245401. [7] Y. Meir, A. Golub, Phys. Rev. Lett. 88 (2002) 116802. [8] A. Golub, Phys. Rev. B 97 (2006) 016602; E. Sela, Y. Oreg, F.v. Oppen, J. Koch, Phys. Rev. Lett. 97 (2006) 086601. [9] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [10] L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79 (1997) 2526. [11] R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature (London) 389 (1997) 162. [12] C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 72 (1994) 724. [13] P. Roche, M. Kociak, S. Gueron, A. Kasumov, B. Reulet, H. Bouchiat, Eur. Phys. J. B 28 (2002) 217. [14] K.-V. Pham, F. Pichon, K.-I. Imura, P. Lederer, Phys. Rev. B 68 (2003) 205110. [15] B. Trauzettel, I. Safi, F. Dolcini, H. Grabert, Phys. Rev. Lett. 92 (2004) 226405. [16] E. Onac, F. Balestro, B. Trauzettel, C.F.J. Lodewijk, L.P. Kouwenhoven, Phys. Rev. Lett. 96 (2006) 026803. [17] N.Y. Kim, P. Recher, W.D. Oliver, Y. Yamamoto, J. Kong, H. Dai, Phys. Rev. Lett. 99 (2007) 036802. [18] P. Recher, N.Y. Kim, Y. Yamamoto, Phys. Rev. B 74 (2006) 235438. [19] L.G. Herrmann, T. Delattre, P. Morfin, J.-M. Berroir, B. Placais, D.C. Glattli, T. Kontos, Phys. Rev. Lett. 99 (2007) 156804. [20] A. Braggio, R. Fazio, M. Sassetti, Phys. Rev. B 67 (2003) 233308; E. Onac, F. Balestro, B. Trauzettel, C.F.J. Lodewijk, L.P. Kouwenhoven, Phys. Rev. Lett. 96 (2006) 026803. [21] J.U. Kim, J.M. Kinaret, M.-S. Choi, J. Phys.: Condens. Matter 17 (2005) 3815. [22] S. Kawaguchi, J. Phys.: Condens. Matter 21 (2009) 395303. [23] A. Komnik, A.O. Gogolin, Phys. Rev. B 68 (2003) 235323. [24] K.-H. Yang, Y.-P. Wu, Y.-L. Zhao, Phys. Lett. A 374 (2010) 917. [25] S. Takei, Y.B. Kim, A. Mitra, Phys. Rev. B 72 (2005) 075337. [26] W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, L.P. Kouwenhoven, Rev. Mod. Phys. 75 (2003) 1. [27] L.P. Kouwenhoven, L.L. Sohn, G. Schön, Mesoscopic Electron Transport, first ed., Kluwer Academic, Dordrecht, 1997. [28] Q.-F. Sun, J. Wang, T.-H. Lin, Phys. Rev. B 61 (2000) 13032. [29] G.B. Lesovik, L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538; G.B. Lesovik, R. Loosen, JETP Lett. 65 (1997) 295. [30] G.B. Lesovik, T. Martin, J. Torrès, Phys. Rev. B 60 (1999) 11935; J. Torrès, T. Martin, G.B. Lesovik, Phys. Rev. B 63 (2001) 134517. [31] R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 78 (1997) 3370. [32] S.R. Yang, Solid State Commun. 81 (1992) 375. [33] R.J. Schoelkopf, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 80 (1998) 2437. [34] A.A. Kozhevnikov, R.J. Schoelkopf, D.E. Prober, Phys. Rev. Lett. 84 (2000) 3398. [35] A. Crépieux, P. Devillard, T. Martin, Phys. Rev. B 69 (2004) 205302. [36] M. Guigou, A. Popoff, T. Martin, A. Crpieux, Phys. Rev. B 76 (2007) 045104. [37] K.-H. Yang, C.-Z. Liu, Eur. Phys. J. B 72 (2009) 409. [38] F. Cheng, G.-H. Zhou, Phys. Rev. B 73 (2006) 125335. [39] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature (London) 391 (1998) 156; S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281 (1998) 540; J. Schmid, J. Weis, K. Eberl, K.v. Klitzing, Physica (Amsterdam) B 256–258 (1998) 182; F. Simmel, R.H. Blick, J.P. Kotthaus, W. Wegscheider, M. Bichler, Phys. Rev. Lett. 83 (1999) 804. [40] J.M. Kinaret, Y. Meir, N.S. Wingreen, P. Lee, X.-G. Wen, Phys. Rev. B 45 (1992) 9489; J.M. Kinaret, Y. Meir, N.S. Wingreen, P.A. Lee, X.-G. Wen, Phys. Rev. B 46 (1992) 4681. [41] H. Haug, A.P. Janho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, 1996. [42] A. Furusaki, Phys. Rev. B 57 (1998) 7141.
