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Fano–Kondo shot noise in a quantum dot embedded interferometer irradiated with microwave fields
Physics Letters A 379 (2015) 389–395
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Physics Letters A www.elsevier.com/locate/pla
Fano–Kondo shot noise in a quantum dot embedded interferometer irradiated with microwave fields Hong-Kang Zhao ∗ , Wei-Ke Zou School of Physics, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 3 September 2014 Received in revised form 19 November 2014 Accepted 24 November 2014 Available online 27 November 2014 Communicated by R. Wu Keywords: Dynamic shot noise Fano–Kondo resonance Quantum dot embedded interferometer Current correlation
a b s t r a c t The shot noise of a quantum dot embedded Aharonov–Bohm (AB) interferometer under the perturbation of microwave fields is investigated by employing equation of motion method. The frequency-dependent shot noise formula is derived, and the photon-assisted Fano–Kondo resonance, the suppression of Kondo peak are presented with increasing the direct tunneling strength. The interference and correlation of electrons induce the asymmetric resonant peak–valley behavior of shot noise. The enhancement and suppression of shot noise are resulting from the competition of incoherent correlation and destructive interference effects, and super-Poissonian and sub-Poissonian noise can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. The periodic oscillation versus AB phase with period 2π appears to show plateaus and flat valleys. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In an ultra-small quantum dot (QD) at low temperature, a new state of many-body character known as Kondo effect is created at its Fermi level, and the zero-bias conductance resonance contains its intrinsic feature [1–5]. A spin singlet is formed by the localized spins on the QD, and conduction electrons tunnel through different terminals. The resonant peak is split associated with the spin–orbit interaction and spin polarization in ferromagnetic systems [6–8]. The Fano effect results from the interference between two electron waves, one passing through the QD and the other traveling along the direct channel characterized by its continuous spectrum. The interference between a resonant channel and the non-resonant channel leads to asymmetric Fano line shape in the differential conductance, where the formula of Fano line shape of transmission coefficient is presented [9]. The closed and open interferometers have been discussed theoretically, in which one path contains either simple QD or a decorated QD [10]. The QD embedded two-terminal Aharonov–Bohm (AB) interferometer was proposed to measure the phase of the transmission through the QD [11]. The Coulomb modified Fano resonance has been observed in a tunable Fano interferometer consisting of a QD [12]. The Fano resonances appear in the electron transport through the QDs formed by magnetic double barriers in quantum wires [13]. The nonlinear
*
Corresponding author. Tel.: +86 10 68914975. E-mail address: [email protected] (H.-K. Zhao).
http://dx.doi.org/10.1016/j.physleta.2014.11.050 0375-9601/© 2014 Elsevier B.V. All rights reserved.
Fano resonance has been investigated experimentally and theoretically in the self-assembled QDs [14]. Because of the existence of bridge channel, the current tunneling through the device can be enhanced obviously [15,16]. The interplay between the Kondo and Fano effects causes the suppression of Kondo plateau with increasing transmission from the bridge channel [17–19]. Shot noise is originated from higher order behavior of particle transport to describe the nonequilibrium property of discrete particles at low temperature, which may supply additional information beyond the current–voltage characteristics and conductance [20–22]. The charge of a quasi-particle can be determined by the measurement of shot noise, for instance, the measurement of Cooper pair charge in quantum coherent superconductor– semiconductor junctions [23], and the fractional charges of quasiparticles in fractional quantum Hall systems [24,25]. Shot noise is determined by the time-dependent current correlation, and it is sensitive to time-dependent perturbations since electrons are highly correlated in time [26]. The suppression and enhancement of shot noise associated with the sub-Poissonian and superPoissonian noise types can be induced by the irradiation of photons due to multi-particle correlation. The super-Poissonian noise is related to the inelastic photon-assisted cotunneling, and photonsuppressed shot noise is associated with the elastic photonassisted coherent tunneling current correlation [27–32]. As a quantum device is applied with an oscillating external field, the information of the external field is transferred to the tunneling current, and hence to the current correlation. The conduction electrons can change energies by absorbing or emitting photons, and, therefore,
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H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395 †
Fig. 1. (Color online.) The schematic diagram of our system. A QD is connected to the left and right terminals, while a direct coupling forms a bridge channel to connect the two terminals. A magnetic field B is threaded through the closed circle to compose an interferometer with AB flux Φ .
the coherent and incoherent transport appears in the current and current correlation. In Ref. [30], the theoretical analysis of photon-assisted shot noise with Fano resonance in a noninteracting QD has been reported, and the variations of Fano profiles with the strengths of non-resonant transmissions are presented. However, if the QD reaches ultra-small sizes, Coulomb interaction will be very strong, and the noninteracting QD model is invalid, where Kondo resonance appears in the investigated interferometer. As the single electron device is constructed to form a interferometer applied with time-oscillating fields, revealing the feature of shot noise in the strong Coulomb interaction may provide more information about the nature of a system. In this Letter, we investigate the shot noise of the photon irradiated AB interferometer where an interacting QD is embedded in one arm of the AB ring. The tunneling electrons are driven out of equilibrium by dc bias voltage, as well as time-dependent external fields. The photon-assisted Fano– Kondo tunneling characteristics in the shot noise has been derived through evaluating the current correlation. The asymmetric peak– valley structure exhibits the enhancement and suppression of shot noise, and super-Poissonian and sub-Poissonian can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. 2. Model and formalism The system is composed of a QD coupling to left and right leads, and a direct coupling forms a bridge channel to connect the two terminals directly. We display the system schematically in Fig. 1. Generally, the system is perturbed by microwave fields with same frequency ω on the two terminals and the central QD, which generate ac potentials with cosine form in the dipole approximation. The energy spectra of the two leads are modified by the (0) external fields to form the ac potentials εαk (t ) = εαk + Δα cos ωt, and the energy spectrum of central QD is modified as εd (t ) = εd − eV g + Δd cos ωt, where V g is the gate voltage applied on the (0)
QD, εαk , and εd are the isolated energies of electrons in the leads and QD. The magnitudes of the dipole potentials induced by the ac fields on the α th terminal and the central QD are denoted by Δα , Δd . A magnetic flux Φ is threaded through the central regime of the system to induce AB effect. The system Hamiltonian is expressed by the second quantization formalism
H=
εαk (t )cα† ,kσ cα ,kσ +
αkσ
+
kσ
k
1
εd (t )d†σ dσ + U nσ nσ¯
αkσ
