AC shot noise through a quantum dot in the Kondo regime

Physics Letters A 375 (2011) 3037–3043

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Physics Letters A www.elsevier.com/locate/pla

AC shot noise through a quantum dot in the Kondo regime Kai-Hua Yang ∗ , Yan-Ju Wu, Yang Chen 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 24 April 2011 Received in revised form 18 June 2011 Accepted 27 June 2011 Available online 30 June 2011 Communicated by R. Wu

a b s t r a c t The photon-assisted shot noise through a quantum dot in the Kondo regime is investigated by applying time-dependent canonical transformation and non-crossing approximation technique. A basic formula for the photon-assisted shot noise is obtained. The rich dependence of the shot noise on the external ac field and temperature is displayed. At low temperature and low frequencies, the differential shot noise exhibits staircase behavior. When the temperature increases, the steps are rounded. At elevated frequencies, the photon-assisted tunneling becomes more obvious. We have also found that the Fano factor is enhanced as the ac frequency is enhanced. © 2011 Elsevier B.V. All rights reserved.

1. Introduction As techniques in manufacturing devices of nanometer scale develop, understanding the strong electronic correlations far from thermal equilibrium is one of the most interesting problems in contemporary mesoscopic physics. A primary example is the outof-equilibrium Kondo effect [1], which has been recently observed in the transport of ultrasmall quantum dots [2]. The Kondo effect, which is caused by the screening of a local spin by coupling to conduction electrons, has become a paradigm for strongly correlated systems as a quite simple model that exhibits many-body correlations [3]. It has been confirmed that many phenomena, which characterize strongly correlated metals and insulators, are present in the quantum dot (QD) systems. In particular, the tunable realization in mesoscopic quantum dots has triggered a renewed interest in Kondo physics probed by transport measurements. Experiments [4–6] have exploited the tunable physical characteristics of the quantum dot to yield important information on Kondo systems. A promising experimental tool to study Kondo physics is current noise measurement. Current noise is caused by the time-dependent current fluctuation. It is interesting that noise in mesoscopic structures can reveal the kinetics of electrons [7]. The measurements of shot noise can yield additional information of transport properties which are unaccessible by conductance measurements alone [8,9]. For example, it can provide a direct estimate of the Kondo temperature [10]. Therefore, the study of current noise through a QD has recently become an emerging topic of theoretical and experimental interest in mesoscopic physics [7,10–15]. Especially, many theoretical methods have been introduced to investigate the shot noise for strongly correlated system. Meir and

*

Corresponding author. Tel.: +86 010 67392201. E-mail address: [email protected] (K.-H. Yang).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.059

Golub [10] studied the shot noise by several complementary approaches from the low-voltage Kondo regime to the high-voltage Coulomb-blockade regime. Dong and Lei [13] reported the shot noise of Kondo-QD systems. Up to now, these theoretical studies on shot noise in mesoscopic systems have concentrated on steadystate transport phenomena in Kondo regime [10–14]. On the other hand, more rich physics could be exploited if the QD device is subject to a microwave (MW) irradiation field. The perturbations of ac fields can give rise to some very interesting phenomena, such as photon–electron pumping effect, the turnstile effect, the sideband effect, and photon-assisted tunneling (PAT) [16–18]. It has been reported that the MW spectroscopy is a possible tool to probe the energy spectrum of small quantum systems [19]. So the PAT could provide a new way of understanding the electron–electron influence on the transport properties of the dot. Indeed, the influence of the ac field on the current–voltage characteristics in the strongly correlated interaction model was discussed by some authors. The essential effect of PAT on transport properties is that the electrons tunneling through the system can exchange energy with MW fields, opening new inelastic tunneling channels and introducing many effects. The measurement of the shot noise at ac field frequency in the order of GHz regime demonstrated that the shot noise is suppressed when the sourcedrain bias and thermal energy are comparable with the frequency [20]. Up to now, most of the theoretical studies on shot noise in mesoscopic systems irradiated with ac fields were concentrated on non-interacting electrons. To our knowledge, there have been some attempts to study the shot noise under the ac field in the Kondo regime [21–24]. For example, Ding and Ng [21] have investigated the time-dependent shot noise in the strongly correlated Kondo regime based on the equation of motion method (EOM). They only found the first-order behavior of shot noise. Zhao and Wang [22] have studied the shot noise in the Kondo regime under the MW irradiated field employing the EOM. They revealed the