K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
[43] [44] [45] [46]
[47] [48] [49] [50] [51] [52] [53] [54]
C. Bruder, H. Schoeller, Phys. Rev. Lett. 72 (1994) 1076. B. Wang, J. Wang, H. Guo, Phys. Rev. Lett. 82 (1999) 398. Y.X. Xing, B. Wang, Y.D. Wei, B.G. Wang, J. Wang, Phys. Rev. B 70 (2004) 245324. H.K. Zhao, Z. Phys. B 102 (1997) 415; H.K. Zhao, Phys. Lett. A 226 (1997) 105; H.K. Zhao, Phys. Rev. B 63 (2001) 205327; H.K. Zhao, Eur. Phys. J. B 64 (2008) 237; H.K. Zhao, J. Wang, Phys. Rev. B 64 (2001) 094505. H.Ch. Yong, K.H. Yang, G.S. Tian, Commun. Theor. Phys. 48 (2007) 1107. Q.F. Sun, J. Wang, T.-H. Lin, Phys. Rev. B 59 (1999) 13126. M. Grifoni, P. Hanggi, Phys. Rep. 304 (1998) 229; G. Platero, R. Aguado, Phys. Rep. 395 (2004) 1. Y.F. Yang, T.H. Lin, Phys. Rev. B 64 (2001) 233314. G.D. Mahan, Many-Particle Physics, second ed., Plenum Press, New York, 1990. L.V. Keldysh, JETP Lett. 20 (1965) 1018. G.B. Lesovik, L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538. R.J. Schoelkopf, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 80 (1998) 2437.
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Physics Letters A www.elsevier.com/locate/pla
Photon-assisted shot noise through a quantum dot coupled to Luttinger liquid Kai-Hua Yang ∗ , Yan-Ju Wu, Yu-Peng Wu, Ya-Liang Zhao College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 2 July 2010 Received in revised form 2 December 2010 Accepted 10 December 2010 Available online 17 December 2010 Communicated by R. Wu
a b s t r a c t Photon-assisted shot noise through a quantum dot coupled to Luttinger liquid leads is considered using nonequilibrium-Green-function-method. We find that the effect of ac field on the differential shot noise is different for different intralead electron interaction. The inelastic channels associated with photonassisted-tunneling can dominate electron transport for some ac parameters. © 2010 Elsevier B.V. All rights reserved.
Keywords: Luttinger liquid Quantum dot Electronic transport
1. Introduction The study of shot noise, nonequilibrium current fluctuation with time due to the discreteness of charge carriers, has become one of the most active topics in condensed matter physics. Unlike thermal noise, which is related to the linear conductance by the Fluctuation– Dissipation theorem, shot noise provides valuable information not available only in an average current measurement. For example, shotnoise experiments can determine the kinetics of electrons and reveal information about the correlation of electronic wave functions. Therefore, shot noise has been extensively studied in a wide variety of systems [1–8]. For uncorrelated carriers described by a Poissonian distribution, the well-known Schottky’s formula S P = 2e I is referred to Poissonian shot noise, where e is the charge of the carrier and I is the average current. In mesoscopic systems, the nonequilibrium shot noise can be suppressed or enhanced either by the Pauli exclusion principle or by Coulomb interactions between electrons with respect to the Poissonian value. Recently, the advances in nanofabrication techniques have opened a new path in studying electric correlation effects. One of special interests in this respect is the shot noise of one-dimensional interacting electron systems described by the Luttinger liquid (LL) model [9]. Shot noise in LL system is a powerful tool to extract important information on the charge and the mutual correlations of the particles. For example, the nonequilibrium shot noise of the edge states has been used to measure the effective charge of quasiparticles in fractional quantum Hall system [10,11]. In single walled carbon nanotubes, shot noise is expected to be a valuable tool for studying the physics of charged elementary excitations [12–19]. For this reason, shot noise has been a subject of a great number of theoretical and experimental works in one-dimensional correlated system. Experimentally, shot noise has been studied by Kim et al. [17] who found power law dependence at low bias voltages as well as oscillations as a function of bias voltage. Similar investigation was also carried by Onac [16]. There are also several theoretical investigations on the shot noise of the LL system with different method [18,20–25]. So far, these studies on shot noise in LL system have concentrated on steady-state transport phenomena. Shot noise involving time-dependent transport phenomena has received less attention. As far as strongly correlated LL systems are concerned, one can imagine that more rich physics could be exploited if the device is subject to a microwave irradiation field. It has been reported that microwave spectroscopy is a possible tool to probe the energy spectrum of small quantum systems and to measure the decoherence time of the quantum states [26]. The essential effect of photonassisted tunneling (PAT) on transport properties is that the electrons tunneling through the system can exchange energy with microwave field, opening new inelastic tunneling channels and introducing many effects such as the sideband effect, the turnstile effect, photonelectron pumping, etc., [27,28]. Therefore the PAT could provide a new way of understanding more essence of the transport properties. In particular, the photon-assisted shot noise [29,30] allows one to extract information on the finite frequency noise, which was measured
*
Corresponding author. E-mail address: [email protected] (K.-H. Yang).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.12.024