The time-dependent energies in the terminals and QD become (0) time-independent ones as εαk (t ) → εαk , and εd (t ) → E d = εd − eV g , while the time-independent interaction strengths τγ d and τγ β are transformed to the time-dependent ones as τγ d (t ) = τγ d exp(i Λγ sin ωt ), τγ β (t ) = τγ β exp(i Λγ β sin ωt ), Λγ d = (Δγ − Δd )/¯hω , Λγ β = Λγ − Λβ , where Λγ = Δγ /¯hω , γ = β . The tunneling current operator of our system can be derived by employing the continuity equation and Heisenberg equation [33]. The current operator in the γ th terminal is given by
ˆI γ (t ) = ie τγ d (t )cγ† ,kσ dσ − τγ∗d (t )d†σ cγ ,kσ h¯ kσ
† , τγ β (t )cγ† ,kσ cβ,k σ − τγ∗β (t )cβ, c + k σ γ ,kσ
(3)
k
for γ = β . This formula indicates that the electron tunneling from the left terminal may transport through the system by two ways: the resonant transport expressed by the first term, and the direct tunneling expressed by the second term. We consider the situation as Λγ β = 0, Λγ d = Λβ d = Λ in the follows. Since the sub-systems are coupled to each other, the electron operators in different terminals and QD are involved in the separate equations of motion (EOMs). From the Heisenberg equation i h¯ ∂∂t c γ ,kσ (t ) = [c γ ,kσ (t ), H ], we obtain an EOM of the electron operator c γ ,kσ (t ), which involves the electron operator of the different terminal c β,kσ (t ), (γ = β ), and the electron operator of coupled QD dσ (t ). We make the Fourier transformation over the EOMs, and then express the operator c γ ,kσ ( ) through substituting the operator c β,kσ ( ) to give
c γ ,kσ ( ) =
1
1+W
J n (Λ) g γr ,kσ ( )
n
r × τγ β τβ d g β, ( ) + τ dσ ( + nh¯ ω) γ d k1 σ k1
+ τγ β
g γr ,kσ ( )ˆc β,k1 σ ( ) + cˆ γ ,kσ ( ) ,
(4)
k1
(1)
α
where α ∈ { L , R }, and τγ β = e i ϕ t γ β , (γ = β ). The left and right electrodes are described by the creation and annihilation operators
ργ =
(0)
δ( − εγ k ) and g γr ,kσ ( ) = (0) 1/[ − εγ k + i η], (η → 0), respectively. cˆ γ ,kσ ( ) is the correspondterminal are expressed by
ταd cα† ,kσ dσ + H.c. ,
Δd † † ˆ U (t ) = exp −i Λα ckα ,σ ckα ,σ + dσ dσ sin ωt . (2) h¯ ω σ
where W = π 2 |τγ β |2 ργ ρβ , (γ = β ), in the wide-band limit. The density of state and Green’s function of free electron in the γ th
2
σ
τγ β cγ† ,kσ cβ,k σ +
of electrons c α ,kσ and c α ,kσ . The Hamiltonian of the QD is de† scribed by the Anderson impurity model, where dσ and dσ are the † creation and annihilation operators, and nσ = dσ dσ is the number operator of local electrons in the central QD. U is the intra-dot electron–electron Coulomb interaction energy. ταd is the interaction strength of electrons between the α th lead and the central QD. t γ β is the interaction strength of electrons between the γ and β leads. The AB phase ϕ = 2π Φ/Φ0 , where Φ is the enclosed flux, and Φ0 = h/e denotes the flux quantum. We choose a gauge in which the AB flux Φ appears in the tunnel matrix for the direct transmission. The spin subscript σ in the Hamiltonian takes the value σ = ±1, which denotes the situation for spin-up ↑, and spin-down ↓. We make the gauge transformation by choosing the unitary operator
k
ing annihilation operator of free electron in the isolated γ th terminal without the perturbation of external fields. It satisfies the en† semble expectation ˆc γ ,kσ ( )ˆc γ ,k σ ( ) = f (εγ k )δγ γ δkk δσ σ . The
Fermi-distribution function is given by f (εγ k ) = 1/{exp[(εγ k −
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
μγ )/k B T ] + 1}, where μγ is the chemical potential of the γ th ter-
minal. J n (Λ) (n = 0, ±1, ±2, . . .) are the Bessel functions of the first kind. Eq. (4) presents that the operator of the γ th terminal involves the electron operators of QD and the other terminal due to the couplings τβ d and τγ β . When the direct coupling is disconnected (τγ β → 0), the electron operator approaches the res
onant situation as c γ ,kσ ( ) → n J n (Λ) g γr ,kσ ( )τγ d dσ ( + nh¯ ω) +
cˆ γ ,kσ ( ). Correspondingly, the electron operator of our coupled QD dσ ( ) can be expressed by
dσ ( )
= gdr σ ( ) τα∗d J n (Λαd )cα ,kσ ( − nh¯ ω) + U Aˆ σ ( ) α ∈{ L , R } kn
+ dˆ σ ( ),
where ) is the free Green’s function of the isolated QD. dˆ σ (t ) is the isolated electron operator of the QD, which has no contribution to the tunneling current, and we neglect it in the following ˆ σ ( ) = nσ¯ ( )dσ ( ) is contributed by derivations. The operator A the Coulomb interaction of electrons in the QD. For the special situation of noninteracting QD system, the Coulomb interaction is ˆ σ ( ) is not involved in Eq. (5). zero as U → 0, and the operator A This circumstance insures that the electron operators c γ ,kσ ( ) and dσ ( ) can be solved exactly by using Eqs. (4) and (5) to give
dσ ( ) =
where we have defined the Green’s function of QD associated with the Coulomb energy U by gdr σ (t , t ) = − h¯i θ(t − t ) exp[− h¯i ( E d +
ˆ α ,kσ (t ), U )(t − t )]. In order to determine the operators Bˆ α ,kσ (t ), D and Qˆ α ,kσ (t ), we continue to write out their EOMs through the Heisenberg equation, and employ the approximation procedure as follows to truncate the equation chain by setting † † [34–37] c α ,kσ c α ,k σ dσ ≈ c α ,kσ c α ,k σ dσ ≈ f (αk )δαα δkk δσ σ dσ , c α ,kσ c α ,k σ dσ ≈ c α ,kσ c α ,k σ dσ ≈ 0. Substituting concrete expression of c γ ,kσ ( ) into the EOM of dσ ( ) we arrive at the operator of the QD dσ ( ) in the Fourier transformation expressed by the form of Eq. (6), however, here the Coulomb interaction effect is involved in the Green’s function of QD. In the strong Coulomb interaction circumstance U → ∞ and taking the wide-band limit, the retarded Green’s function of QD is given as
(5)
gdr σ (
1 1+W
r τα∗d J n (Λ)G dr σ ( ) τα β gα ,kσ ( ) α =β
nk
G dr σ (
− E −χ ( )
ΓαA ( ) =
) =
− E d − Σ r ( )
(6)
where Σ r ( ) is the retarded self-energy of the noninteracting QD defined by
1
r
Σ ( ) =
1+W
∗
2 r αd J n (Λ) g α ,kσ (
τ
− nh¯ ω)
α =β nk
× τα β τβ d
r g β, ¯ ω) + ταd . k1 σ ( − nh
1
χ ( ) =
2π α 1
× 1+
γ ( ) =
2√
π
2 +η2
,
W cos(ϕ )
d1 1 + f α1 (1 )
m
and
2 Jm (Λ)Γα1
J n2 (Λ)
α th ter-
d2 γ
γ m − 1
n − 2 ,
n
η → 0. The occupation number of electron nσ
is determined by solving the equation
nσ =
1 2π
Im
d G d<σ ( )
2
2 Jm (Λγ )ΓγA ( ) f γ ( − mh¯ ω)G dr σ ( ) .
(10)
mγ
ˆ α ,kσ (t ) = c α ,kσ¯ (t )d (t )dσ (t ), Qˆ α ,kσ (t ) = Bˆ α ,kσ (t ) = c α ,kσ (t )nσ¯ (t ), D σ¯ † c α ,kσ¯ (t )dσ (t )dσ¯ (t ) are involved in the same equation as presented by the form †
dt 1 gdr σ (t , t 1 ) τα∗d (t 1 ) α ∈{ L , R } k
ταd (t 1 ) Qˆ α ,kσ (t 1 ) , × Bˆ α ,kσ (t 1 ) − Dˆ α ,kσ (t 1 ) +
1 2 (Λ) f α1 ( + mh¯ ω) , Γα1 J m Γα 1 + 1 − nσ¯
Γ mα 1
G d<σ ( ) = i
The noninteracting model is suitable for a relatively larger QD, which usually provides a proper description of transport associated with the relatively larger QDs possessing weak Coulomb interaction. In an ultra-small QD at low temperature, a new state of many-body character is created at its Fermi level, and the zerobias conductance resonance contains its intrinsic feature [1–5]. In order to investigate the transport nature of a strongly correlated electron devises, one has to consider the role of Coulomb interaction in different regime. In this paper, we assume the large U situation, and solve the strongly correlated electron operator of QD from Eqs. (4) and (5). Since Eq. (5) contains the unknown ˆ σ ( ), we have to derive the expression of this operaoperator A ˆ σ (t ), we find that new operators tor. On deriving the EOM of A
ˆ σ (t ) = A
self-consistently, where the Keldysh Green’s function is determined by
k1
√
α = Γ = α Γα . The linewidth of the where Γ minal is Γα = 2πρα |ταd |2 . We have defined the function
(7)
,
(9)
,
1
) is the retarded Green’s function of the cou-
1
A ( ) w σ ( ) + 2i Γ
d where w σ ( ) = 1− + W cos(ϕ )ΓA ( ), and ΓA = α ΓαA . nσ¯
In the above formulas, the modified linewidth of our system is expressed by
G dr σ (
In the formula, pled QD given by
1
G dr σ ( ) =
Γα , and 1+ W
k1
× cˆ β,k1 σ ( − nh¯ ω) + cˆ α ,kσ ( − nh¯ ω) (γ = β).