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interaction can either enhance or reduce the shot noise affected by ac field. The EOM method is known not adequate to go deep into the Kondo regime. Recently, the noise of a Kondo dot with the ac gate modulation was also studied by combining the Floquet theorem and the non-crossing approximation (NCA) [24] by Wu and Cao. The numerical results show that the photon-assisted shot noise of a Kondo dot reveals no singularity at low frequencies. This implies that the Kondo resonance is actually not influenced by an ac field when the energy scale of the ac parameters is the order of the Kondo temperature. Naturally, one would like to ask whether there are other factors (for instance, weak versus strong coupling between the QD and the leads) that affect the photon-assisted noise in the Kondo regime. Compared with the non-interacting case, a clear picture of the Kondo resonance in the presence of weak ac field is less obvious. And a generally explicit formula of the photon-assisted shot noise in the ac Kondo regime has been still lacking. In this Letter, we will apply a time-dependent canonical transformation [25] to remove the time-dependence of the gate voltage of the QD that it causes an ac current through the dot. Then, we introduce the slave-boson representation for the on-dot fermion operators and the so-called NCA which will be used to rewrite the on-dot Coulomb interaction term in the Hamiltonian. A clear formula for the time-averaged shot noise of an interacting dot in ac field is developed. The numerical results indicate that explicit steps of the photon-assisted shot noise appear in the strongly correlated Kondo regime at the low temperature and low frequency. As the temperature is increased, step-like structures in the differential shot noise are gradually washed away. At higher frequencies, however, side peaks at multiples of ac frequency Ω can be clearly resolved. We have found that the Fano factor, or the shot noise to current ratio, is enhanced as the ac frequency is increased, implying that the rate of reduction of the current is greater than that of the shot noise. This result also implies the possibility of controlling the Fano factor of the transistor by tuning the ac frequency. 2. Model and method The QD system consists of a dot of nanometer scale in the presence of time-dependent gate voltage and two conducting leads. They are connected via electron tunneling. The Hamiltonian of this system can be described by the Anderson single impurity model

H=



σ

α ,k,σ

+



†

(kα − V α )ckσ α ckσ α +



† V kα ckσ α dσ

d (t )d†σ dσ + U d†↑d↑d†↓ d↓

 + V kα dσ ckσ α , ∗

†

α = L ( R ) is an index for the left and right conductive leads. †

Correspondingly, ckσ α and ckσ α denote the creation and annihilation operators which creates or destroys an electron of energy k and spin σ in lead α . Moreover, V α is the chemical potential of electron in conductive lead α and the quantity V = ( V L − V R )/e (−e is the electron charge) is defined to be the voltage bias im†

 U (t ) = exp

i h¯

t









 †

dt  cos Ω t d d

(2)

−∞

to Hamiltonian (1). Under this transformation, we obtain

 H (t ) = U (t ) H (t )U −1 (t ) − ih¯ U (t )∂t U −1 (t )   † (kα − V α )ckσ α ckσ α + 0 d†σ dσ + U d†↑d↑d†↓ d↓ = α k,σ

+

 

σ

†  V kα (t )ckσ α dσ

† + V k∗α (t )dσ ckσ α .

(3)

α k,σ

In Hamiltonian (3), the time-dependence of the gate voltage removed. Instead, the electron tunnel couplings

  V kα (t ) = V kα exp −

i h¯

t

d is



  dt  cos Ω t  

(4)

−∞

is now time-dependent. The physics of the system is unchanged under the canonical transformation. To eliminate the Coulomb interaction term on the dot, we introduce the slave-boson representation for the fermion operators dσ †

and dσ . In the limit U → ∞, the double occupancy of electron on the QD is completely removed. In other words, the electronic energy level 0 is either empty or occupied by just one electron. This restriction eliminates the Coulomb interaction term. But, it makes also difficult to use the standard diagrammatic technique in calculating the Green functions since the well-known anticommutation relations do not hold for the on-dot fermion operators. Technically, this problem can be solved by applying the slave-boson representation. Following Coleman [27], we introduce the bosonic operators ˆ which creates (annihilates) an unoccupied on-dot energy bˆ † (b),

level ¯0 , and fermionic operators ˆf σ ( ˆf σ ), which creates (annihilates) a single fermion of spin σ in the same energy level. In terms † of these operators, the original electron operators dˆ σ and dˆ σ can be expressed as †