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K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
experimentally in diffusive metallic wires [31] and theoretically investigated for ballistic systems [32]. In practice, it is challenging on finite frequency measurements in experiment. However, information on the finite frequency noise spectrum can be extracted from zerofrequency noise, provided that the system has an added, external finite frequency perturbation. Experimentally, photon-assisted shot noise has been measured in diffusive metallic conductor [33], diffusive junctions and quantum point contacts [34]. Photon-assisted shot noise was also studied theoretically by many authors [6,28–30,35–38]. The transport properties for a LL wire in the presence of a time-dependent impurity were studied only at zero temperature by Fang Cheng and Guanghui Zhou [38]. For normal metals, the differential shot noise as the function of bias voltage exhibits a series of steps, whose width is Ω and heights are proportional to Bessel function, where Ω is the microwave field frequency [28,35]. One naturally expects that this step behavior is modified by the electron–electron interacting in the leads, i.e. a smoothing of the steps due to electronic correlations. There have been many studies on the photon-assisted shot noise of a quantum dot (QD) system coupled to Fermi liquid (FL) leads irradiated by microwave fields [6,28,37]. However, the photon-assisted shot noise is not still investigated through a QD coupled to LL leads. QD have often been used to study various transport properties of strongly correlated systems, such as most notably the Kondo effect [39]. The experimental control over the properties of the QD allows a more detailed investigation of the strongly correlated states than that accessible in bulk systems. An experimental study of LL behavior in QD has been suggested by Kinaret et al. [40] who theoretically investigated tunneling into a QD in a strong magnetic field. In this Letter, we will study photon-assisted shot noise through a QD coupled to LL. A general photon-assisted shot noise formula is firstly derived by applying the nonequilibrium Green function approach [41]. We in detail study the effect of the intralead electron interaction and ac parameters on the transport properties. We find that, for strong intensity of the microwave field, the inelastic channels associated with photon-assisted electron tunneling can control over electron transport. For a weak electron interaction, the differential conductance and differential shot noise display resonant-like behavior as a function of bias and gate voltages. In the limit of strong interaction, resonant behavior disappears due to strong electronic correlations and the differential conductance and differential shot noise scale as a power law in bias voltage. In strong coupling between a QD and LL, the differential shot noise as a function of voltage exhibits steps at integer values of the ac frequency Ω when no electron–electron interaction in the leads. Besides, the numerical results show the width of the dip in Fano factor becomes broader with the increase of the intensities of the ac field. The steps are smoothed out due to the fact that electrons tunnel in a strongly correlated one-dimensional system. The Letter is organized as follows: in Section 2, we introduce the bosonized representation Hamiltonians of the LL leads, which are diagonal to obtain the shot noise formula of the LL-QD-LL system by nonequilibrium Green function technique. Section 3 discusses our numerical results for differential conductance and differential shot noise. Finally, we conclude in Section 4. 2. Model and formulation We start out by introducing a model for a QD coupled to the LL leads under a time-dependent gate voltage. The Hamiltonian of the system can be described as follows
H = H leads + H D + H T
(1)
where H leads = H L + H R represents the Hamiltonians of the left and right LL leads and its standard form [42] is H leads = h¯ v c
∞ 0
†
ak ak k dk,
H D = ε (t )d d is the Hamiltonian of the QD, with {d , d} the creation/annihilation operators of the electron in the QD, ε (t ) = ε + cos Ω t, ε is the time-independent single electron energy of the QD without MW fields, and Ω are the amplitude and frequency of the ac gate †
†
voltage, respectively. It causes an alternating current through the dot. H T is the tunneling Hamiltonian and can be written as
HT =
t α d† ψα + h.c.
(2)
α †
where t α is the electron tunneling constant and {ψα , ψα }(α = L / R ) are the Fermi operators at the end points of the left/right lead. The operator ψα could be written in a “bosonized” form [42]
ψα =
∞
2
πα
exp(−α k/2) † dk akα − akα 2kρ k
exp 0
(3)
where α is a short-distance cutoff of the order of the reciprocal of the Fermi wave number k F , and kρ = (2/ g − 1)−1 is the interaction parameter in the “fermionic” form of the LL Hamiltonian, it defines the LL parameter g which is varied between 0 and 1: the case g = 1 describes the noninteracting (FL) leads, while in the case g → 0 the interaction in the leads goes to infinity. Firstly, we apply a time-dependent canonical transformation [43] (hereafter h¯ = 1) to Hamiltonian H D
t U (t ) = exp i
†
dt cos Ω t d d .
(4)
−∞
Under this transformation, we obtain H D (t ) = U (t ) H (t )U −1 (t ) − iU (t )∂t U −1 (t ) = εd† d, the time-dependence of the gate voltage removed. Instead, the electron tunnel coupling
t
t α (t ) = t α exp −i −∞
is now time-dependent.