391
(8)
The current operator is given by substituting the electron operators Eqs. (4) and (6) into Eq. (3), and it is expressed as
i nm ˆI γ (t ) = e d1 d2 exp ε12 t h nmσ h¯ α1 =β1 α2 =β2
γβ † × J n (Λ) J m (Λ) A α1 α2 1n , 2m cˆ α1 σ (1 )ˆc α2 σ (2 ),
(11)
nm n = + nh¯ ω, and ε12 = 1 − 2 + (n − m)¯hω . The transγβ port coefficient A α1 α2 (1 , 2 ) is determined by the retarded and
where
advanced Green’s functions of QD. Without loss of generally, we only consider the symmetric coupling Γ L = Γ R = Γ /2 in the following evaluations. We obtain the explicit expression γβ
A α1 α2 (1 , 2 ) 1
γβ = − ΓΓA G adσ (1 )G dr σ (2 ) + i ξα1 G dr σ (2 ) 4 γ β∗ − i ξα2 G adσ (1 ) + δΛ0 T b (δα1 γ δα2 γ − δα1 β δα2 β ) + i S b [δα1 γ δα2 β − δα1 β δα2 γ ] .
(12)
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H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
In the transport coefficient above, we have defined the function √ γβ ξα1 = 12 Γ[δα1 γ − (δα1 γ − δα1 β )( T b + i cos(ϕ ) S b )], where S b = 2 T b (1 − T b ), and T b = 4W /(1 + W ) . The quantities T b and S b are the transmission coefficient and shot noise of the direct tunneling for the special case without the resonant tunneling through QD. By taking ensemble and time average over the current operator (11), we arrive at the tunneling current
Iγ =
e h
nσ
γ β n n , f α1 ( ),
d J n2 (Λ) A α1 α1
(13)
α1 =β1
for γ = β . We therefore obtain the Landauer–Büttiker-like formula of current in the γ th terminal explicitly as
Iγ = for
e h
d J n2 (Λ) T σ
n
f γ ( ) − f β ( ) ,
(14)
of absorption and emission of photons presents different behaviors of shot noise in our system. Two cases are specified as: the balanced absorption where n = m, n = m ; the unbalanced absorption where m = n − m + n . The balanced absorption indicates that the absorbed photons in one of the correlated current are emitted completely to form coherent current correlation. For the unbalanced absorption, the absorbed photons are emitted to the other correlated currents to form incoherent current correlation. Eq. (16) is exactly suitable for the noninteracting systems where one-body correlation functions are exactly derived, and all dynamical many-body correlations vanish. For our interacting QD system, this formula evaluates the shot noise approximately by substitut-
nσ
S γ γ (0) =
γ = β . The transmission coefficient T σ ( ) is defined by 4
4
√
Sb
where we have defined the function Y σ ( ) = [1 − 2T b + A × Γ ( ) w σ ( ) cos(ϕ )]. The transmission coefficient T σ ( ) describes the parallel tunneling through the resonant and nonresonant paths, where the strong Coulomb interaction is involved to govern the coherent and interferential transport. Although the form of Eq. (14) is generally valid for the whole regime of Coulomb interaction (U = 0–∞), the concrete transmission coefficient here is only valid for the strong Coulomb interaction when U → ∞, where the Fermi distribution function is involved int the self-energy of Green’s function of QD. For this circumstance, one novel electron tunneling channel emerges on the Fermi level, and the Kondo effect appears in the differential conductance. The current correlation of two current operators ˆI γ (t ) and
ˆI γ (t ) at different times t and t is settled by Πγ γ (t , t ) = δ ˆI γ (t )δ ˆI γ (t ) + δ ˆI γ (t )δ ˆI γ (t ) , where δ ˆI γ (t ) = ˆI γ (t ) − ˆI γ (t ) is the current fluctuation operator. The symbol · · · in above formula
denotes the ensemble average over the system. Substituting the current operator (11) into the definition of current correlation, we have the current correlation [38] associated with the two times t and t as
2 e h
d1 d2 exp
× exp γβ
i h¯
i h¯
nm n m α1 α2 σ
nm ε12 t
nm ε21 t J n (Λ) J m (Λ) J n Λ J m Λ
γ β m
n
m
where the joint Fermi distribution function is involved by the definition F α β ( , ) = f α ( )[1 − f β ( )] + [1 − f α ( )] f β ( ). The shot noise spectrum S γ γ (Ω) is determined by making Fourier transformation over the current correlation [20,21] Πγ γ (t , t ) via the re-
lation S γ γ (Ω)δ(Ω + Ω ) = 21π Πγ γ (Ω, Ω ). Correspondingly, the shot noise spectrum is evaluated as
e2 h
h
nm n m
d J n (Λ) J m (Λ) J n Λ
σ
d J n (Λ) J m (Λ) J n Λ
nm n m α1 α2 σ
γβ
γ β × J m Λ A α1 α2 n , ˜ n A α2 α1 ˜ m , m
× F α1 α2 , ˜ (n−m) ,
2
α
1
m G r n 2 G r m 2 + W n W dσ dσ 4 2 + Sb W n G dr σ n δΛ 0 + S b δΛ0 δΛ 0
(n−m) , × Fγ β ,
(17)
√
[(1 − 2T b ) A ( )] where the notation W ( ) = Γ w σ ( ) − S b cos(ϕ )Γ is utilized in the formula. Here δΛ0 signifies that the magnitude of applied field is zero in this term, i.e., J n (Λ)δΛ0 = 1. As direct tunneling is disconnected by setting W = 0, we obtain the photon-assisted resonant transport shot noise. As the resonant tunneling is disconnected by setting Γ = 0, we have the direct tunneling shot noise. Eq. (17) is the main result of the shot noise for our strongly correlated system, where the Coulomb interaction is U → ∞. It cannot be reduced to the shot noise of the noninteracting QD system due to the overlap and interactions between the local electrons in the QD and tunneling electrons from the terminals. The Kondo and Fano effects are combined efficiently resulting from Kondo resonant tunneling and Fano interfering correlation of electrons. Since the occupation number and Fermi distribution function are involved in the noise formula, one should solve the formula self-consistently. 3. Numerical results
× A α1 α2 1n , 2 A α2 α1 2 , 1 F α1 α2 (1 , 2 ), (15)
S γ γ (Ω) =
1 n m
× J m Λ F αα , (n−m) Tσ Tσ
1
Πγ γ t , t =
2e 2
ΓA ( )G r ( )2 Y σ ( ) + T b δΛ0 , T σ ( ) = Γ dσ
γ β
γβ
ing the explicit coefficients A α1 α2 ( n , ˜ n ) and A α2 α1 (˜ m , m ) given in Eq. (12) into the noise formula (16). We now consider the zerofrequency shot noise in the same terminal by setting Ω = 0 to find
(16)
where ˜ n = n + h¯ Ω , and n − m + n − m = 0 is satisfied to indicate the relation of absorption and emission of photons. The way
The numerical calculations are performed in this section for the symmetric situation as Γ L = Γ R = Δ, Λ L = Λ R = 1.0, and we take Δ as the energy scale by choosing the common parameters as εd = 0, k B T = 0.0008Δ. We set Δ = 1 in the numerical evaluations and the following expressions. The differential conductance, shot noise, and derivative of shot noise are scaled by G 0 = 2e 2 /h, S 0 = 4e 2 Δ/h, and 4e 3 /h, respectively. We depict the differential conductance versus source-drain bias in Fig. 2 to investigate the influence of applied ac fields and the direct tunneling strength. Fig. 2(a) shows the variation of differential conductance G with respect to different photon energy h¯ ω as W = 0.05. Single Kondo resonant peak is located at eV = 0 when h¯ ω = 0. The side resonant peaks appear around the Kondo resonant peak due to the absorption and emission of photons, and the height of Kondo peak is suppressed due to the photon absorption. The suppression of the Kondo effect in a resonant tunneling QD by external irradiation has been reported by Kaminski et al.
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
393
Fig. 2. (Color online.) The conductance versus source-drain bias eV at ϕ = 0. (a) The conductance at eV g = 3.0 for different photon energy h¯ ω = 0, 0.4, 0.8. (b) The conductance at h¯ ω = 0.8 for different W as W = 0.05, 0.2, 0.4, 0.6, 0.8, 1.0.
Fig. 4. (Color online.) The shot noise S, derivative of shot noise dS /deV g , and Fano factor F versus gate bias eV g for the unbalanced case. The other parameters are chosen as ϕ = 0, W = 0.05, eV = 1.0.
