Tunable supercurrent in a triangular triple quantum dot system

Physics Letters A 374 (2010) 2584–2588

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

Tunable supercurrent in a triangular triple quantum dot system Long Bai a,b , Qingyun Zhang a , Liang Jiang a , Zhengzhong Zhang a , R. Shen a,∗ a b

National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China College of Science, China University of Mining and Technology, Xuzhou 221008, China

a r t i c l e

i n f o

Article history: Received 3 March 2010 Accepted 9 April 2010 Available online 18 April 2010 Communicated by R. Wu Keywords: Supercurrent Quantum dot

a b s t r a c t The supercurrent in a triangular triple quantum dot system is investigated by using the nonequilibrium Green’s function method. It is found that the sign of the supercurrent can be changed from positive to negative with increasing the strength of spin-flip scattering, resulting in the π -junction transition. The supercurrent and the π -junction transition are also modulated by tuning the system parameters such as the gate voltage and the interdot coupling. The tunable π -junction transition is explained in terms of the current carrying density of states. These results provide the ways of manipulating the supercurrent in a triple quantum dot system. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Quantum interference effects in electronic transport through coupled quantum dot (QD) systems are of great interest from both theoretical and potential application points of view. The important feature of preserved coherence of electrons in the QD systems, such as the Aharonov–Bohm (AB) effect and the Fano effect, has been widely observed [1–5]. Recently, an AB interferometer containing two QDs or more than two QDs, has attracted much attention [6–10]. Different from the AB interferometer with single QD, the QDs are coupled here and their energy levels are mixed, which makes the quantum transport phenomena rich and varied. It is possible to manipulate each of the QDs separately, which enlarges the dimension of the parameter space for the transport properties in contrast to the single QD AB interferometer. More recently, a triangular triple QD (TTQD) system in magnetic field has been widely studied both theoretically and experimentally [11,12]. Delgado et al. put forward a theory of spin-selective AB oscillation in the TTQD system where the current depends not only on the flux but also on the relative orientation of the spin of the incoming and the localized electrons [13]. Kuzmenko and coworkers analyzed the physics of tunneling in a TTQD structure under a perpendicular magnetic field and demonstrated a unique combination of Kondo and AB properties owing to an interplay between the continuous symmetry, the discrete symmetry and the U (1) gauge invariance [14]. Dinu et al. studied the interplay between the Kondo correlation and the interference effect in a TTQD system, which gives rise to different temperature behaviors depending on the gate voltage [15]. Consequently, the TTQD system includes rich physics and may be an important molecule device in the future.

*

Corresponding author. Fax: +86 (25) 83595535. E-mail address: [email protected] (R. Shen).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.031

On the other hand, it is well known that the Josephson effect is an important macroscopic quantum phenomenon including phase coherence between two superconductors (SCs). The Josephson junction can be applied as superconducting qubit and as nanoscale quantum interference device for detecting weak magnetic flux change. In recent years, there has been a growing interest in the mesoscopic SC/normal-metal/SC Josephson junctions both experimentally and theoretically [16–25]. At the normalmetal/SC interface, when an electron is incident from the normal metal with energy below the superconducting gap, a Cooper pair is transferred into the SC with a hole reflected, known as Andreev reflection (AR). In mesoscopic SC/QD/SC system, a supercurrent is conducted through the central region by the AR process, where two superconducting electrodes serve as two ‘mirrors’, reflecting an electron into a hole and a hole into an electron. In the ballistic systems, Andreev bound states can be formed [26] and each Andreev bound state carries a supercurrent in the positive or negative direction at a given phase difference ϕ between the two SCs. When two SCs are weakly linked, the current-phase relation (CPR) is I = I c sin ϕ with I c being the critical current. In some cases, the sign of I c can be reversed, leading to the so-called π junction [27–32]. Pan and Lin studied the supercurrent through parallel-coupled double QDs connected to two SC leads and found that the sign of the critical current can be controlled by tuning the interdot coupling, the dot energy levels and the magnetic flux threading the ring connecting dots and leads [33]. In addition, the π -junction transition is also related to the spin-flip scattering and the Coulomb repulsion in the QD system [34]. Nevertheless, the supercurrent in the TTQD system connected by two SC leads has not been studied in detail. The knowledge of the 0–π transition and the modulation of the system parameters on the supercurrent in a TTQD system is still insufficient. In this work, the Josephson current in an SC/TTQD/SC system is investi-

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where σˆ z is a 12 × 12 matrix with the Pauli matrix σz as its diagonal components. The 12 × 12 Nambu representation is utilized to include the physics of the Andreev reflection. The retarded and lesser Green’s functions are defined as G r (t , t  ) = −i θ(t − t  ) × {Ψ (t ), Ψ † (t  )} and G < (t , t  ) = i Ψ † (t  )Ψ (t ), respectively, with †

†

†

†

†

the operator Ψ † = (d1↑ , d1↓ , d1↓ , d1↑ , d2↑ , d2↓ , d2↓ , d2↑ , d3↑ , d3↓ ,

Fig. 1. A schematic diagram of the TTQD system connected with two SC leads. The magnetic flux threading the ring connecting three dots is Φ .