dt cos Ω t
ε is
(5)
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The current operator which describes tunneling from the L lead into the QD at time t is found to be: (in units of h¯ = 1)
ˆI L (t ) = ie t L (t )ψ † d − t ∗L (t )d† ψ L . L
(6)
Using nonequilibrium-Green-function technique and Langreth theorem of analytic continuation, the current can then be expressed as:
I L (t ) = −2e Re
∗ < a dt 1 t L (t ) G r (t , t 1 ) g < L (t 1 , t ) + G (t , t 1 ) g L (t 1 , t ) t L (t 1 )
(7)
where G r (t , t 1 ) and G < (t , t 1 ) are the Green’s function of the QD. We have defined G r (t , t 1 ) = −i θ(t − t 1 ){d(t ), d† (t 1 )} H and all the other <,a Green functions with the standard forms [41]. g L are the lesser (advanced) Green functions of the left leads without coupling to the QD. The retarded Green function G r (t , t 1 ) and lesser Green function G < (t , t 1 ) can be calculated from the following Dyson equation:
dτ dτ g r (t , τ )Σ r
G r (t , t 1 ) = g r (t , t 1 ) +
τ , τ G r τ , t1
(8)
and the Keldysh equation
dτ dτ G r (t , τ )Σ <
G < (t , t 1 ) =
τ , τ G a τ , t1
(9)
where g r (t , t 1 ) is the free retarded Green function of isolated dot which depends only on the time difference t − t 1 . Σ r /a,< (τ , τ ) = r /a,< ∗ (τ , τ )t α (τ ) is the self-energy. We now make Fourier transformation over the two times t and t which switches from α = L , R t α (τ ) g α the time-domain into energy representation through a double-time Fourier-transform defined as [44,45]
F (ω, ω1 ) = and
dt dt 1 F (t , t 1 )e i ωt e −i ω1 t1
dω dω1
F (t , t 1 ) =
2π 2π
(10)
F (ω, ω1 )e −i ωt e i ω1 t1 .
(11)
Then from Eq. (8), one can derive the Dyson equation
dω dω G r
ω, ω e iωt e−iω t1 =
dω g r (ω)e i ω(t −t1 ) +
1
(2π
)2
Next we take the time average over the above equation with respect to
F (t , t 1 ) = lim
T →∞
T
1
F
2T
dω dω1 dω g r (ω)Σ r (ω, ω1 )G r
ω1 , ω e iωt e−iω t1 .
(12)
τ = (t + t 1 )/2 and let t = t − t 1 .
τ + t /2 , τ − t /2 d τ .
(13)
−T
With the help of the time average Eq. (13), we finally obtain following expression for Dyson equation
dω G r (ω)e i ω(t −t1 ) =
dω g r (ω)e i ω(t −t1 ) +
1
(2π )2
dω dω1 g r (ω)Σ r (ω, ω1 )G r (ω1 , ω)e i ω(t −t1 ) .
(14)
Since we are interested in the time-averaged tunneling current, we only need to calculate the Green functions related to the Fourier transformed versions in diagonal variable forms by using the resonant approximation described in Refs. [6,46] to find
G r (ω, ω1 ) = 2π G r (ω)δ(ω − ω1 ).
(15)
This signifies that the electrons tunneling through the QD possess resonant structure. The similar approximation method has been also adopted by the other authors [47,48]. Now we can obtain the Dyson equation for the QD as a single time function: G r (ω) = g r (ω) + g r (ω)Σ r (ω)G r (ω). Then the retarded Green’s function is related to
G r (a) (ω) =
1
(16)
ω − ε − Σ r (a) (ω)
where the retarded self-energy Σ r (ω) can be obtained by Σ r (ω) = (Σ > (ω) − Σ < (ω))/2. The greater (lesser) self-energy Σ > (ω) (Σ < (ω)) can be obtained by the time-averaged double-time self energy as
Σ
>,<
(t 1 , t 2 ) =
∗
>,<
t α (t 1 ) g α
α=L/ R
(t 1 , t 2 )t α (t 2 ) =
α = L / R ,nm
Jn
Ω
Jm
|t α |2 g α>,< (t 1 − t 2 )e inΩ t1 e −imΩ t2 Ω
(17)
∞
where J m ( z) is the mth order Bessel function. In deriving this equation, we used identity exp(iz sin θ) = m=−∞ J m ( z)e imθ , which is a signature of photon-assisted processes [49]. With above the same method, the time-averaged self-energy can be obtained Σ >,< (ω) =
2 2 >,< α = L / R ,n J n ( Ω )|t α | g α (ω + nΩ). Similarly, using above the same procedure to the lesser Green function, after Fourier transform and time average, one obtains the following expression for the lesser Green function from Eq. (9)
G < (ω) = G r (ω)Σ < (ω)G a (ω).