Fig. 3. (Color online.) The shot noise S, derivative of shot noise dS /deV g , and Fano factor F versus gate bias eV g for the balanced case. The other parameters are chosen as ϕ = 0, W = 0.05, eV = 1.0.
in Ref. [39], where the shape of resonant peak in the conductance has been studied. Fig. 2(b) exhibits the modification of differential conductance with respect to changing the amplitude of direct transmission strength W . The Kondo resonant peak is obviously suppressed, and the Fano resonance becomes more evident by increasing W from W = 0 to 1.0. As W is strong enough (W ≈ 0.6) the Kondo peak can be completely smeared, while the Fano dip appears obviously around eV = −5. This indicates that the electron tunneling through QD induces the Kondo resonance effect, while the direct tunneling results in non-resonant tunneling. The competition of interference and Kondo resonance generates the Fano–Kondo cooperated effect in the differential conductance. We present the variations of shot noise S, derivative of shot noise dS /deV g , and Fano factor F = S /(2eI ) versus gate bias eV g for the balanced case in Fig. 3. The asymmetric behavior exhibits obviously in the shot noise, where the photon induced resonant
peak–valley structure is located around the main valley around eV g = 0. Away from the central valley regime, the shot noise exhibits left lower and right higher asymmetric structure. The value of central dip is about S ≈ 0.02S 0 , while the maximum value on the right peak is about S ≈ 0.18S 0 with the variation rate 89% shown by diagram Fig. 3(a). This asymmetric effect is generated from the interference and correlation of photon-assisted electron tunneling in different arms. Electrons traveling in the resonant arm are adjusted by the QD and possess resonant energy level with the QD, while electrons traveling through the direct tunneling arm possess continue spectrum with the terminals. When the electrons with deferent phase characteristics meet in the device, destructive and constructive interference occur, and the asymmetric behavior of shot noise appears. The derivative of shot noise versus eV g shown in Fig. 3(b) exhibits photon-assisted peaks corresponding to the drastic variation of shot noise in Fig. 3(a). The main peak is located around eV g = 0, and the side photon-induced peaks located at the positions eV g = ±¯hω display asymmetric structure. The Fano factor shown in Fig. 3(c) is suppressed to F ≈ 0.1 at the central valley around eV g = 0. The side peak–valley structure displayed around the central valley is induced by the photon absorption and emission effect. The saturate values of the Fano factor show the left higher and right lower appearance. The Fano factor exhibits sub-Poissonian noise feature with F < 1 in the whole regime of eV g . This means that coherent current correlation for the balanced case suppresses the shot noise, and the competition of Fano–Kondo resonant tunneling results in the asymmetric characteristics of shot noise and Fano factor. Fig. 4 presents the shot noise, derivative of shot noise, and Fano factor versus gate bias eV g for the unbalanced case. The resonant peak appears at eV g = 0 with height S ≈ 0.5 ∼ 0.7 when the photon energy reduces from 5 to 3 shown in Fig. 4(a). The left and right valleys exhibits asymmetric behavior, where the right valley is suppressed to zero as h¯ ω = 3. This signifies that the destructive interference effect is considerably affects the appearance of shot noise, where the photon energy takes important role. The derivative of unbalanced shot noise displays the peak–valley structure
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H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
Fig. 5. (Color online.) The shot noise S and Fano factor F versus AB phase ϕ for the balanced case. The other parameters are chosen as eV = 1.0, eV g = 0, and h¯ ω = 3.0.
shown in Fig. 4(b), and the appearance of peak–valley structure is left- and right-shifted due to increasing photon energy. The enhancement of shot noise results in the Fano factor exhibiting a resonant peak F ≈ 2.7 at about eV g = −2 for h¯ ω = 3 shown in Fig. 4(c), and the suppression of shot noise exhibits the Fano factor valley F = 0 at about eV g = 5 (the dotted line). This signifies that the enhancement and complete suppression of shot noise can be achieved by adjusting the gate voltage and photon energy h¯ ω . The enhancement and suppression of shot noise are contained in this case due to the competition of incoherent and destructive interference effects. For the unbalanced case, the absorbed photons in one current are not equal to the emitted photons in the other current. The currents correlate to induce energy accumulation in the joint Fermi function F γ β ( , (n−m) ), and this effect generates the excess shot noise even when the source–drain bias is removed at zero temperature. We present the variation of balanced shot noise and Fano factor with respect to AB phase ϕ in Fig. 5. The periodic oscillation with period 2π appears in both of the shot noise and Fano factor. The peaks are located at ϕ = (2n + 1)π , and the valleys are located at ϕ = 2nπ (n = 0, ±1, ±2, . . .). The shot noise and Fano factor do not increase monotonically with increasing the value of W . One observes that as W is strong (W = 0.6), the peaks are suppressed to show plateaus S ≈ 0.2, F ≈ 0.28, and flat valleys S ≈ 0, F ≈ 0 with wide regime around | ϕ |< 0.7π in a period. When W is given by a moderate value as W = 0.2, we have large oscillation peaks as S ≈ 0.52, and F ≈ 0.8. This suppression and enhancement of shot noise indicates that by adjusting W and ϕ one can control the magnitude and shape of shot noise to obtain zero and large noise circumstances. We display the shot noise S and Fano factor with respect to W for the balanced case by choosing separate photon energy in Fig. 6. The shot noise and Fano factor increase as W increases rapidly to the maximum values, and then they decline mildly to zero. The maximum values exhibit at about W = 0.18 for the shot noise, and at W = 0.12 for the Fano factor. This effect signifies that the direct tunneling may induce the enhancement and suppression of shot noise and Fano factor due to changing the parameter W . The maximum values also increase with increasing of photon energy from h¯ ω = 3.0 to h¯ ω = 5.0. Similar appearances display in the shot noise and Fano factor corresponding to changing W with different magnetic phase
Fig. 6. The shot noise S and Fano factor F versus W for the balanced case with different photon energy h¯ ω . The other parameters are chosen as eV = 1.0, eV g = 0, and ϕ = 0.
Fig. 7. The shot noise S and Fano factor F versus W for the balanced case with different AB phase ϕ . The other parameters are chosen as eV = 1.0, eV g = 0, and h¯ ω = 3.0.
ϕ shown in Fig. 7. As the magnetic phase ϕ increases from ϕ = 0 to π , the shot noise increases from S ≈ 0.1 to 0.5, while
the Fano factor increases from F ≈ 0.2 to 0.85 with the changing rates 80% and 76%, respectively. The direct tunneling takes main contribution to shot noise and Fano factor when W = 1, while the resonant tunneling is the sole contribution when W = 0. The competition of resonant and direct tunneling currents correlation arouses significant noise variation and configuration. This indicates that by adjusting W one can obtain extreme and minimum values of shot noise and Fano factor for noise controlling in electronic device designing. Compared with the shot noise of the noninteracting QD system (see Ref. [30]), the peak–valley structure behaves quite differently here due to the strong Coulomb interaction. The modification of
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
shot noise structure appears intimately related to the Kondo effect in finite temperature, where novel channel emerges on the Fermi level. The current correlation is associated with the interference of electrons from this channel to the electrons with continuous spectrum, which results compound effect as the Fano–Kondo resonance. The symmetry of phase locking as ϕ → −ϕ is kept, however, the plateaus and flat valleys with period of 2π may be generated. Shot noise reflects the higher order particle transport generated from the current correlation, and it reveals the statistical features of particles as the super-Poissonian and sub-Poissonian distributions. The time-averaged current is zero for our system, but the current correlation is nonzero due to the perturbation of external ac fields. This also induces the strong correlations of tunneling electrons in the Kondo channel and the continuous channels. The Fano–Kondo appearance of shot noise reflects the hidden physics, which does not displays in the current characteristics. The deviation of current and shot noise is significantly revealed by the characteristics of Fano factor. 4. Concluding remarks The shot noise and Fano factor of a QD embedded AB interferometer under the perturbation of microwave fields have been investigated. The frequency-dependent shot noise formula is derived explicitly by evaluating the current operator correlation through EOM method. The competition of Kondo resonant and non-resonant direct tunnelings results in photon-assisted Fano– Kondo resonant transport, and the suppression of Kondo peak appears due to increasing the direct tunneling strength. The interference and correlation of photon-assisted electrons induce the asymmetric resonant peak–valley behavior of shot noise. The enhancement and suppression of shot noise are derived due to the competition of incoherent and destructive interference effects, and super-Poissonian and sub-Poissonian noise can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. The periodic oscillation with period 2π appears in both of the shot noise and Fano factor, where plateaus and flat valleys are observed. The direct tunneling may induce the enhancement and suppression of shot noise and Fano factor due to changing the direct tunneling parameter. Acknowledgement We gratefully acknowledge the support by the National Natural Science Foundation of China under the Grant No. 11175015.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Fano–Kondo shot noise in a quantum dot embedded interferometer irradiated with microwave fields Hong-Kang Zhao ∗ , Wei-Ke Zou School of Physics, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 3 September 2014 Received in revised form 19 November 2014 Accepted 24 November 2014 Available online 27 November 2014 Communicated by R. Wu Keywords: Dynamic shot noise Fano–Kondo resonance Quantum dot embedded interferometer Current correlation
a b s t r a c t The shot noise of a quantum dot embedded Aharonov–Bohm (AB) interferometer under the perturbation of microwave fields is investigated by employing equation of motion method. The frequency-dependent shot noise formula is derived, and the photon-assisted Fano–Kondo resonance, the suppression of Kondo peak are presented with increasing the direct tunneling strength. The interference and correlation of electrons induce the asymmetric resonant peak–valley behavior of shot noise. The enhancement and suppression of shot noise are resulting from the competition of incoherent correlation and destructive interference effects, and super-Poissonian and sub-Poissonian noise can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. The periodic oscillation versus AB phase with period 2π appears to show plateaus and flat valleys. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In an ultra-small quantum dot (QD) at low temperature, a new state of many-body character known as Kondo effect is created at its Fermi level, and the zero-bias conductance resonance contains its intrinsic feature [1–5]. A spin singlet is formed by the localized spins on the QD, and conduction electrons tunnel through different terminals. The resonant peak is split associated with the spin–orbit interaction and spin polarization in ferromagnetic systems [6–8]. The Fano effect results from the interference between two electron waves, one passing through the QD and the other traveling along the direct channel characterized by its continuous spectrum. The interference between a resonant channel and the non-resonant channel leads to asymmetric Fano line shape in the differential conductance, where the formula of Fano line shape of transmission coefficient is presented [9]. The closed and open interferometers have been discussed theoretically, in which one path contains either simple QD or a decorated QD [10]. The QD embedded two-terminal Aharonov–Bohm (AB) interferometer was proposed to measure the phase of the transmission through the QD [11]. The Coulomb modified Fano resonance has been observed in a tunable Fano interferometer consisting of a QD [12]. The Fano resonances appear in the electron transport through the QDs formed by magnetic double barriers in quantum wires [13]. The nonlinear
*
Corresponding author. Tel.: +86 10 68914975. E-mail address: [email protected] (H.-K. Zhao).