(1)

α ,k,σ

where

its non-perturbative part H 0 should not contain any interacting term. Otherwise, the Wick theorem cannot be applied [26]. Apparently, Hamiltonian (Eq. (1)) of the QD system does not satisfy this requirement. We notice that the on-dot Hamiltonian H D , which should be a part of H , has the electron–electron interaction and transfer their effects into the electron tunneling Hamiltonian. Therefore, before calculating the relevant Green functions, we have to introduce some canonical transformations which eliminate these on-dot interactions. In the first step, we apply a time-dependent canonical transformation [25] in order to conveniently handle the problem,

posed on the system. Similarly, dσ , dσ and d (t ) represent electron operators and their degenerate energy on the dot imposing a timedependent gate voltage on the system. (For simplicity, one usually assumes that only one of the on-dot energy levels is active.) d (t ) = 0 +  cos Ω t,  and Ω are the amplitude and frequency of the ac gate voltage, respectively. It causes an ac through the dot. Parameters U > 0 and V kα stand for the on-site electron Coulomb interaction and the tunneling amplitudes between the central dot and conductive lead α . In order to apply the Feynman diagram method, the Hamiltonian of the system must be in a standard form. More precisely,

dσ = b † f σ ,

†

†

dσ = f σ b .

(5)

Obviously, the electronic Hilbert space on the dot is now enlarged. To recover the original configurations, we impose the following condition † † Qˆ = bˆ † bˆ + ˆf ↑ ˆf ↑ + ˆf ↓ ˆf ↓ = 1

(6)

on these operators. By substituting Eq. (5) into H T and H D , we finally obtain the following Hamiltonian

H=

  † (kα − V α )ckσ α ckσ α + 0 f σ† f σ α k,σ

+

  α k,σ

σ †  V kα (t )ckσ α



fσ b +  V k∗α (t )bf σ ckσ α . †

†

(7)

K.-H. Yang et al. / Physics Letters A 375 (2011) 3037–3043

It is worthwhile pointing out that the first two terms, which represent the Hamiltonians of conductive leads and the QD, are quadratic in both boson and fermion operators. Therefore, the Wick theorem is applicable. Consequently, we are able to apply the standard diagrammatic technique to compute the time-ordered Green function under the constraint condition (6). To go further, we apply the so-called NCA. It is based on the following observation: If the spin of electron on the dot is assumed to have N components and N is large (in the real materials, N = 2), then the time-ordered Green function can be expanded as an infinite series of 1/ N. Furthermore, to the order of O (1/ N ), the dominant contribution is from the sum of diagrams which have no crossing bosonic and fermionic lines. In other words, the diagrams with vertex corrections can be ignored to this order. At the lowest order the boson self-energy involves the fermion propagator while the fermion self-energy involves the boson propagator in perturbation theory. By using the two relations self-consistently, one obtains a set of coupled integral equations, which can be solved numerically. Solving these self-consistent equations corresponds to sum a subset of diagrams to all orders in the hopping matrix element. More detailed explanation on this point can be found on page 878 of Ref. [28]. Within the NCA, the on-dot electronic retarded Green † function G dr σ (t ) = −i θ(t )dσ (t ), dσ (0) is expressed in terms of the full propagators for slave-boson and fermion. The Fourier transform G dr σ (ω) can be written within NCA as [29]

G dr σ (ω) =

∞

1 4π

2Z



dω  D >







ω G
−∞

(8)

with the normalization factor Z which is obtained using the constraint condition

i 2π



∞

dω D < (ω) −



G< f σ (ω) .

(9)

σ

−∞

To determine the lesser and greater Green functions in the above equations, firstly we can apply the Keldysh equations [30] <

<

r

a

>

<

Σ rf σ (ω) = i

<

Σ> f σ (ω) = i



2 Jm

m

×



 Ω

<

Π > (ω) = −i



2 Jm

m

×



dω  2π

<   <  | V kα |2 gk>α ω − ω + mΩ D > ω ,

α k,σ

 Ω



3. Transport of a quantum dot in the Kondo regime According to the continuity equation and Heisenberg equation, we can obtain the operator of current flowing into the QD from the α lead. The current operators of the lead α to the dot is expressed as

I α (t ) = ie





The noise power spectra between the leads α and α  can be expressed as the Fourier transform of the current–current correlation function,



    I α (t ),  I α  t  ,

(15)

where  I α (t ) = I α (t ) −  I α (t ) . The symbol · · · in above formula denotes the quantum expectation over the electron state and the ensemble average over the system. In order to evaluate the shot noise, we substitute the current operator equation (14) into Eq. (15). Following the standard procedure described in Ref. [30] and expressing these quantum statistical (non-equilibrium) averages in terms of the non-equilibrium Green function. 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. (15) by the expression

∞ ∞ dt 1 dt 2 e i ω1 t1 e i ω2 t2 S (t 1 , t 2 ).