(18)
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Once the G r ,a is calculated, the lesser Green’s function is obtained from the Keldysh equation. After using Langreth theorem of analytic T /2 continuation, and taking Fourier transformation over the current equation (7) and the time average, F (t ) = lim T →∞ T1 − T /2 F (t ) dt, the time-averaged tunneling current can be expressed as
IL =
e 2π
2 2 Jm J n |t L |2 |t R |2
> > < dω G dr G ad g < Ln (ω) g Rm (ω) − g Ln (ω) g Rm (ω)
(19)
m,n
with g Lm/ Rm (ω) = g L / R (ω + mΩ). This formula describes the time-averaged current through the LL-QD-LL system in the presence of ac >,< >,< fields. g L and g R are the greater (lesser) Green functions of the left and right leads without the couplings to the dot, and can be directly quoted from [42]
T
<,>
g L / R (ω) = ±i
exp ∓(ω − μ L / R )/2T
|t L / R |2
where T is the temperature. μ L / R = μ F + eV L / R , system. γ L / R (ω) is defined as
γL / R (ω) =
ΓL/ R 2π T
πT Λ
1 / g −1
γL / R (ω − μL / R )
(20)
μ F is the chemical potential of LL leads, V L − V R = V is the bias voltage imposed on the
|(1/2g + i ω/2π T )|2 (1/ g )
(21)
here (x) is the Gamma function, g is the interaction parameter characterizing the left/right LL leads, which should not be confused with the Green functions with the same symbols, Γ L / R = 2π ct 2L / R /α Λ describe the effective level broadening of the dot, c is a dimensionless constant of order 1, and Λ is the high energy cutoff or a band width [42,50]. If we introduce the electron occupation number for interacting |[1/2g +i (ω−μ L / R )/2π T ]|2 which is analogous to the Fermi distribution f (ω) for noninteracting (1/ g ) 2 |[ 1 / 2g + i ( ω − μ )/ 2 π T ]| L/ R 1 (ω−μ L / R )/2T π T 1/ g −1 e (Λ) is analogous to 1 − f (ω) for FL leads, then we finally obtains the 2π (1/ g )
electrons F L/ R = PAT current
I =e
2 2 Jm Jn
m,n
dω 2π
> < < > Γ L Γ R G r G a F Rm (ω) F Ln (ω) − F Rm (ω) F Ln (ω)
(22)
with F Lm/ Rm (ω) = F L / R (ω + mΩ). The spectral density of shot noise is defined by the Fourier transformation of the current correlation
S αα t , t = ˆI α (t ) ˆI α t + ˆI α t ˆI α (t )
(23)
where ˆI α (t ) = ˆI α (t ) − ˆI α (t ). By substituting the current operator Eq. (6) into Eq. (23), the shot noise can be expressed in terms of †
the four-operator Green functions which is related to the electrons operators in the leads ψα (ψα ) and in the QD d† (d). The Hamiltonian H Eq. (1) is noninteracting because the bosonized form H L / R have been mapped to a noninteracting boson-type model with renormalized parameters. We can factorize the four-operator Green functions into a product of the two-operator Green functions with an annihilation electron operator and a creation electron operator by using the Wick theorem [51]. There appear the off-diagonal functions < † G d<α (t , t ) ≡ i ψα (t )d(t ), G < αd (t , t ) ≡ i d (t )ψα (t ) and G α ,β (t , t ) ≡ i ψβ (t )ψα (t ) in the shot noise expression. In order to obtain these Green functions, as the starting point, we construct the contour-ordered Green function and derive functions equation of motion (EOM). And then, via analytical continuation rules, we obtain these off-diagonal Green functions which are expressed in terms of both the Green functions of the QD and the bare Green functions of the leads. With the help of the Keldysh Green functions [52], after a straightforward calculation and derivation via EOM, we obtain the double-time shot noise S (t , t ) under the dc bias and time-dependent voltage field. The more details of derivation process on shot noise can be found in Ref. [24]. Usually, we are interested in the spectral density of current noise in the pseudoequilibrium state, which is defined as the Fourier transformed current correlation of Eq. (23) by the expression: †
†
†
†
∞ ∞ S (ω1 , ω2 ) =
dt 1 dt 2 e i ω1 t1 e i ω2 t2 S (t 1 , t 2 ).
(24)
−∞ −∞
In this Letter, we consider the spectral density of shot noise in the zero-frequency limit due to the fact that the presence of the timedependent field perturbation mimics a finite frequency shot noise measurement. The shot noise spectral density contains the effect of photon absorption and emission induced by the microwave fields. In experiment, in order to obtain the observed shot noise spectral density, the absorption and emission of photon numbers must be equal, which comes from the requirement of Fourier transformation of the current correlation function in the pseudoequilibrium state. In fact, it is the selection rule for the absorption and emission of photons associated with the energy conservation, i.e., the total absorbed photon energy equals the total emitted photon energy in the system. In this Letter, we are interested in the time-averaged zero-frequency shot noise for the balanced absorption case where the absorption and emission of photon numbers are equal in the same lead, and there exist no correlations between the sidebands of photon between different terminal currents. Since the shot noise spectrum possesses the property S LL (ω) = S R R (ω), we present only the noise on the left terminal, and denote it as S = S LL (0). Therefore, following standard mathematical procedure, making the Fourier transformation over the two times t and t and the time average over τ = (t + t )/2, we finally obtain the time-averaged spectral density of shot noise in the zero-frequency limit for the balanced absorption case:
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751
Fig. 1. Differential conductance (a,b,c) and differential shot noise (d,e,f) as a function of bias voltage for several different microwave intensities. Here we put Γ = 0.02Ω , Ω = 1.0, ε = 2.0, here = 0.7 (a,d), = 1.0 (b,e) and = 1.7 (c,f). Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from bottom to top. For 1/2 < g 1, the main resonance peak is accompanied by a set of satellite resonant peaks and when > 1 the first satellite peak is higher than the main resonant peak. In the limit of the strong interaction, resonance behavior disappears and shows bias-voltage-dependent power law scalings.
S = e2
2 2 Jm Jn
m,n
− e2
dω 2π
> < < > Γ L Γ R G r G a F Rm (ω) F Ln (ω) + F Rm (ω) F Ln (ω)
2 2 2 2 Jm Jn J p Jq
m,n, p ,q
dω 2π
ΓL Γ R G r G a
2
> < > < > < > < F Rq (ω) F Lm (ω) − F Lm (ω) F Rq (ω) F Rp (ω) F Ln (ω) − F Ln (ω) F Rp (ω) .