http://dx.doi.org/10.1016/j.physleta.2014.11.050 0375-9601/© 2014 Elsevier B.V. All rights reserved.
Fano resonance has been investigated experimentally and theoretically in the self-assembled QDs [14]. Because of the existence of bridge channel, the current tunneling through the device can be enhanced obviously [15,16]. The interplay between the Kondo and Fano effects causes the suppression of Kondo plateau with increasing transmission from the bridge channel [17–19]. Shot noise is originated from higher order behavior of particle transport to describe the nonequilibrium property of discrete particles at low temperature, which may supply additional information beyond the current–voltage characteristics and conductance [20–22]. The charge of a quasi-particle can be determined by the measurement of shot noise, for instance, the measurement of Cooper pair charge in quantum coherent superconductor– semiconductor junctions [23], and the fractional charges of quasiparticles in fractional quantum Hall systems [24,25]. Shot noise is determined by the time-dependent current correlation, and it is sensitive to time-dependent perturbations since electrons are highly correlated in time [26]. The suppression and enhancement of shot noise associated with the sub-Poissonian and superPoissonian noise types can be induced by the irradiation of photons due to multi-particle correlation. The super-Poissonian noise is related to the inelastic photon-assisted cotunneling, and photonsuppressed shot noise is associated with the elastic photonassisted coherent tunneling current correlation [27–32]. As a quantum device is applied with an oscillating external field, the information of the external field is transferred to the tunneling current, and hence to the current correlation. The conduction electrons can change energies by absorbing or emitting photons, and, therefore,
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H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395 †
Fig. 1. (Color online.) The schematic diagram of our system. A QD is connected to the left and right terminals, while a direct coupling forms a bridge channel to connect the two terminals. A magnetic field B is threaded through the closed circle to compose an interferometer with AB flux Φ .
the coherent and incoherent transport appears in the current and current correlation. In Ref. [30], the theoretical analysis of photon-assisted shot noise with Fano resonance in a noninteracting QD has been reported, and the variations of Fano profiles with the strengths of non-resonant transmissions are presented. However, if the QD reaches ultra-small sizes, Coulomb interaction will be very strong, and the noninteracting QD model is invalid, where Kondo resonance appears in the investigated interferometer. As the single electron device is constructed to form a interferometer applied with time-oscillating fields, revealing the feature of shot noise in the strong Coulomb interaction may provide more information about the nature of a system. In this Letter, we investigate the shot noise of the photon irradiated AB interferometer where an interacting QD is embedded in one arm of the AB ring. The tunneling electrons are driven out of equilibrium by dc bias voltage, as well as time-dependent external fields. The photon-assisted Fano– Kondo tunneling characteristics in the shot noise has been derived through evaluating the current correlation. The asymmetric peak– valley structure exhibits the enhancement and suppression of shot noise, and super-Poissonian and sub-Poissonian can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. 2. Model and formalism The system is composed of a QD coupling to left and right leads, and a direct coupling forms a bridge channel to connect the two terminals directly. We display the system schematically in Fig. 1. Generally, the system is perturbed by microwave fields with same frequency ω on the two terminals and the central QD, which generate ac potentials with cosine form in the dipole approximation. The energy spectra of the two leads are modified by the (0) external fields to form the ac potentials εαk (t ) = εαk + Δα cos ωt, and the energy spectrum of central QD is modified as εd (t ) = εd − eV g + Δd cos ωt, where V g is the gate voltage applied on the (0)
QD, εαk , and εd are the isolated energies of electrons in the leads and QD. The magnitudes of the dipole potentials induced by the ac fields on the α th terminal and the central QD are denoted by Δα , Δd . A magnetic flux Φ is threaded through the central regime of the system to induce AB effect. The system Hamiltonian is expressed by the second quantization formalism
H=
εαk (t )cα† ,kσ cα ,kσ +
αkσ
+
kσ
k
1
εd (t )d†σ dσ + U nσ nσ¯
αkσ
The time-dependent energies in the terminals and QD become (0) time-independent ones as εαk (t ) → εαk , and εd (t ) → E d = εd − eV g , while the time-independent interaction strengths τγ d and τγ β are transformed to the time-dependent ones as τγ d (t ) = τγ d exp(i Λγ sin ωt ), τγ β (t ) = τγ β exp(i Λγ β sin ωt ), Λγ d = (Δγ − Δd )/¯hω , Λγ β = Λγ − Λβ , where Λγ = Δγ /¯hω , γ = β . The tunneling current operator of our system can be derived by employing the continuity equation and Heisenberg equation [33]. The current operator in the γ th terminal is given by
ˆI γ (t ) = ie τγ d (t )cγ† ,kσ dσ − τγ∗d (t )d†σ cγ ,kσ h¯ kσ
† , τγ β (t )cγ† ,kσ cβ,k σ − τγ∗β (t )cβ, c + k σ γ ,kσ
(3)
k
for γ = β . This formula indicates that the electron tunneling from the left terminal may transport through the system by two ways: the resonant transport expressed by the first term, and the direct tunneling expressed by the second term. We consider the situation as Λγ β = 0, Λγ d = Λβ d = Λ in the follows. Since the sub-systems are coupled to each other, the electron operators in different terminals and QD are involved in the separate equations of motion (EOMs). From the Heisenberg equation i h¯ ∂∂t c γ ,kσ (t ) = [c γ ,kσ (t ), H ], we obtain an EOM of the electron operator c γ ,kσ (t ), which involves the electron operator of the different terminal c β,kσ (t ), (γ = β ), and the electron operator of coupled QD dσ (t ). We make the Fourier transformation over the EOMs, and then express the operator c γ ,kσ ( ) through substituting the operator c β,kσ ( ) to give
c γ ,kσ ( ) =
1
1+W
J n (Λ) g γr ,kσ ( )
n