(16)

By substituting the current operator (14) into Eq. (15), the shot noise can be expressed in terms of the four-operator Green functions which is related to the electrons operators in the leads †

ckα (ckα ), boson operators b† (b) and Fermi operators f † ( f ). The Hamiltonian H (Eq. (7)) is non-interacting. We can apply the Wick theorem [26] to factorize these operators. The statistical expectation of Eq. (14) yields the current through  the QD given by I α (t ) = 2e kσ  V k (t )G < † (t , t ), where the lesser kb f

obtain the current and shot noise formula, as a starting point, we construct the complex time Green’s function 



<







ω − ω + mΩ G f σ ω , >

(11)

where J m ( z) is the mth-order Bessel function. In deriving this ∞ equation, we used identity exp(iz sin t ) = m=−∞ J m ( z)e imt . Here, >

the lower-case Green functions gk<α (ω) are defined by



G kb† f







 

† τ , τ  = −i T c b† (τ ) f (τ )ˆckL τ

  †   † = −i T C S c b I (τ ) f I σ (τ )ˆc IkL σ τ  con ,

(12)

(17)

where T c is the contour time-ordered operator and τ (τ  ) is the complex times running along a complex contour [30]. The subscript of the boson and Fermi operator I is in the interaction picture and the S-matrix operator S c has been given by

gk>α (ω) = −2π i 1 − f α (ω) δ(ω − k ), gk<α (ω) = 2π i f α (ω)δ(ω − k ).

(14)

kσ

kb f

α k,σ





b† f σ −  V k∗α (t ) f σ bckσ α . †

kσ α

†

2π >

†  V kα (t )c

Green’s function G < † (t 1 , t 2 ) = i ckα (t 1 )b† (t 2 ) f (t 2 ) . In order to

dω 

| V kα |2 gk<α

(13)

ω − ω + i η

2π

−∞ −∞

(10)

Then, as explained in Ref. [10] by the NCA, we have also the following relations



Π > (ω ) , 2π ω − ω  + i η  >  dω Σ f σ (ω )

hold true. Together with the well-known relations D r (ω) = [ω − Π r (ω)]−1 and G rf σ (ω) = [ω − Σ rf σ (ω)]−1 , we have now a complete set of equations, which can be numerically solved.

<

r > a G> f σ (ω) = G f σ (ω)Σ f σ (ω) G f σ (ω).

dω 

S (ω1 , ω2 ) =

D (ω) = D (ω)Π (ω) D (ω), >



Π r (ω) = i

S αα  t , t  =

   

− D < ω G >f σ ω + ω

Z=

However, these equations are not closed. We still need to find connections between the greater self-energy functions and the retarded Green functions. In fact, it can be proved [10] that the following equations





3039



S c (−∞, −∞) = exp −i c



  dt  H T t 

(18)

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K.-H. Yang et al. / Physics Letters A 375 (2011) 3037–3043

around the Keldysh contour. ˆ T , where the We expand S with the tunneling Hamiltonian H lowest-order non-zero term in the expansion is n = 1. By applying Wick’s theorem [26] to the lead part and the Langreth analytic continuation rules [31], we obtain the lesser Green functions  G < † (t , t  ) which is related to G <† † (t , t ). kb f σ

b fσ fσ b



  dt   V k∗α t  G r †

< G kb † f (t 1 , t 2 ) =

†

†

Σ

(t 1 , t 2 ) =

(t , t  ), con-

<,r ,a  V kα (t 1 ) gkα (t 1 , t 2 ) V k∗α (t 2 ).

α ,k



<  V k (t )G kb † f (t , t ),

+ G <†



G <†

+G



<

t1 , t

†

b† f σ f σ b





gkaα



   t 1 , t  gk<α t  , t 2



t , t2 . 