(25)
Eq. (25) describes the photon-assisted shot noise through the QD coupled to the LL leads under the alternating field, which contains the thermal noise and shot noise. Eq. (25) is a basic and general formula for the time-averaged zero-frequency shot noise through a QD. It is the central result of this work, and is different from Eq. (30) in Ref. [38] which was obtained only at zero temperature. From Eq. (25) we can find that the effect of photon absorption and emission induced by the microwave fields is included in the shot noise spectral density. The expression for the noise becomes more complicated due to the opening of multiple inelastic tunneling channels. In the absence of the ac gate voltage (i.e., = 0), Eq. (25) can be expressed in a standard form as Eq. (21) without the ac field [37]. More details of the shot noise formula can be found in the literature [37]. When g = 1, Eq. (25) is reduced into the formula (23) in Ref. [24]. In order to observe qualitative effect of the intralead electron interaction on the transport properties, we consider the weakly coupling between QD and LL. After taking a contour integral, Eq. (22) reduces to the
I =e
> (ε ) F < (ε ) − F < (ε ) F > (ε )) ( F Rm Ln Rm Ln 2 2 Jm J n ΓL Γ R 2 > (ε ) + F < (ε )] J [ F m α , m α , m α ,m m,n
(26)
and Eq. (25) becomes
S = e2
> < < > 2 2 ( F Rm (ε ) F Ln (ε ) + F Rm (ε ) F Ln (ε )) Jm Jn 2 > < α ,m J m [ F α ,m (ε ) + F α ,m (ε )] m,n
− e2
m,n, p ,q
2 2 2 2 Jm Jn J p Jq
> (ε ) F < (ε ) − F > (ε ) F < (ε ))( F > (ε ) F < (ε ) − F > (ε ) F < (ε )) ( F Rq Lm Lm Rq Rp Ln Ln Rp .
2 > (ε ) + F < (ε )]3 J [ F α ,m α ,m m α ,m
(27)
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Fig. 2. Differential shot noise as a function of bias voltage. The parameters are Γ L = 40.0, Γ R = 20.0, Ω = 1.0, = 1.7,
ε = 2.0, g = 1.0.
Fig. 3. Differential conductance (a,b,c) and differential shot noise (d,e,f) as a function of the level energy ε . V = 5.0, all the other parameters are the same as Fig. 1. Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from bottom to top. For g 1, the differential shot noise exhibits a double peak, however, only a single peak shows up in differential conductance.
The validity of the approximation has been given by the authors [50]. This evaluation validates only when T = 0. Here we no longer restate the approximation. 3. Numerical results and discussion In this section, we present the numerical results of the photon-assisted zero frequency differential shot noise and Fano factor. For simplicity, we only consider the symmetric systems Γ L = Γ R and symmetric bias V L = − V R = V /2. The Fermi energy E F = 0 and temperature T = 0.1. We take the microwave field frequency Ω as the energy unit throughout the Letter. Moreover, we set h¯ = e = 1.
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Fig. 4. Fano factor as a function of the level energy ε with V = 5.0 (a,b,c) and the bias voltage with ε = 5.0 (d,e,f). Different curves correspond to g = 0.2, 0.4, 0.6, 0.8 and 1.0 from top to bottom. The width of dip becomes broader with the increase of /Ω in (a,b,c). Strong intralead electron interaction smears out the steps in Fano factor.
First in Fig. 1, we plot the differential shot noise as a function of the bias voltage for different strengths on the basis of Eqs. (26), (27). We can find the following features in this figure: firstly, the main resonant peak is suppressed while the satellite peak connected with the photon-absorption processes grows up gradually with the increase of ac intensity for 1/2 < g 1. And the main resonance peak is now accompanied by a set of satellites spaced by δ = h¯ Ω . Secondly, when the amplitude becomes larger than one, = 1.7, the first satellite peak related to the photon-absorption processes is higher than the main resonant peak. It implies that increasing intensity of the ac field will enhance the photon-assisted processes and the inelastic channels associated with photon-assisted electron tunneling can dominate electron transport. This is the most interesting phenomenon in this Letter. For moderate interaction, the sideband peaks become indistinct, and finally all the resonance peaks disappear for LL leads with strong enough electron interactions. It is due to the fact that the tunneling density of states on the LL is modified by the Coulomb interactions in the LL. They scale as a power law in higher bias voltage, dI /dV ∝ n J n2 ( Ω )| V + nΩ|1/ g −2 and dS /dV ∝ n J n2 ( Ω )| V + nΩ|1/ g −2 , respectively, in accordance with the LL prediction in the absence of an ac gate voltage. These features are related directly to the power law behavior of the density of state. In order to further check our model