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Transport properties of a double quantum dot molecule in the presence of impurity effects Y.S. Liu a,∗ , X.F. Yang b , X. Zhang a , Y.J. Xia a a College of Physics and Engineering, Qufu Normal University, Qufu 273165, People’s Republic of China b National Lab for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, 500 Yutian Road,

200083 Shanghai, People’s Republic of China Received 15 January 2008; accepted 25 January 2008 Available online 1 February 2008 Communicated by V.M. Agranovich

Abstract In this Letter, we studied the electronic transport through a parallel-coupled double quantum dot (DQD) molecule including impurity effects at zero temperature. The linear conductance can be calculated by using the Green’s function method. An obvious Fano resonance arising from the impurity state in the quantum dot is observed for the symmetric dot-lead coupling structure in the absence of the magnetic flux through the quantum device. When the magnetic flux is presented, two groups of conductance peaks appear in the linear conductance spectra. Each group is decomposed into one Breit–Wigner and one Fano resonances. Tuning the system parameters, we can control effectively the shapes of these conductance peaks. The Aharonov–Bohm (AB) oscillation for the magnetic flux is also studied. The oscillation period of the linear conductance with π , 2π or 4π may be observed by tuning the interdot tunneling coupling or the dot-impurity coupling strengths. © 2008 Elsevier B.V. All rights reserved. PACS: 73.23.-b; 73.63.Kv; 73.40.Gk; 85.35.Be Keywords: Double quantum dot molecule; Fano effects; Impurity effects

1. Introduction The transport properties of an Aharonov–Bohm (AB) ring including one quantum dot on each arm have attracted wide attention in recent years. An oscillating current as a function of the magnetic flux has been detected experimentally [1]. Today, the Fano resonance arising from quantum interference between resonant and non-resonant processes is used to be a good probe for the phase coherence [2]. A lot of theoretical works on Fano effects in the parallel double quantum dots system have been reported [3–7]. The interdot tunneling coupling always results in one weakly-coupled electronic state and one strongly-coupled electronic state when the double quantum dots are coupled to leads. A Fano resonance happens when the Fermi energy sweeps the weakly-coupled electronic state, and a Breit–Winger resonance is centered at the strongly-coupled electronic state [3]. A swap effect between the Breit–Winger resonance and the Fano resonance can be developed by tuning the magnetic flux through the quantum ring consisting of two parallel-coupled DQD molecule [6]. Such a quantum model has become interest candidate in quantum computation and quantum information domains. Especially, a quantum controlled-NOT (CNOT) is proposed by using the electron orbit states in a DQD system [8]. It is very difficult to fabricate two clean quantum dots in the experiments due to irregularities and defect in the QD system. Some localized states often appear in the quantum dot system, which are hybridized with the quantum-dot levels. But they are not * Corresponding author. Tel.: +86 0537 4446096 321.

E-mail address: [email protected] (Y.S. Liu). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.01.049

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Fig. 1. Schematic plot of double quantum dot system including impurity effects with interdot tunneling t and tunneling coupling between the j th quantum dot and the lead α Γjα (j = 1, 2; α = L, R). λ represents the dot-impurity coupling strengths.

coupled directly to the leads. We shall call the localized states as impurity states. Recently, Wu et al. [9] studied the phonon-assisted through a single quantum dot with impurity scattering effects. The results show that impurity states have weak influence on the suppression of the shot noise. More recently, Tanaka et al. [10] demonstrated the trapping of a conduction electron between two identical impurities in a quantum-dot array. To date, the AB ring consisting of two clean parallel-coupled quantum dots have been studied widely [3,6]. In this Letter, we study the transport properties of the parallel-coupled DQD molecule including impurity effects in the case of symmetric dot-lead coupling. For simplicity, we may ignore the intradot and interdot Coulomb interactions. The transport properties of the quantum system are modified by the impurity scattering effects in the quantum dot. At the zero-magnetic-case, the linear conductance spectra can be decomposed into one Breit–Wigner peak and one Fano peak. Two groups of conductance peaks appear in the conductance spectra when the magnetic flux through the quantum device is presented. Each group consists of one Breit– Winger peak and one Fano peak due to quantum interference effects. These conductance peaks can be effectively controlled by using the interdot tunneling coupling or the interacting strength of dot-impurity. The AB oscillation for the magnetic flux is also studied. The oscillation period of the linear conductance with π , 2π or 4π may be observed by tuning the interdot tunneling coupling or the dot-impurity coupling strengths. 2. Model and method In this Letter, we consider a parallel-coupled double-dot model enclosing a magnetic flux Φ including impurity effects as shown in Fig. 1. For simplicity, we take only one quantum level into account, and only one isolated impurity state in each quantum dot is included. The total Hamiltonian describing the quantum system is written as H = Hleads + HDQD + HT ,