(20)

We now make Fourier transformation over the two times t and t  which switches from the time-domain into energy representation through a double-time Fourier-transform defined as [32,33]:



dt dt 1 F (t , t 1 )e i ωt e −i ω1 t1 ,

F (ω, ω1 ) = and

F (t , t 1 ) =



1

dt dt 1 F (ω, ω1 )e −i ωt e i ω1 t1 .

(2π )2

(21)

(22)

In experiments, it is actually the averaged current that is free from the capacitive contributions and hence, is directly relevant [30]. Therefore, we take the time average over the above equation with respect to the time t





T /2

1

I = I (t ) ≡ lim

T →∞

T

I (t ) dt .

I (t ) = 2e



Jm

kσ m,n

× Gr †

†

(23)

− T /2

The time-averaged current is

 Ω



Jn

     1 Re dω1  V k2α  2 Ω (2π )

(ω1 , ω1 + mΩ − nΩ) gk<α (ω1 + mΩ)

(ω1 , ω1 + mΩ − nΩ) gkaα (ω1 + mΩ) .

b fσ fσ b

+ G <†

†

b fσ fσ b

(24)

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 Ref. [22] to find

Gr †

†

b fσ fσ b

(ω1 , ω1 + mΩ − nΩ)

= 2π G r †

†

b fσ fσ b

(ω) = G r †

(ω1 )δ(ω1 − ω1 + mΩ − nΩ).

†

b fσ fσ b

with

Σ < (ω) =



2 Jm

mα



2 Jm

 Ω

(ω)Σ < (ω)G a †

†

b fσ fσ b

(ω)

  < g (ω + mΩ) Ω kα

 ∞

 †

b fσ fσ b

kσ

†

b fσ fσ b

mσ

  dt   V kα (t ) V k∗α t  G r †

Re

(25)

With the help of the Keldysh equations [36]

the current can be written as

I (t ) = 2e

†

b fσ fσ b

I =e

kσ



(ω1 ) gk<α (ω1 + mΩ)

(ω1 ) gkaα (ω1 + mΩ) .

and NCA method where the slave-boson and pseudofermion operators can be separated, we can obtain all the relevant Green function. After a lengthy but straightforward derivation, one obtains the average current through the Kondo-QD

Substituting Eq. (19) into the current formula

I α (t ) = 2e

†

  dω1  2   V  Re kα Ω 2π

b fσ fσ b

tinuing to utilize the above procedure and comparing it to the Dyson equation, we obtain the time-ordered self-energy Σ τ (t 1 , t 2 ). All the self-energy functions Σ <,r ,a (t 1 , t 2 ) can be given by the analytic continuation of the time-ordered self-energy Σ τ (t 1 , t 2 ). The self-energy functions are obtained by



2 Jm

kσ m

(19) b fσ fσ b



× Gr †

   t 1 , t  gk<α t  , t 2

In order to solve the lesser Green function G <†

<,r ,a

I (t ) = 2e



b fσ fσ b     < G † t 1 , t  gkaα t  , t 2 . † b fσ fσ b

+

This signifies that the electrons tunneling through the QD possess resonant structure. The similar approximation method has been also adopted by the other authors [34,35]. The time-averaged tunneling current can be expressed finally as

−∞

dε 2π



f L (ε + mΩ) − f R (ε + mΩ) T (ε ). (26)

Here T (ε ) = ε ε) represents the transmission coefficient of an electron tunneling from one lead to another. Here, Γ L / R is the line-width function and G r (a) (ε ) is the Fourier transform of the QD retarded (advanced) Green function. With the standard procedure as above and after a tedious derivation, we can obtain the spectral density of current noise. In experiments, 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 to the total emitted photon energy in the system. In particular, Schoelkopf et al. [37] has detected that finite-frequency shot noise has a singularity at h¯ Ω = eV , which has been theoretically investigated for ballistic systems [38]. In practice the finite-frequency measurements turn out to be challenging. As an alternative to a finite-frequency measurement, the second derivative of the zero frequency shot noise with respect to the voltage d2 S /dV 2 exhibits peaks at h¯ Ω = eV for normal metals. Therefore, information on the finite-frequency noise spectrum can be extracted from zero frequency noise. Thus 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 obtain the time-averaged zero-frequency shot noise can be expressed in a standard form as follows

Γ L G dr (

S=

2e 2  h

×



2 Jm

mn



 Ω

)Γ R G ad (



J n2



 Ω



dε F th T + F sh (1 − T ) T



.