and theory, we plot dS /dV vs the bias voltage in the case when Γ Ω in the Fig. 2 based on Eq. (25). At g = 1, the differential shot noise exhibits steps whose spacing is an integer value of Ω and the step heights are proportional to J n2 (n = ±1, ±2, . . .). This is in complete agreement with the theoretical results obtained by Lesovik and Levitov [53] and experiment result obtained by Schoelkopf et al. [54] for a FL, respectively. Discussion of the strongly coupling between QD and Luttinger leads is beyond the scope of this Letter, and the effects of strongly coupling on transport under ac field will be investigated in our future publication. In Fig. 3, to further understand the PAT, the differential shot noise and differential conductance are plotted as a function of level energy ε , or, equivalently, as a function of gate voltage V g . The bias voltage is V = 5.0. For g = 1, the main resonance peaks correspond to the level positions at ε = ± V /2. Comparing Fig. 1 and Fig. 2, one can see that the some similarities are preserved for different the intensity of ac field. For example, when = 1.7, the first satellite peak connected is higher than the main resonant peak. On the other hand, with the intralead electron interaction increasing, all the peaks become blurring and then merge into a single central peak for g = 0.2 in differential conductance. These features are due to the power law behavior of the density of states and the suppression of the electron tunneling near the Fermi energies of LL leads. Correspondingly, the differential shot noise power exhibits two peaks located symmetrically around the conductance peak position. This central peak in differential conductance and differential shot noise entirely narrows for g = 0.2. They come from the contribution of the direct tunneling process and many PAT processes. In Fig. 4, we study the variation of the Fano factor F as the function of both the bias voltage and the level energy of the QD at a given bias V = 5.0 with different strengths and frequencies of the external fields, respectively. From the Fig. 4 we find that, the steps appear for noninteraction in the leads. For a weak interaction, the sideband steps become blurring and finally disappear for strong interaction
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and exhibit a simple behavior. In particular, we observe that, in Fig. 4a, the width of the dip becomes broader with the increase of /Ω , originated from the contributions of the multi-photon-assisted tunneling processes and the summation of the Bessel function. 4. Summary In conclusion, we have investigated the photon-assisted transport properties of QD coupled to LL by applying nonequilibrium Green function method. A general expression for the photon-assisted shot noise and current is obtained, which includes the arbitrary ac field and the Coulomb effects in the leads. Numerical results have shown that the resonant-like behavior of the photon-assisted shot noise for weak electron interaction in leads cannot survive in the strong Coulomb interaction leads. For strong interaction, photon-assisted shot noise has a power-law dependence with the voltage. This conclusion is qualitatively in agreement with a recent study [36] where the step structure of the ac noise of LL in carbon nanotube is found to be smoothed out by the Coulomb interaction. These are due to the fact that power law behavior of the density of states of the electrons in a strongly correlated one-dimensional system. We reproduce the result of the step-like behavior of the photon-assisted noise in noninteracting leads for the case of Γ Ω . In different parameter region, the effect of the electron interaction and ac field on the transport properties is different. These results may shed light on the understanding of the behavior of LL. We expect that the work can be tested in experiment. In addition, photon-assisted transport in this work is a first step for studying the behavior of mesoscopic systems in nonstationary situations, which is relevant to the understanding of the finite-frequency shot noise simulated by a zero-frequency noise measurement operated on a nanotube in presence of PAT processes. Acknowledgements This work was supported by the Beijing Novel Program No. 2005B11 and Fund of Beijing Municipal Education Commission. References [1] A. Hewson, D. Newns, J. Phys. C 13 (1980) 4477. [2] Y.M. Blanter, M. Buttiker, Phys. Rep. 336 (2000) 1. [3] K.E. Nagaev, Phys. Rev. B 57 (1998) 4628; Y. Wei, B. Wang, J. Wang, H. Guo, Phys. Rev. B 60 (1999) 16900. [4] B. Dong, X.L. Lei, J. Phys.: Condens. Matter 14 (2002) 4963. [5] N.Y. Kim, P. Recher, W.D. Oliver, Y. Yamamoto, J. Kong, H.