r × τγ β τβ d g β, ( ) + τ dσ ( + nh¯ ω) γ d k1 σ k1
+ τγ β
g γr ,kσ ( )ˆc β,k1 σ ( ) + cˆ γ ,kσ ( ) ,
(4)
k1
(1)
α
where α ∈ { L , R }, and τγ β = e i ϕ t γ β , (γ = β ). The left and right electrodes are described by the creation and annihilation operators
ργ =
(0)
δ( − εγ k ) and g γr ,kσ ( ) = (0) 1/[ − εγ k + i η], (η → 0), respectively. cˆ γ ,kσ ( ) is the correspondterminal are expressed by
ταd cα† ,kσ dσ + H.c. ,
Δd † † ˆ U (t ) = exp −i Λα ckα ,σ ckα ,σ + dσ dσ sin ωt . (2) h¯ ω σ
where W = π 2 |τγ β |2 ργ ρβ , (γ = β ), in the wide-band limit. The density of state and Green’s function of free electron in the γ th
2
σ
τγ β cγ† ,kσ cβ,k σ +
of electrons c α ,kσ and c α ,kσ . The Hamiltonian of the QD is de† scribed by the Anderson impurity model, where dσ and dσ are the † creation and annihilation operators, and nσ = dσ dσ is the number operator of local electrons in the central QD. U is the intra-dot electron–electron Coulomb interaction energy. ταd is the interaction strength of electrons between the α th lead and the central QD. t γ β is the interaction strength of electrons between the γ and β leads. The AB phase ϕ = 2π Φ/Φ0 , where Φ is the enclosed flux, and Φ0 = h/e denotes the flux quantum. We choose a gauge in which the AB flux Φ appears in the tunnel matrix for the direct transmission. The spin subscript σ in the Hamiltonian takes the value σ = ±1, which denotes the situation for spin-up ↑, and spin-down ↓. We make the gauge transformation by choosing the unitary operator
k
ing annihilation operator of free electron in the isolated γ th terminal without the perturbation of external fields. It satisfies the en† semble expectation ˆc γ ,kσ ( )ˆc γ ,k σ ( ) = f (εγ k )δγ γ δkk δσ σ . The
Fermi-distribution function is given by f (εγ k ) = 1/{exp[(εγ k −
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
μγ )/k B T ] + 1}, where μγ is the chemical potential of the γ th ter-
minal. J n (Λ) (n = 0, ±1, ±2, . . .) are the Bessel functions of the first kind. Eq. (4) presents that the operator of the γ th terminal involves the electron operators of QD and the other terminal due to the couplings τβ d and τγ β . When the direct coupling is disconnected (τγ β → 0), the electron operator approaches the res
onant situation as c γ ,kσ ( ) → n J n (Λ) g γr ,kσ ( )τγ d dσ ( + nh¯ ω) +
cˆ γ ,kσ ( ). Correspondingly, the electron operator of our coupled QD dσ ( ) can be expressed by
dσ ( )
= gdr σ ( ) τα∗d J n (Λαd )cα ,kσ ( − nh¯ ω) + U Aˆ σ ( ) α ∈{ L , R } kn
+ dˆ σ ( ),
where ) is the free Green’s function of the isolated QD. dˆ σ (t ) is the isolated electron operator of the QD, which has no contribution to the tunneling current, and we neglect it in the following ˆ σ ( ) = nσ¯ ( )dσ ( ) is contributed by derivations. The operator A the Coulomb interaction of electrons in the QD. For the special situation of noninteracting QD system, the Coulomb interaction is ˆ σ ( ) is not involved in Eq. (5). zero as U → 0, and the operator A This circumstance insures that the electron operators c γ ,kσ ( ) and dσ ( ) can be solved exactly by using Eqs. (4) and (5) to give
dσ ( ) =
where we have defined the Green’s function of QD associated with the Coulomb energy U by gdr σ (t , t ) = − h¯i θ(t − t ) exp[− h¯i ( E d +
ˆ α ,kσ (t ), U )(t − t )]. In order to determine the operators Bˆ α ,kσ (t ), D and Qˆ α ,kσ (t ), we continue to write out their EOMs through the Heisenberg equation, and employ the approximation procedure as follows to truncate the equation chain by setting † † [34–37] c α ,kσ c α ,k σ dσ ≈ c α ,kσ c α ,k σ dσ ≈ f (αk )δαα δkk δσ σ dσ , c α ,kσ c α ,k σ dσ ≈ c α ,kσ c α ,k σ dσ ≈ 0. Substituting concrete expression of c γ ,kσ ( ) into the EOM of dσ ( ) we arrive at the operator of the QD dσ ( ) in the Fourier transformation expressed by the form of Eq. (6), however, here the Coulomb interaction effect is involved in the Green’s function of QD. In the strong Coulomb interaction circumstance U → ∞ and taking the wide-band limit, the retarded Green’s function of QD is given as
(5)
gdr σ (
1 1+W
r τα∗d J n (Λ)G dr σ ( ) τα β gα ,kσ ( ) α =β
nk
G dr σ (
− E −χ ( )
ΓαA ( ) =
) =
− E d − Σ r ( )
(6)
where Σ r ( ) is the retarded self-energy of the noninteracting QD defined by
1
r
Σ ( ) =
1+W
∗
2 r αd J n (Λ) g α ,kσ (
τ
− nh¯ ω)
α =β nk
× τα β τβ d
r g β, ¯ ω) + ταd . k1 σ ( − nh
1
χ ( ) =
2π α 1
× 1+
γ ( ) =
2√
π
2 +η2
,
W cos(ϕ )
d1 1 + f α1 (1 )
m
and
2 Jm (Λ)Γα1
J n2 (Λ)
α th ter-
d2 γ
γ m − 1
n − 2 ,
n
η → 0. The occupation number of electron nσ
is determined by solving the equation
nσ =
1 2π
Im
d G d<σ ( )
2
2 Jm (Λγ )ΓγA ( ) f γ ( − mh¯ ω)G dr σ ( ) .
(10)
mγ
ˆ α ,kσ (t ) = c α ,kσ¯ (t )d (t )dσ (t ), Qˆ α ,kσ (t ) = Bˆ α ,kσ (t ) = c α ,kσ (t )nσ¯ (t ), D σ¯ † c α ,kσ¯ (t )dσ (t )dσ¯ (t ) are involved in the same equation as presented by the form †
dt 1 gdr σ (t , t 1 ) τα∗d (t 1 ) α ∈{ L , R } k
ταd (t 1 ) Qˆ α ,kσ (t 1 ) , × Bˆ α ,kσ (t 1 ) − Dˆ α ,kσ (t 1 ) +
1 2 (Λ) f α1 ( + mh¯ ω) , Γα1 J m Γα 1 + 1 − nσ¯
Γ mα 1
G d<σ ( ) = i
The noninteracting model is suitable for a relatively larger QD, which usually provides a proper description of transport associated with the relatively larger QDs possessing weak Coulomb interaction. In an ultra-small QD at low temperature, a new state of many-body character is created at its Fermi level, and the zerobias conductance resonance contains its intrinsic feature [1–5]. In order to investigate the transport nature of a strongly correlated electron devises, one has to consider the role of Coulomb interaction in different regime. In this paper, we assume the large U situation, and solve the strongly correlated electron operator of QD from Eqs. (4) and (5). Since Eq. (5) contains the unknown ˆ σ ( ), we have to derive the expression of this operaoperator A ˆ σ (t ), we find that new operators tor. On deriving the EOM of A
ˆ σ (t ) = A
self-consistently, where the Keldysh Green’s function is determined by
k1
√
α = Γ = α Γα . The linewidth of the where Γ minal is Γα = 2πρα |ταd |2 . We have defined the function
(7)
,
(9)
,
1
) is the retarded Green’s function of the cou-
1
A ( ) w σ ( ) + 2i Γ
d where w σ ( ) = 1− + W cos(ϕ )ΓA ( ), and ΓA = α ΓαA . nσ¯
In the above formulas, the modified linewidth of our system is expressed by
G dr σ (
In the formula, pled QD given by
1
G dr σ ( ) =
Γα , and 1+ W
k1
× cˆ β,k1 σ ( − nh¯ ω) + cˆ α ,kσ ( − nh¯ ω) (γ = β).