† d 3↓ , d 3↑ ) .

gated theoretically by using the nonequilibrium Green’s function techniques. Compared with the single QD structure, the TTQD system provides more tunable system parameters to control the electronic transport behaviors. The spin-flip effects on the supercurrent has been considered. It is shown that the π -junction transition can be controlled by tuning the strength of the spin-flip scattering, the gate voltage on the QD, the interdot coupling and the magnetic flux. The π -junction transition is also interpreted on grounds of the current carrying density of states. Our results provide the ways of manipulating the supercurrent in the TTQD system. The rest of this Letter is organized as follows. The formulism is present in Section 2 and the numerical results is discussed in Section 3. Finally, we briefly summarize our results in Section 4.

Iν =

In the following, we need to obtain the retarded Green’s function G r ( ) and lesser Green’s function G < ( ). The retarded Green’s function can be calculated by using the Dyson equation

2. Model description

G r ( ) = G r0 + G r ( )Σ r ( )G r0 ,

The SC/TTQD/SC system considered in this Letter is shown in Fig. 1. The system Hamiltonian is given by

where G r0 is the retarded Green’s function for the isolated TTQD, and Σ r = Σ Lr + Σ Rr is the self-energy. In the Nambu representation, G r0 can be written as



H=

Hν + H D + H T .

(1)

ν =L, R

The first term H ν (ν = L , R) is the standard BCS Hamiltonian for the superconducting leads with the macroscopic phase ϕν , the energy gap , and the single particle energy ν ,k measured from the Fermi level, which reads

Hν =



†

ν ,k C ν ,kσ C ν ,kσ +



k,σ

e

− i ϕν

†



†

C ν ,k↑ C ν ,−k↓ + H.c. .

(2)

k

By taking the Fourier transformation, the current formula may be written as

2e h¯

HD =

i ,σ

−





† te i φ d1σ d2σ

† + td2σ d3σ

  † + td3σ d1σ + H.c.

σ

+R

† d 1↑ d 1↓

† + d 2↑ d 2↓

† + d 3↑ d 3↓

 + H.c. .

(3)

Here, i is the energy level of dot i (i = 1, 2, 3), t is the interdot hopping, phase φ = 2π Φ/φ0 comes from the magnetic flux Φ threading the ring connecting three dots and φ0 = h/e is the unit quantum flux, and R is the strength of the spin-flip scattering. The third term in Eq. (1), H T , is the tunneling Hamiltonian and is given by

HT =





†

V ν C ν ,kσ d1σ + H.c. ,

(4)

ν ,kσ

where V ν describes the tunneling coupling between dot 1 and lead ν . According to the standard nonequilibrium Green’s function techniques, the supercurrent flowing through the lead ν can be expressed in terms of the Green’s function of the QDs as

I ν (t ) =

2e h¯



2π



 















dt  Tr σˆ z G < t , t  Σνa t , t  + G r t , t  Σν< t , t 







Tr σˆ z Re G < ( )Σνa ( ) + G r ( )Σν< ( ) .

I=

(I L − I R ) 2     e d   <  a = Tr σˆ z G Σ L − Σ Ra + G r Σ L< − Σ R< . h¯ 2π



 −1 ) =



G r0 (

A1 C† B

C A2 B

B B A3

(7)

(8)



(9)

,

where A i (i = 1, 2, 3), B, and C indicate 4 × 4 matrices,

Ai

⎛

⎜ =⎜ ⎝

 − i + i0+ 0 −R 0

0

−R

0 R

 − i + i0+

 + i + i0+

0 R 0

0

0

 + i + i0+

⎞ ⎟ ⎟, ⎠ (10)

⎞

t 0 0 0 ⎜ 0 −t 0 0 ⎟ B =⎝ ⎠, 0 0 t 0 0 0 0 −t ⎛ iφ te 0 0 0 ⎜ 0 −te −i φ C =⎝ 0 0 te i φ 0 0 0