-J. Dai, Phys. Rev. Lett. 99 (2007) 036802. [6] H.-K. Zhao, J. Wang, Phys. Rev. B 74 (2006) 245401. [7] Y. Meir, A. Golub, Phys. Rev. Lett. 88 (2002) 116802. [8] A. Golub, Phys. Rev. B 97 (2006) 016602; E. Sela, Y. Oreg, F.v. Oppen, J. Koch, Phys. Rev. Lett. 97 (2006) 086601. [9] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [10] L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79 (1997) 2526. [11] R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature (London) 389 (1997) 162. [12] C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 72 (1994) 724. [13] P. Roche, M. Kociak, S. Gueron, A. Kasumov, B. Reulet, H. Bouchiat, Eur. Phys. J. B 28 (2002) 217. [14] K.-V. Pham, F. Pichon, K.-I. Imura, P. Lederer, Phys. Rev. B 68 (2003) 205110. [15] B. Trauzettel, I. Safi, F. Dolcini, H. Grabert, Phys. Rev. Lett. 92 (2004) 226405. [16] E. Onac, F. Balestro, B. Trauzettel, C.F.J. Lodewijk, L.P. Kouwenhoven, Phys. Rev. Lett. 96 (2006) 026803. [17] N.Y. Kim, P. Recher, W.D. Oliver, Y. Yamamoto, J. Kong, H. Dai, Phys. Rev. Lett. 99 (2007) 036802. [18] P. Recher, N.Y. Kim, Y. Yamamoto, Phys. Rev. B 74 (2006) 235438. [19] L.G. Herrmann, T. Delattre, P. Morfin, J.-M. Berroir, B. Placais, D.C. Glattli, T. Kontos, Phys. Rev. Lett. 99 (2007) 156804. [20] A. Braggio, R. Fazio, M. Sassetti, Phys. Rev. B 67 (2003) 233308; E. Onac, F. Balestro, B. Trauzettel, C.F.J. Lodewijk, L.P. Kouwenhoven, Phys. Rev. Lett. 96 (2006) 026803. [21] J.U. Kim, J.M. Kinaret, M.-S. Choi, J. Phys.: Condens. Matter 17 (2005) 3815. [22] S. Kawaguchi, J. Phys.: Condens. Matter 21 (2009) 395303. [23] A. Komnik, A.O. Gogolin, Phys. Rev. B 68 (2003) 235323. [24] K.-H. Yang, Y.-P. Wu, Y.-L. Zhao, Phys. Lett. A 374 (2010) 917. [25] S. Takei, Y.B. Kim, A. Mitra, Phys. Rev. B 72 (2005) 075337. [26] W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, L.P. Kouwenhoven, Rev. Mod. Phys. 75 (2003) 1. [27] L.P. Kouwenhoven, L.L. Sohn, G. Schön, Mesoscopic Electron Transport, first ed., Kluwer Academic, Dordrecht, 1997. [28] Q.-F. Sun, J. Wang, T.-H. Lin, Phys. Rev. B 61 (2000) 13032. [29] G.B. Lesovik, L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538; G.B. Lesovik, R. Loosen, JETP Lett. 65 (1997) 295. [30] G.B. Lesovik, T. Martin, J. Torrès, Phys. Rev. B 60 (1999) 11935; J. Torrès, T. Martin, G.B. Lesovik, Phys. Rev. B 63 (2001) 134517. [31] R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 78 (1997) 3370. [32] S.R. Yang, Solid State Commun. 81 (1992) 375. [33] R.J. Schoelkopf, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 80 (1998) 2437. [34] A.A. Kozhevnikov, R.J. Schoelkopf, D.E. Prober, Phys. Rev. Lett. 84 (2000) 3398. [35] A. Crépieux, P. Devillard, T. Martin, Phys. Rev. B 69 (2004) 205302. [36] M. Guigou, A. Popoff, T. Martin, A. Crpieux, Phys. Rev. B 76 (2007) 045104. [37] K.-H. Yang, C.-Z. Liu, Eur. Phys. J. B 72 (2009) 409. [38] F. Cheng, G.-H. Zhou, Phys. Rev. B 73 (2006) 125335. [39] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature (London) 391 (1998) 156; S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281 (1998) 540; J. Schmid, J. Weis, K. Eberl, K.v. Klitzing, Physica (Amsterdam) B 256–258 (1998) 182; F. Simmel, R.H. Blick, J.P. Kotthaus, W. Wegscheider, M. Bichler, Phys. Rev. Lett. 83 (1999) 804. [40] J.M. Kinaret, Y. Meir, N.S. Wingreen, P. Lee, X.-G. Wen, Phys. Rev. B 45 (1992) 9489; J.M. Kinaret, Y. Meir, N.S. Wingreen, P.A. Lee, X.-G. Wen, Phys. Rev. B 46 (1992) 4681. [41] H. Haug, A.P. Janho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, 1996. [42] A. Furusaki, Phys. Rev. B 57 (1998) 7141.
K.-H. Yang et al. / Physics Letters A 375 (2011) 747–755
[43] [44] [45] [46]
[47] [48] [49] [50] [51] [52] [53] [54]
C. Bruder, H. Schoeller, Phys. Rev. Lett. 72 (1994) 1076. B. Wang, J. Wang, H. Guo, Phys. Rev. Lett. 82 (1999) 398. Y.X. Xing, B. Wang, Y.D. Wei, B.G. Wang, J. Wang, Phys. Rev. B 70 (2004) 245324. H.K. Zhao, Z. Phys. B 102 (1997) 415; H.K. Zhao, Phys. Lett. A 226 (1997) 105; H.K. Zhao, Phys. Rev. B 63 (2001) 205327; H.K. Zhao, Eur. Phys. J. B 64 (2008) 237; H.K. Zhao, J. Wang, Phys. Rev. B 64 (2001) 094505. H.Ch. Yong, K.H. Yang, G.S. Tian, Commun. Theor. Phys. 48 (2007) 1107. Q.F. Sun, J. Wang, T.-H. Lin, Phys. Rev. B 59 (1999) 13126. M. Grifoni, P. Hanggi, Phys. Rep. 304 (1998) 229; G. Platero, R. Aguado, Phys. Rep. 395 (2004) 1. Y.F. Yang, T.H. Lin, Phys. Rev. B 64 (2001) 233314. G.D. Mahan, Many-Particle Physics, second ed., Plenum Press, New York, 1990. L.V. Keldysh, JETP Lett. 20 (1965) 1018. G.B. Lesovik, L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538. R.J. Schoelkopf, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 80 (1998) 2437.
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