(1)

where the first term Hleads in Eq. (1) describes the left and right leads in the noninteracting electron approximation † Hleads = αk aαk aαk ,

(2)

α=L,R;k † (aαk ) denotes the creation (annihilation) operator for an electron with energy αk in lead α. The second term in Eq. (1) dewhere aαk scribes the dynamics of parallel-coupled double quantum dot molecule including impurity scattering effects, which can be modelled by using a four-site Hamiltonian † dj cj + cj† dj , j dj† dj + fj cj† cj − t d1† d2 + d2† d1 − λ HDQD = (3) j

where

dj†

j

j

(dj ) creates (annihilates) an electron with the energy j in the j th quantum dot. t describes the interdot tunneling

coupling, for convenience, which is taken as a real number. cj† (cj ) denotes the creation (annihilation) operator of an impurity state with energy fj in quantum dot j , and λ represents the coupling strength between the quantum dot energy level and the impurity

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state. For simplicity, we consider the case of the same dot-impurity coupling strength in each quantum dot. The third term in Eq. (1) describes the tunneling coupling between two quantum dots and the leads, which is divided into two parts HT = HTL + HTR .

(4)

HTα describes the tunneling coupling between the isolate double coupled quantum dots and the electrode α HTα = Vα1 d1† + Vα2 d2† aαk + H.c. .

(5)

k,α

A phase factor is attached to Vαj (j = 1, 2; α = L, R) in the presence of the magnetic flux, and they can be written as VL1 = |VL1 |eiφ/4 , VL2 = |VL2 |e−iφ/4 , VR1 = |VR1 |e−iφ/4 , VR2 = |VR2 |eiφ/4 with the AB phase φ = 2πΦ/Φ0 and the flux quantum Φ0 = h/e. In the presence of the magnetic flux, the linewidth matrices Γ L and Γ R are given L Γ Γ1L Γ2L eiφ/2 1 L Γ = (6) Γ1L Γ2L e−iφ/2 Γ2L and

ΓR =

Γ1R

Γ1R Γ2R e−iφ/2

Γ1R Γ2R eiφ/2

(7)

Γ2R

with the linewidth matrix Γjα = k |Vαj |2 2πδ( − αk ) (α = L, R; j = 1, 2). The total linear conductance G is related to the total transmission T () by the following formula [11] e2 2e2 L r (8) T () = Tr Γ G ()Γ R Ga () . h h In order to obtain the linear conductance G, we need to know the retarded Green’s function Gr (). Gr () is the Fourier transform of the retarded Green’s function of the double quantum dot molecule Gr (t − t ), which is defined by

Gr (t − t ) = −iθ(t − t ) Ψ (t), Ψ † (t ) , (9) G=

where the operator Ψ † = (d1† , d2† ). g r is defined as the Fourier-transformed retarded Green’s function of the center region without the coupling to two leads and impurity effects, which is written as r −1 − 1 + i0+ t = . g () (10) t − 2 + i0+ By employing the matrix Dyson’s equation, the retarded Green’s function Gr () can be written as −1

−1 Gr () = g r () − ΣTr () − ΣIrM ()

(11)

where ΣTr is the retarded self-energy matrix from the tunneling coupling between the DQD molecule and the leads. Under the wide-bandwidth approximation, one can obtain i L Γ +ΓR 2 denotes the effective self-energy matrix from impurity effects in quantum dots λ2 0 + −f +i0 1 ΣIrM () = . λ2 0 −f +i0+ ΣTr () = −

ΣIrM

(12)

(13)

2

In this work, we only consider the case of the symmetric tunneling coupling between the double quantum dots and leads. So we set system parameter Γjα = Γ0 (j = 1, 2; α = L, R). Combine Eqs. (10), (11) and (12), we arrive at Gr11 () =

− 2 − ( − 1 −

Gr12 () = Gr21 () =

λ2 −f1

+ iΓ0 )( − 2 −

λ2 −f2 λ2 −f2

+ iΓ0 + iΓ0 ) − [t + iΓ0 cos(φ/2)]2

,

−t − iΓ0 cos(φ/2) [ − 1 −

λ2 −f1

+ iΓ0 ][ − 2 −

λ2 −f2

+ iΓ0 ] − [t + iΓ0 cos(φ/2)]2

(14) (15)

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Fig. 2. The linear conductance as a function of EF in the absence of the magnetic flux. (a) 1 = 2 = f1 = f2 = 0; (b) 1 = 2 = f1 = f2 = 0; (c) 1 = 2 = 0; Other system parameters are shown in Fig. 2.

and Gr22 () =

− 1 − [ − 1 −

λ2 −f1

+ iΓ0 ][ − 2 −

λ2 −f1 λ2 −f2

+ iΓ0 + iΓ0 ] − [t + iΓ0 cos(φ/2)]2

.