391
(8)
The current operator is given by substituting the electron operators Eqs. (4) and (6) into Eq. (3), and it is expressed as
i nm ˆI γ (t ) = e d1 d2 exp ε12 t h nmσ h¯ α1 =β1 α2 =β2
γβ † × J n (Λ) J m (Λ) A α1 α2 1n , 2m cˆ α1 σ (1 )ˆc α2 σ (2 ),
(11)
nm n = + nh¯ ω, and ε12 = 1 − 2 + (n − m)¯hω . The transγβ port coefficient A α1 α2 (1 , 2 ) is determined by the retarded and
where
advanced Green’s functions of QD. Without loss of generally, we only consider the symmetric coupling Γ L = Γ R = Γ /2 in the following evaluations. We obtain the explicit expression γβ
A α1 α2 (1 , 2 ) 1
γβ = − ΓΓA G adσ (1 )G dr σ (2 ) + i ξα1 G dr σ (2 ) 4 γ β∗ − i ξα2 G adσ (1 ) + δΛ0 T b (δα1 γ δα2 γ − δα1 β δα2 β ) + i S b [δα1 γ δα2 β − δα1 β δα2 γ ] .
(12)
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H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
In the transport coefficient above, we have defined the function √ γβ ξα1 = 12 Γ[δα1 γ − (δα1 γ − δα1 β )( T b + i cos(ϕ ) S b )], where S b = 2 T b (1 − T b ), and T b = 4W /(1 + W ) . The quantities T b and S b are the transmission coefficient and shot noise of the direct tunneling for the special case without the resonant tunneling through QD. By taking ensemble and time average over the current operator (11), we arrive at the tunneling current
Iγ =
e h
nσ
γ β n n , f α1 ( ),
d J n2 (Λ) A α1 α1
(13)
α1 =β1
for γ = β . We therefore obtain the Landauer–Büttiker-like formula of current in the γ th terminal explicitly as
Iγ = for
e h
d J n2 (Λ) T σ
n
f γ ( ) − f β ( ) ,
(14)
of absorption and emission of photons presents different behaviors of shot noise in our system. Two cases are specified as: the balanced absorption where n = m, n = m ; the unbalanced absorption where m = n − m + n . The balanced absorption indicates that the absorbed photons in one of the correlated current are emitted completely to form coherent current correlation. For the unbalanced absorption, the absorbed photons are emitted to the other correlated currents to form incoherent current correlation. Eq. (16) is exactly suitable for the noninteracting systems where one-body correlation functions are exactly derived, and all dynamical many-body correlations vanish. For our interacting QD system, this formula evaluates the shot noise approximately by substitut-
nσ
S γ γ (0) =
γ = β . The transmission coefficient T σ ( ) is defined by 4
4
√
Sb
where we have defined the function Y σ ( ) = [1 − 2T b + A × Γ ( ) w σ ( ) cos(ϕ )]. The transmission coefficient T σ ( ) describes the parallel tunneling through the resonant and nonresonant paths, where the strong Coulomb interaction is involved to govern the coherent and interferential transport. Although the form of Eq. (14) is generally valid for the whole regime of Coulomb interaction (U = 0–∞), the concrete transmission coefficient here is only valid for the strong Coulomb interaction when U → ∞, where the Fermi distribution function is involved int the self-energy of Green’s function of QD. For this circumstance, one novel electron tunneling channel emerges on the Fermi level, and the Kondo effect appears in the differential conductance. The current correlation of two current operators ˆI γ (t ) and
ˆI γ (t ) at different times t and t is settled by Πγ γ (t , t ) = δ ˆI γ (t )δ ˆI γ (t ) + δ ˆI γ (t )δ ˆI γ (t ) , where δ ˆI γ (t ) = ˆI γ (t ) − ˆI γ (t ) is the current fluctuation operator. The symbol · · · in above formula
denotes the ensemble average over the system. Substituting the current operator (11) into the definition of current correlation, we have the current correlation [38] associated with the two times t and t as
2 e h
d1 d2 exp
× exp γβ
i h¯
i h¯
nm n m α1 α2 σ
nm ε12 t
nm ε21 t J n (Λ) J m (Λ) J n Λ J m Λ
γ β m
n
m
where the joint Fermi distribution function is involved by the definition F α β ( , ) = f α ( )[1 − f β ( )] + [1 − f α ( )] f β ( ). The shot noise spectrum S γ γ (Ω) is determined by making Fourier transformation over the current correlation [20,21] Πγ γ (t , t ) via the re-
lation S γ γ (Ω)δ(Ω + Ω ) = 21π Πγ γ (Ω, Ω ). Correspondingly, the shot noise spectrum is evaluated as
e2 h
h
nm n m
d J n (Λ) J m (Λ) J n Λ
σ
d J n (Λ) J m (Λ) J n Λ
nm n m α1 α2 σ
γβ
γ β × J m Λ A α1 α2 n , ˜ n A α2 α1 ˜ m , m
× F α1 α2 , ˜ (n−m) ,
2
α
1
m G r n 2 G r m 2 + W n W dσ dσ 4 2 + Sb W n G dr σ n δΛ 0 + S b δΛ0 δΛ 0
(n−m) , × Fγ β ,
(17)
√
[(1 − 2T b ) A ( )] where the notation W ( ) = Γ w σ ( ) − S b cos(ϕ )Γ is utilized in the formula. Here δΛ0 signifies that the magnitude of applied field is zero in this term, i.e., J n (Λ)δΛ0 = 1. As direct tunneling is disconnected by setting W = 0, we obtain the photon-assisted resonant transport shot noise. As the resonant tunneling is disconnected by setting Γ = 0, we have the direct tunneling shot noise. Eq. (17) is the main result of the shot noise for our strongly correlated system, where the Coulomb interaction is U → ∞. It cannot be reduced to the shot noise of the noninteracting QD system due to the overlap and interactions between the local electrons in the QD and tunneling electrons from the terminals. The Kondo and Fano effects are combined efficiently resulting from Kondo resonant tunneling and Fano interfering correlation of electrons. Since the occupation number and Fermi distribution function are involved in the noise formula, one should solve the formula self-consistently. 3. Numerical results
× A α1 α2 1n , 2 A α2 α1 2 , 1 F α1 α2 (1 , 2 ), (15)
S γ γ (Ω) =
1 n m
× J m Λ F αα , (n−m) Tσ Tσ
1
Πγ γ t , t =
2e 2
ΓA ( )G r ( )2 Y σ ( ) + T b δΛ0 , T σ ( ) = Γ dσ
γ β
γβ
ing the explicit coefficients A α1 α2 ( n , ˜ n ) and A α2 α1 (˜ m , m ) given in Eq. (12) into the noise formula (16). We now consider the zerofrequency shot noise in the same terminal by setting Ω = 0 to find
(16)
where ˜ n = n + h¯ Ω , and n − m + n − m = 0 is satisfied to indicate the relation of absorption and emission of photons. The way
The numerical calculations are performed in this section for the symmetric situation as Γ L = Γ R = Δ, Λ L = Λ R = 1.0, and we take Δ as the energy scale by choosing the common parameters as εd = 0, k B T = 0.0008Δ. We set Δ = 1 in the numerical evaluations and the following expressions. The differential conductance, shot noise, and derivative of shot noise are scaled by G 0 = 2e 2 /h, S 0 = 4e 2 Δ/h, and 4e 3 /h, respectively. We depict the differential conductance versus source-drain bias in Fig. 2 to investigate the influence of applied ac fields and the direct tunneling strength. Fig. 2(a) shows the variation of differential conductance G with respect to different photon energy h¯ ω as W = 0.05. Single Kondo resonant peak is located at eV = 0 when h¯ ω = 0. The side resonant peaks appear around the Kondo resonant peak due to the absorption and emission of photons, and the height of Kondo peak is suppressed due to the photon absorption. The suppression of the Kondo effect in a resonant tunneling QD by external irradiation has been reported by Kaminski et al.
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
393
Fig. 2. (Color online.) The conductance versus source-drain bias eV at ϕ = 0. (a) The conductance at eV g = 3.0 for different photon energy h¯ ω = 0, 0.4, 0.8. (b) The conductance at h¯ ω = 0.8 for different W as W = 0.05, 0.2, 0.4, 0.6, 0.8, 1.0.
Fig. 4. (Color online.) The shot noise S, derivative of shot noise dS /deV g , and Fano factor F versus gate bias eV g for the unbalanced case. The other parameters are chosen as ϕ = 0, W = 0.05, eV = 1.0.