(11)

⎞

0 0 ⎟ ⎠. 0 − iφ −te

(12)

We assume the wide-band limit approximation, the linewidth functions Γν = L , R are independent of energy and thus the retarded self-energy can be written as

Σνr ( ) i

= − Γν ρ ( ) 2 ⎛ ⎛ 1  i ϕν ⎜ ⎜⎜−  e ⎜⎝ ⎜ 0 ×⎜ ⎜ 0 ⎝

− i ϕν − e

0

1

0

0

1

0

 i ϕν e

0 0



,

(5)

(6)

1

⎛

† i di σ di σ



d

Since I = I L = − I R in the stationary transport, the current formula can be further reduced to

The second term in Eq. (1), H D , is the Hamiltonian of the TTQD system, which can be written as





⎞

⎞ ⎟ 0 ⎟ ⎟  − i ϕν ⎠ 0 0 ⎟ ⎟ e  ⎟, ⎟ 1 ⎠ 0

0 0 0 0

(13) where the factor

ρ ( ) is defined as

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Fig. 2. The gate voltage V g dependence of the supercurrent I at macroscopic phase difference

ρ ( ) =

⎧ | | ⎨√ 2

,

 −2 ⎩ √  , i 2 − 2

| | > , (14)

| | < .

From the fluctuation-dissipation theorem, the lesser Green’s function G < ( ) can be obtained as





G < ( ) = f ( ) G a ( ) − G r ( ) ,

(15)

where G a ( ) = [G r ( )]† and f ( ) = 1/(e β  + 1) is the Fermi–Dirac function. Similarly, the lesser self-energy can be derived as

Σ < ( ) = Σ L< ( ) + Σ R< ( ),

(16)

with

  Σν< ( ) = f ( ) Σνa ( ) − Σνr ( ) ,

(17)

where Σνa ( ) = [Σνr ( )]† . Then, the Josephson current may be rewritten as

I=

=

e



d

h¯ 2π  e d h¯

2π











Tr σˆ z Re G a Σ La − Σ Ra − G r Σ Lr − Σ Rr



f ( ) j ( ),

(18)

where j ( ) = Tr(σˆ z Re[G a (Σ La − Σ Ra ) − G r (Σ Lr − Σ Rr )]) is the current carrying density of states (CCDOS), giving the information of the supercurrent carried by each of the Andreev bound states. The Josephson current can be divided into two parts, contributed by the continuous spectrum for | | >  and by the discrete spectrum for | | < ,

I = Ic + Id

 − ∞  e = + d f ( ) j ( ) + d f ( ) j ( ). e

h

−∞

h



(19)

−

3. Results and discussions In this section, the numerical results of the supercurrent and its dependence on the gate voltages, the magnetic flux, and the strength of the spin-flip scattering are discussed in detail. We perform the calculations at zero temperature in units of h = e = 1, the energy gap of the superconductor is fixed as  = 1. We also set ϕL = −ϕ R = ϕ /2, ΓL = Γ R = Γ = 0.1, and 1 = 2 = 3 = 0 = 0.01 for the symmetric and weak-coupling case.

ϕ = π /2 and t = 0.1. (a) The magnetic flux φ = 0. (b) The magnetic flux φ = π .

Firstly, we consider the gate voltage V g dependence of the supercurrent I at the macroscopic phase difference ϕ = π /2 with different spin-flip strength R and different magnetic flux φ . The current is calculated with the interdot coupling t = 0.1 in Fig. 2. For the case of φ = 0 and R = 0, the Hamiltonian H D can be diagonalized to give three eigenvalues as λ1 = λ2 = 0 + t and λ3 = 0 − 2t, corresponding to two energy levels. Therefore, there are only two peaks in the voltage dependence of current, as shown by the solid line in Fig. 2(a). When R = 0, the supercurrent is always positive, indicating a 0-junction. From the dashed line and the dotted line in Fig. 2(a), one finds that, with increasing the spin-flip strength, the supercurrent exhibits a negative valley, indicating that the π -junction transition occurs due to the spin-flip effects. When R becomes strong enough (R = 0.5), the π -junction exists in a rather large range of the gate voltage (−0.37–0.31). In the case of φ = π , three eigenvalues of H D are obtained as λ1 = λ2 = 0 − t and λ3 = 0 + 2t, which are nearly opposite to those in the case of φ = 0. Consequently, the gate voltage dependence of supercurrent shows the asymmetric shape compared with the zero magnetic flux case as shown in Fig. 2(b). Such a swap effect resulting from the different magnetic flux can be utilized to design a switch. The supercurrent is also plotted as a function of R for different gate voltages as shown in Fig. 3(a). In the case of V g = 0, as R increasing, the supercurrent first increases and then decreases into the negative region, which indicates that the π -junction transition occurs. One finds that the junction is always a 0-junction with small spin-flip scattering and a π -junction with strong spinflip scattering. The spin-flip strength needed to take the π -junction transition increases as the gate voltage increasing. The dependence of the supercurrent I on the interdot couping for various spin-flip strengths R is depicted in Fig. 3(b). For the case of small interdot coupling, similar to those shown in Fig. 3(a), the junction is a 0-junction for small spin-flip scattering and a π junction for large spin-flip scattering. As the interdot coupling increases, the spin-flip strength needed for the π -junction transition increases rapidly. For large interdot coupling, the junction remains a 0-junction with rather large spin-flip scattering, which indicates a competition between the interdot coupling and the spin-flip scattering for the 0–π transition. The oscillations of the supercurrent as a function of magnetic flux φ and macroscopic phase difference ϕ between two SC leads are exhibited in Fig. 4. In the hybrid system, the interferent channels include two paths stemming from the AR. One is the incident electron from the left SC lead to the right one, the other is the reflecting hole from the right SC lead to the left one. This in-