(16)

When the retarded Green’s function is obtained, matrix elements of the advanced Green’s function can be obtained by the relation Ga ()ij = [Gr ()j i ]∗ . Using Eq. (8), we can calculate the linear conductance of the double quantum dots at zero temperature. 3. Results and discussions In the following numerical calculations, Γ0 is chosen as the energy unit. When the impurity effects on each quantum dot are ignored, such a quantum model has been studied extensively in some previous literatures [3–6]. In this Letter, we pay our attention to the effects of impurity effects on the linear conductance. First we study the linear conductance of the double quantum dots in the absence of the magnetic flux and in the symmetric tunneling coupling (ΓL = ΓR ). Through the Letter, the energy level, the Fermi energy, the tunneling coupling strength are measured by using units of Γ0 . For convenience, we set the energy levels in the quantum dots and energy levels arising from impurity state are aligned with one another. When impurity effects in the quantum dots are ignored, the width of the Fano resonance becomes zero. The reason is that the antibonding state is decoupled from the two leads due to the destructive quantum interference [4]. So we see that only a Breit–Wigner peak with the symmetric structure is centered at −t in the linear conductance spectra as shown in Fig. 2(a) (solid line). But the clean quantum dot is difficult to be fabricated in the experimentations. We need to pay some attention to impurity effects on the transport properties. Fig. 2(a) shows the linear conductance G as a function of the Fermi energy EF with several different dot-impurity coupling strengths. The results show that a striking linear conductance dip (G = 0) is observed

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Fig. 3. (a) The linear conductance as a function of EF in the absence of the magnetic flux under several different values of f2 . (b) Energies of four molecule states as a function of f2 . Form low to top, the cures corresponding to the ground state, the first excitated state, the second excitated state and the third excitated state, respectively. Other parameters are chosen as t = λ = Γ0 and 1 = 2 = f1 = 0.

when the Fermi energy sweeps across the impurity states. The conductance dip arises from quantum interference effects between the two different paths. For example, due to impurity states, the electron has a quantum dot. Two √ two different paths through √ conductance peaks with the unitary limit value 2e2 / h appear around (−t − t 2 + 4λ2 )/2 and (−t + t 2 + 4λ2 )/2, respectively. With λ increasing, two conductance peaks move in the opposite direction and the zero conductance point is invariable. In Fig. 2(b), the linear conductance as a function of the Fermi energy EF is plotted under several different interdot tunneling couplings with the fixed λ. We note that the conductance peak move in the same direction with t increasing, which is different from the case shown in Fig. 2(a). The left Breit–Wigner peak becomes wider, while the right Fano peak becomes shaper. The zero conductance point is still fixed at the original position. Fig. 2(c) shows the case when the energy levels of the impurity states are not aligned with the energy levels of the quantum dots. With the energy levels of the impurity states increasing, the conductance dip (zero conductance point) moves in the right direction. The results show that the conductance dip(G = 0) appears when the Fermi energy sweeps across the energy levels of the impurity states in this case. The linear conductance G as a function of EF under several values of f2 is plotted in the left plane of Fig. 3. For simplicity, we only tune the energy of the impurity state in the 2th quantum dot. The results show that the linear conductance spectra is decomposed into one Breit–Wigner peak and three Fano peaks when f2 = 0. When the energy of the impurity state in the 2th quantum dot is increased, the ground state energy has almost no changes. The first excitated state energy and the second excitated state energy increase slightly, while the third excitated state energy increases very fast as shown in the right plane in Fig. 3. When the Fermi energy EF sweeps across four molecule states with different effective couplings to leads, one Breit–Wigner peak and three Fano peaks appear in turn. The reason is that there are four different electron transmission paths in the device, Fano interference effects happen when EF sweeps across weakly-coupled molecule states. From the left plane in Fig. 3, one can see that Breit–Wigner peak almost never changes, and the first Fano and the second Fano peaks move slightly in the right direction. The third Fano peak has an obvious motion in the right direction. Now we study the effects of the magnetic flux Φ on the linear conductance spectra of the double quantum dots model including the impurity effects. The magnetic flux parameter φ is chosen as 0.2π , and other parameters are same with Fig. 2(a). It is noted that the linear conductance spectra may be decomposed into two groups of conductance peaks, of one √ √ and each group consists 2 2 2 2 Breit–Wigner peak and one Fano √ peak. In this case, we may obtain four molecule states (−t − t + 4λ )/2, (t − t + 4λ )/2, √ 2 2 2 2 (−t + t + 4λ )/2 and (t + t + 4λ )/2, respectively. To obtain a clear insight into the influences of system parameters λ and t in the presence of the magnetic flux. In Fig. 4(a), we plot the linear conductance G as a function of EF under several different dot-impurity coupling strengths with the fixed interdot tunneling coupling strengths. With λ increasing, two group peaks move in the opposite direction, and the distance between the Breit–Wigner peak and Fano peak in each group is invariable. From the above given four molecule states, we can derive easily that the distance between the Breit–Wigner peak and Fano peak is about t. Fig. 4(b) shows the linear conductance as a function of the Fermi energy EF under several different interdot coupling strengths with the fixed dot-impurity coupling strengths. The results show that the Breit–Wigner peak and the Fano peak in each group move in the opposite direction with t increasing. It is well known that the magnetic flux through the quantum ring results in an oscillating behavior for the linear conductance G. An oscillating current had been detected experimentally by tuning the magnetic flux [1]. In some theoretical works, 2π -period oscillation without the interdot tunneling coupling and 4π -period oscillation with the interdot tunneling coupling have been reported. The linear conductance G as a function of the magnetic flux φ is plotted in Fig. 5. For convenience, we set the energy levels in the