Fig. 3. (Color online.) The shot noise S, derivative of shot noise dS /deV g , and Fano factor F versus gate bias eV g for the balanced case. The other parameters are chosen as ϕ = 0, W = 0.05, eV = 1.0.
in Ref. [39], where the shape of resonant peak in the conductance has been studied. Fig. 2(b) exhibits the modification of differential conductance with respect to changing the amplitude of direct transmission strength W . The Kondo resonant peak is obviously suppressed, and the Fano resonance becomes more evident by increasing W from W = 0 to 1.0. As W is strong enough (W ≈ 0.6) the Kondo peak can be completely smeared, while the Fano dip appears obviously around eV = −5. This indicates that the electron tunneling through QD induces the Kondo resonance effect, while the direct tunneling results in non-resonant tunneling. The competition of interference and Kondo resonance generates the Fano–Kondo cooperated effect in the differential conductance. We present the variations of shot noise S, derivative of shot noise dS /deV g , and Fano factor F = S /(2eI ) versus gate bias eV g for the balanced case in Fig. 3. The asymmetric behavior exhibits obviously in the shot noise, where the photon induced resonant
peak–valley structure is located around the main valley around eV g = 0. Away from the central valley regime, the shot noise exhibits left lower and right higher asymmetric structure. The value of central dip is about S ≈ 0.02S 0 , while the maximum value on the right peak is about S ≈ 0.18S 0 with the variation rate 89% shown by diagram Fig. 3(a). This asymmetric effect is generated from the interference and correlation of photon-assisted electron tunneling in different arms. Electrons traveling in the resonant arm are adjusted by the QD and possess resonant energy level with the QD, while electrons traveling through the direct tunneling arm possess continue spectrum with the terminals. When the electrons with deferent phase characteristics meet in the device, destructive and constructive interference occur, and the asymmetric behavior of shot noise appears. The derivative of shot noise versus eV g shown in Fig. 3(b) exhibits photon-assisted peaks corresponding to the drastic variation of shot noise in Fig. 3(a). The main peak is located around eV g = 0, and the side photon-induced peaks located at the positions eV g = ±¯hω display asymmetric structure. The Fano factor shown in Fig. 3(c) is suppressed to F ≈ 0.1 at the central valley around eV g = 0. The side peak–valley structure displayed around the central valley is induced by the photon absorption and emission effect. The saturate values of the Fano factor show the left higher and right lower appearance. The Fano factor exhibits sub-Poissonian noise feature with F < 1 in the whole regime of eV g . This means that coherent current correlation for the balanced case suppresses the shot noise, and the competition of Fano–Kondo resonant tunneling results in the asymmetric characteristics of shot noise and Fano factor. Fig. 4 presents the shot noise, derivative of shot noise, and Fano factor versus gate bias eV g for the unbalanced case. The resonant peak appears at eV g = 0 with height S ≈ 0.5 ∼ 0.7 when the photon energy reduces from 5 to 3 shown in Fig. 4(a). The left and right valleys exhibits asymmetric behavior, where the right valley is suppressed to zero as h¯ ω = 3. This signifies that the destructive interference effect is considerably affects the appearance of shot noise, where the photon energy takes important role. The derivative of unbalanced shot noise displays the peak–valley structure
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Fig. 5. (Color online.) The shot noise S and Fano factor F versus AB phase ϕ for the balanced case. The other parameters are chosen as eV = 1.0, eV g = 0, and h¯ ω = 3.0.
shown in Fig. 4(b), and the appearance of peak–valley structure is left- and right-shifted due to increasing photon energy. The enhancement of shot noise results in the Fano factor exhibiting a resonant peak F ≈ 2.7 at about eV g = −2 for h¯ ω = 3 shown in Fig. 4(c), and the suppression of shot noise exhibits the Fano factor valley F = 0 at about eV g = 5 (the dotted line). This signifies that the enhancement and complete suppression of shot noise can be achieved by adjusting the gate voltage and photon energy h¯ ω . The enhancement and suppression of shot noise are contained in this case due to the competition of incoherent and destructive interference effects. For the unbalanced case, the absorbed photons in one current are not equal to the emitted photons in the other current. The currents correlate to induce energy accumulation in the joint Fermi function F γ β ( , (n−m) ), and this effect generates the excess shot noise even when the source–drain bias is removed at zero temperature. We present the variation of balanced shot noise and Fano factor with respect to AB phase ϕ in Fig. 5. The periodic oscillation with period 2π appears in both of the shot noise and Fano factor. The peaks are located at ϕ = (2n + 1)π , and the valleys are located at ϕ = 2nπ (n = 0, ±1, ±2, . . .). The shot noise and Fano factor do not increase monotonically with increasing the value of W . One observes that as W is strong (W = 0.6), the peaks are suppressed to show plateaus S ≈ 0.2, F ≈ 0.28, and flat valleys S ≈ 0, F ≈ 0 with wide regime around | ϕ |< 0.7π in a period. When W is given by a moderate value as W = 0.2, we have large oscillation peaks as S ≈ 0.52, and F ≈ 0.8. This suppression and enhancement of shot noise indicates that by adjusting W and ϕ one can control the magnitude and shape of shot noise to obtain zero and large noise circumstances. We display the shot noise S and Fano factor with respect to W for the balanced case by choosing separate photon energy in Fig. 6. The shot noise and Fano factor increase as W increases rapidly to the maximum values, and then they decline mildly to zero. The maximum values exhibit at about W = 0.18 for the shot noise, and at W = 0.12 for the Fano factor. This effect signifies that the direct tunneling may induce the enhancement and suppression of shot noise and Fano factor due to changing the parameter W . The maximum values also increase with increasing of photon energy from h¯ ω = 3.0 to h¯ ω = 5.0. Similar appearances display in the shot noise and Fano factor corresponding to changing W with different magnetic phase
Fig. 6. The shot noise S and Fano factor F versus W for the balanced case with different photon energy h¯ ω . The other parameters are chosen as eV = 1.0, eV g = 0, and ϕ = 0.
Fig. 7. The shot noise S and Fano factor F versus W for the balanced case with different AB phase ϕ . The other parameters are chosen as eV = 1.0, eV g = 0, and h¯ ω = 3.0.
ϕ shown in Fig. 7. As the magnetic phase ϕ increases from ϕ = 0 to π , the shot noise increases from S ≈ 0.1 to 0.5, while
the Fano factor increases from F ≈ 0.2 to 0.85 with the changing rates 80% and 76%, respectively. The direct tunneling takes main contribution to shot noise and Fano factor when W = 1, while the resonant tunneling is the sole contribution when W = 0. The competition of resonant and direct tunneling currents correlation arouses significant noise variation and configuration. This indicates that by adjusting W one can obtain extreme and minimum values of shot noise and Fano factor for noise controlling in electronic device designing. Compared with the shot noise of the noninteracting QD system (see Ref. [30]), the peak–valley structure behaves quite differently here due to the strong Coulomb interaction. The modification of
H.-K. Zhao, W.-K. Zou / Physics Letters A 379 (2015) 389–395
shot noise structure appears intimately related to the Kondo effect in finite temperature, where novel channel emerges on the Fermi level. The current correlation is associated with the interference of electrons from this channel to the electrons with continuous spectrum, which results compound effect as the Fano–Kondo resonance. The symmetry of phase locking as ϕ → −ϕ is kept, however, the plateaus and flat valleys with period of 2π may be generated. Shot noise reflects the higher order particle transport generated from the current correlation, and it reveals the statistical features of particles as the super-Poissonian and sub-Poissonian distributions. The time-averaged current is zero for our system, but the current correlation is nonzero due to the perturbation of external ac fields. This also induces the strong correlations of tunneling electrons in the Kondo channel and the continuous channels. The Fano–Kondo appearance of shot noise reflects the hidden physics, which does not displays in the current characteristics. The deviation of current and shot noise is significantly revealed by the characteristics of Fano factor. 4. Concluding remarks The shot noise and Fano factor of a QD embedded AB interferometer under the perturbation of microwave fields have been investigated. The frequency-dependent shot noise formula is derived explicitly by evaluating the current operator correlation through EOM method. The competition of Kondo resonant and non-resonant direct tunnelings results in photon-assisted Fano– Kondo resonant transport, and the suppression of Kondo peak appears due to increasing the direct tunneling strength. The interference and correlation of photon-assisted electrons induce the asymmetric resonant peak–valley behavior of shot noise. The enhancement and suppression of shot noise are derived due to the competition of incoherent and destructive interference effects, and super-Poissonian and sub-Poissonian noise can be adjusted by the applied photon irradiation, gate bias, and direct tunneling strength. The periodic oscillation with period 2π appears in both of the shot noise and Fano factor, where plateaus and flat valleys are observed. The direct tunneling may induce the enhancement and suppression of shot noise and Fano factor due to changing the direct tunneling parameter. Acknowledgement We gratefully acknowledge the support by the National Natural Science Foundation of China under the Grant No. 11175015.
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