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Fig. 3. (a) The supercurrent I (ϕ = π /2) versus the strength of the spin-flip scattering R with φ = 0 and t = 0.1. (b) The supercurrent I (ϕ = π /2) versus interdot coupling t with φ = 0 and V g = 0.

Fig. 4. (a) The supercurrent I (ϕ = π /2) versus the magnetic flux φ with t = 0.1 and V g = 0. (b) The supercurrent I versus the macroscopic phase difference magnetic flux φ = 0, t = 0.1, and V g = 0.

duces AB oscillation for the supercurrent. Fig. 4(a) presents the dependence of I (ϕ = π /2) on the magnetic flux φ with different spin-flip strength R. The supercurrent is a 2π -periodic function of the magnetic flux. As R increases, it is noticeable that the magnitude and the sign of the supercurrent change, showing that the spin-flip strength has an important influence on the amplitude and the sign of the supercurrent. The CPR of the SC/TTQD/SC structure is shown in Fig. 4(b). With different spin-flip scattering, the curves are quite different. When the spin-flip scattering is absent, the CPR is a sin ϕ -like curve. But for large spin-flip scattering, the current sign is changed from positive to negative leading to the sin(ϕ + π )like CPR, which is referred to as the π -junction. Finally, in order to understand the π -junction transition, the CCDOS j ( ) is plotted for the cases with and without the spin-flip effect in Fig. 5. For the case of R = 0, j ( ) has four δ -function-type discrete spectra in the superconducting gap, which correspond to the four Andreev bound states. The two bound states A 1 and A 2 carrying the positive current, while A −1 and A −2 carrying the negative current. The CCDOS also has two continuous spectra outside the superconducting gap, which is negative for C −1 and positive for C 1 . At zero temperature, only the spectrum of  < 0 has contribution to the supercurrent. The contribution from the discrete spectrum A 1 is dominant and larger than that from the continuous one C −1 , resulting in the positive supercurrent and a 0-junction. When the spin-flip strength R = 0.3, more Andreev bound states

ϕ with the

occur within the superconducting gap. It is found that the contribution from the discrete spectrum ( A 1 , A 2 , A 3 and A −4 ) almost cancel each other. Thus, the contribution from continuous spectrum C −1 is dominant, resulting in the negative current and a π -junction. 4. Summary To conclude, by using the nonequilibrium Green’s function method, the supercurrent in an SC/TTQD/SC structure is investigated. With increasing the spin-flip strength, the sign of the supercurrent can be changed from positive to negative, resulting in the π -junction transition. The 0–π transition is understood in the picture of the CCDOS. It is found that the supercurrent is very sensitive to the system parameters, such as the gate voltage, the interdot coupling and the magnetic flux. Our results can provide the ways of manipulating the supercurrent in a triple quantum dot system. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 10504011 and also by the State Key Program for Basic Researches of China under Grant Nos. 2006CB921803 and 2009CB929504.

2588

Fig. 5. The CCDOS j ( ) versus energy (a) R = 0. (b) R = 0.3.

L. Bai et al. / Physics Letters A 374 (2010) 2584–2588

 with the macroscopic phase difference ϕ = π /2, the gate voltage V g = 0, the magnetic flux φ = 0 and the interdot coupling t = 0.1.

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