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Fig. 4. The linear conductance as a function of EF in the presence of the magnetic flux φ = 0.2π . (a) The case with several different values of dot-impurity coupling strength with the fixed interdot tunneling coupling. (b) The case with several different values of interdot tunneling coupling strength with the fixed dot-impurity coupling strength.

quantum dots and the energy levels arising from impurity state are aligned 1 = 2 = f1 = f2 = 0. The Fermi energy EF is fixed at 3Γ0 . We plot the AB oscillation as a function of φ in the presence of several interdot tunneling coupling strengths with the fixed dot-impurity coupling strength λ = 3Γ0 . The results show that the oscillations have sinusoidal behavior with 2π -period, and the amplitude of the oscillations and the linear conductance G are suppressed with t increasing as shown in the upper plane in Fig. 5. In this case, the linear conductance can be written as G(φ) =

4t 2 Γ02 2e2 . h t 4 + 3t 2 Γ02 + Γ04 cos(φ/2)4 + (t 2 − Γ02 )Γ02 cos(φ)

(17)

From the above equation, we see G is a period for φ with 2π -period. When cos(φ/2) = 0 (φ = (2n + 1)π, n = 0, 1, 2, . . .), the linear conductance is simplified as G=

4t 2 Γ02 2e2 . h t 4 + 3t 2 Γ02 − (t 2 − Γ02 )Γ02

(18)

The results show that the linear conductance reach the maximal value G = 2e2 / h when t = Γ0 . The linear conductance as a function of t with the fixed magnetic flux φ = π is plotted in Fig. 5(b). The dependence of the linear conductance on the interdot tunneling coupling t first increases monotonously, then arrives it is maximal value 2e2 / h when t = Γ0 . The linear conductance is suppressed when t > Γ0 . In the case of t 2 = G(φ) =

16 2e2 . h 17 + cos(2φ)

Γ02 2 ,

Eq. (17) is written as

(19)

This result shows the linear conductance for φ with π -period as shown in Fig. 5(b) (solid line). When the dot-impurity coupling departs from the Fermi energy, the 4π -period oscillation returns back as shown in the low plane in Fig. 5(c). The result offers a method to measure the dot-impurity coupling strengths by using AB oscillation for φ. When the Fermi energy EF is aligned with the dot-impurity coupling strengths, the AB oscillation for the magnetic flux φ has 2π -period. The 4π -period of the linear conductance for the magnetic flux φ is observed in the case of EF = λ.

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Fig. 5. AB oscillation for φ (a) several different the interdot tunneling couplings t Γ0 with fixed dot-impurity coupling and the Fermi energy EF (λ = EF = 3Γ0 ). (b) Several weak interdot tunneling couplings. Other parameters are chosen in Fig. 5. (c) Several different values of dot-impurity coupling strengths.

4. Summary In summary, we investigate theoretically the linear conductance of a parallel-coupled DQD molecule including impurity effects attached to leads by using the nonequilibrium Green’s function method. The transport properties of the quantum system can be modified by the impurity scattering in the quantum dot. At the zero-magnetic-case, the linear conductance spectra is decomposed into one Breit–Winger peak and one Fano peak. Two groups of conductance peaks appear when the magnetic flux is presented. Each group of peaks consists of one Breit–Winger peak and one Fano peak due to quantum interference effects. The AB oscillation for the magnetic flux is also studied. The results show that the AB oscillating period is sensitive to the interdot tunneling coupling, dot-impurity coupling strengths and the Fermi energy. Acknowledgements The work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 10774088. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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