(27)

K.-H. Yang et al. / Physics Letters A 375 (2011) 3037–3043

3041

Fig. 1. (a) The time-averaged density of state ρσ ( ) at V L = V R = 0 and (b) the averaged differential conductance as a function of the bias voltage in the presence of the ac gate modulation for different ac parameters. The ac strength is fixed at /Ω = 1. The inset displays the evolution of the Kondo peak with increasing ac field. For the lowest Ω = T k , the height of the Kondo peak is decreased.

In Eq. (27), F th = f L (ε + nΩ)[1 − f L (ε + mΩ)] + f R (ε + nΩ)[1 − f R (ε + mΩ)] and F sh = [ f L (ε + nΩ) − f R (ε + nΩ)][ f L (ε + mΩ) − f R (ε + mΩ)]. And f L ( R ) (ε ) = 1/{1 + exp[(ε − V L ( R ) )/k B T ]} is the Fermi distribution function of electron on the left (right) lead. The time-averaged spectral density of shot noise (Eq. (27)) contains the effect of photon absorption and emission induced by the MW fields, which describes shot noise through the Kondo-QD under the alternating field. In particular, the shot noise formula (Eq. (27)) is formally equivalent to the version derived for the non-interacting electron systems [39] and is the central result of this work. It can contain all of the possible effects concerning the electron–photon interaction and the external ac fields. In Ref. [40], we give more details of the derivation of the shot noise formula on how to solve these equations. In the absence of the ac gate voltage (i.e.  = 0), Eq. (27) can be expressed in a standard form as follows [40]

S=

2e 2 h



dε





f L (1 − f L ) + f R (1 − f R ) T



 + ( f L − f R )2 (1 − T ) T .

(28)

4. Numerical calculation and results In the following, we show the calculated shot noise and the derivative of shot noise in the Kondo regime at several different frequencies Ω . In this Letter, we focus our attention on the symmetric systems Γ L = Γ R and symmetric bias V L = − V R = V /2. We take the Γ L + Γ R = Γ as the energy unit throughout the Letter. We also set k B = h¯ = e = 1. For definiteness, we let E F = 0 be the Fermi energy of electron in conductive leads. Moreover, we choose 0 = −2.0. All the calculations were done for T = 0.001, well below the Kondo temperature (T k ∼ 0.013). Under these conditions, we calculate first the on-site electronic density of states (DOS) and the differential conductance shown in Fig. 1. We find that, the Kondo peak is suppressed monotonically for low frequencies Ω ∼ T k . It corresponds to an adiabatic modulation of the level which suppresses the Kondo effect on the timeaveraged DOS. With increasing frequency, we observe that the central Kondo peak in the density of states is also accompanied

by several satellites spaced by h¯ Ω which shows the marked nonadiabatic effects. Furthermore, we notice that, away from the central resonant peak, the height of those satellite peaks decreases rapidly. When the ac frequency becomes strong, the central Kondo peak height is greatly reduced. Moreover, the central peak has also a larger width. Qualitatively, our results are in good agreement with the previous works [41,42]. These observations can be explained by the fact that the system has no energy eigenstates with an ac gate voltage being imposed. In other words, according to the uncertainty principle, all the transitions by operators d† and d in the electronic retarded Green function are from a quasiground state to a quasi-excited state, which have finite lifetime. In addition, compared with Ref. [24], they found that the Kondo peak remains robust up to an external ac frequency of Ω = 20T K where they consider a stronger coupling between the dot and the lead. In Ref. [43], Goker has studied the effect of the ac frequency and the temperature on the time-averaged conductance. When h¯ Ω = eV in Ref. [43], the satellite peak was not obvious. From Fig. 2(b) of the present Letter, we observe that the effect of the low ac field Ω = 1T k on the time-averaged conductance is also not obvious. In order to further study the effect of the low ac field on the transport properties, we will investigate the shot noise. It is known that, the shot noise can give a direct measurement of the second moment of the density of states. In order to check how sensitive the shot noise is to the effects of ac field, we plot the differential shot noise as a function of bias voltage in Fig. 2. We can clearly see that a series of steps appear with a step width of Ω whether high frequency or low frequency in Fig. 2. In particular for the lowest frequency Ω = T k , the steps in differential shot nose obviously appear less pronounced. The low frequency photo-assisted shot noise displays frequency-dependent features bearing all the features of PAT. When the temperature increases, the steps are rounded. At higher frequencies, the PAT becomes more obvious. We can directly relate the detection of a tunneling electron to the absorption or emission of the photon. Thus shot noise measurements also lead to a more straightforward determination of the PAT, especially for low frequency, which also demonstrates that the information available in noise measurements in the

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K.-H. Yang et al. / Physics Letters A 375 (2011) 3037–3043

Fig. 2. (a) The time-averaged differential shot noise as a function of the bias voltage for different values of the frequency with /Ω = 1, T = 0.001; (b) The time-averaged differential shot noise as a function of the bias voltage for different ac strength with fixed frequency Ω = 20T k , T = 0.001.

Fig. 3. The time-averaged differential shot noise as a function of the bias voltage for different temperatures. The ac frequency is Ω = T k with ac strength /Ω = 1.

Fig. 4. The Fano factor is plotted as a function of the bias voltage for different frequency with /Ω = 1, T = 0.001.

Kondo regime cannot be obtained in the usual transport experiments. In order to display the detail of non-adiabatic PAT, we also plot the differential shot noise as the function of voltage for different ac power in Fig. 2(b). We find that in positive bias regime, the positive photon-assisted differential shot noise magnitude decreases with the ac field amplitude increasing, while the negative photonassisted differential noise magnitude increases in the negative bias regime. In a word, a crossover from positive-bias maximum to negative-bias maximum with the ac field amplitude increasing. Again, we find that the differential shot noise demonstrates very different behavior from the differential conductance. Therefore, the measurement of the shot noise spectrum can improve our understanding electrons tunneling in the QD systems. The photonassisted shot noise can thus be considered as a probe for the PAT. Fig. 3 shows the effect of the temperature on the differential shot noise for Ω = T k . The differential shot noise is calculated for three different temperature. It can be clearly seen that the differential shot noise exhibits distinct step behavior at low temperature

T = 0.001. As temperature rises, the steps are rounded and these structures are gradually washed away when temperature T ∼ T k . Here, one can observe that the differential shot noise is a smooth function of bias V . This suggests that the smooth noise behavior shown in Fig. 3 is due to the finite-temperature effect. The observed temperature dependence of these structures agrees well with the findings reported in Ref. [44] The Fano factor after subtracting the thermal noise component 4k B T G ( V = 0) is displayed in Fig. 4 for different frequency Ω . First, it is seen that the Fano factor F is suppressed in the linear response regime V → 0 for the symmetric systems. The different curves show the effect of the ac fields on the Fano factor clearly. As the ac frequency increases, the Fano factor is significantly enhanced due to the suppression of the Kondo peak. It implies the possibility of controlling the Fano factor by tuning the ac frequency. The steps also appear when the ac gate field becomes stronger. This opens the way to high-frequency shot noise characterizations of quantum dots in a regime where the Fano factor can be increased.

K.-H. Yang et al. / Physics Letters A 375 (2011) 3037–3043

5. Summary In summary, we have studied the effect of an external alternating gate voltage and temperature on the shot noise through the QD system in the Kondo regime. The time-averaged shot noise formula has been derived by applying the time-dependent canonical transformation and NCA method. The numerical results indicate a rich dependence of the shot noise on the external parameters and temperature in the Kondo regime. At low temperature T = 0.001, the photon-assisted differential shot noise of a Kondo dot as a function of the bias voltage shows up steplike behavior. With the temperature increasing, these steplike structures gradually disappear for low frequency. In particular, the PAT becomes more obvious for higher frequencies since steps become sharper. It is shown that the shot noise yields more information on the PAT in the Kondo regime for low frequency and temperature, unaccessible in traditional transport measurements. We have found that the Fano factor is enhanced as the ac frequency is increased. These results implies the possibility of controlling the Fano factor of the transistor by tuning the ac frequency. We hope that our work will inspire experimental investigation of these phenomena of ac driving applied to Kondo systems.

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Acknowledgements

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This work is supported by Beijing Natural Science Foundation (1112003), the Beijing Novel Program (2005B11) and Foundation of Beijing Municipal Education Commission (006000546311503).

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