Josephson current through a T-shaped double quantum dots: π-junction transition and interdot antiferromagnetic correlations

Physics Letters A 377 (2013) 1127–1133

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

Josephson current through a T-shaped double quantum dots: π -junction transition and interdot antiferromagnetic correlations Guangyu Yi a , Limin An b , Wei-Jiang Gong a,∗ , Haina Wu a , Xiaohui Chen a a b

College of Sciences, Northeastern University, Shenyang 110006, China College of Physics Science and Technology, Heilongjiang University, Harbin 150080, China

a r t i c l e

i n f o

Article history: Received 14 October 2012 Received in revised form 21 February 2013 Accepted 25 February 2013 Available online 28 February 2013 Communicated by R. Wu Keywords: Quantum dot Josephson current Antiferromagnetic coupling

a b s t r a c t By means of the exact diagonalization approach, the Josephson current in a T-shaped double quantum dot structure is theoretically investigated. The ground state is obtained within zero bandwidth approximation in which the superconductors are replaced by effective local pairing potentials. It is found that Josephson current can flow through this structure in the presence of various electron correlations. Furthermore, in the half-filled case, a novel 0–π transition behavior is observed, which arises from the interplay of interdot antiferromagnetic coupling and electron correlations. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The continuous progress in nanofabrication during the last decades made it possible to investigate basic physical effects in a very controlled manner. One example of such highly controllable devices is quantum dot (QD), which is used, amongst various applications, for a detailed and very controlled study of the Kondo effect [1–3]. In comparison with a single QD structure, the coupled doublequantum-dot (DQD) systems provide much more Feynman paths [4] for the electron transmission and possess more tunable parameters to manipulate the electronic transport behaviors [5–10]. In these systems, the interdot electron correlation plays an important role in determining the spin configuration of the ground state. For instance, in serial DQD system, the localized moments on the QDs either form the Kondo singlet with the conduction electrons in the leads or form a local spin singlet, dependent on the interdot hopping [9]. In parallel DQD structures, recent experiments demonstrated that an extraordinary control over the physical properties of DQDs can be achieved, which enables the direct experimental investigations of the competition between the Kondo effect and the interdot exchange interaction between localized moments on the dots [10]. When the leads of the single QD systems are superconductors, the electron correlations can also induce various interesting phenomena, due to the interplay between the Josephson effect and

*

Corresponding author. E-mail address: [email protected] (W.-J. Gong).

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.02.041

electron correlations [11–17]. A paradigmatic example is the prediction about a transition to π junction behavior as a function of some relevant system parameters [14]. In these systems, the ratio of the Kondo temperature T K to the superconducting gap Δ is a key parameter. In the strong coupling limit T K  Δ, the Kondo effect survives even in the presence of the superconductivity; a Cooper pair is broken in order to screen the localized spin in the QD. On the other hand, in the weak coupling limit T K  Δ, the Kondo effect is negligible because a strongly bound Cooper pair cannot be broken. Then, the Cooper pair feels the localized magnetic moment in the QD. Under this situation, when Coulomb interaction is strong inside the QD, the so-called 0–π transition occurs. To be concrete, the dependence of Josephson current I J on the macroscopic phase difference ϕ between the superconductors changes from I J = I c sin(ϕ ) to I J = I c sin(π + ϕ ) = − I c sin(ϕ ), and then the critical current I c becomes negative. But, in the case of coupled-DQD geometries, the situation is more complicated due to the interplay among the Kondo effect, the interdot exchange interaction, and pairing correlations [18–20]. In this case, there are three different energy scales, namely, Kondo temperature T K , interdot antiferromagnetic exchange interaction J , and superconducting gap Δ. Accordingly, this system exhibits a richer magnetic behavior than the single-QD case, due to the competition between Kondo effect and the other two kinds of electron correlations. Recently, R. Žitko et al. investigated the Josephson current through a serial DQDs by use of the numerical renormalization group (NRG) approach [20]. They found that there exists a rich phase diagram of the 0 and π -junction regimes by adjusting the interdot exchange interaction J . In particular, when both the superconductivity and the exchange interaction compete with the Kondo physics

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out the electronic degrees of freedom of the superconducting leads [19]. This procedure leads to an effective low energy theory in which each superconductor is replaced by a single site with an effective pairing potential Δ˜ . Also, the hopping terms between the leads and the QD are replaced by an effective parameter V˜ α . Then, we can write the new expressions of H α and H T , i.e., Fig. 1. Schematic of a S/TDQD/S structure. QD-1 is connected to two s-wave BCS superconductor leads and QD-2 is side coupled to QD-1 with interdot coupling parameter t c .

(Δ ∼ J ∼ T K ), there appears an island of π  phase at large values of the superconducting phase difference. In this work, we investigate the Josephson current in a T-shaped DQDs structure by means of exact diagonalization techniques. The structure that we consider is illustrated in Fig. 1, and hereafter the system is referred to as S/TDQD/S. QD-1 is connected to two s-wave Bardeen–Cooper–Schrieffer (BCS) superconductor leads and QD-2 is side coupled to QD-1 with interdot coupling parameter t c . By adjusting the relevant parameters of such a structure, we observed a rich phase diagram of the 0–π transition caused by the competition among the Kondo effect, the interdot exchange interaction, and pairing correlations. Moreover, we find that there is a novel 0–π transition behavior originating from the interplay between the interdot antiferromagnetic coupling and electron correlations. The rest of this Letter is organized as follows. In Section 2, we introduce the model Hamiltonian of the system and the method of calculation. The numerical results are presented and discussed in Section 3. In Section 4, we give the summary. 2. Model and method In order to describe the system shown in Fig. 1, we use the following Hamiltonian, which can be written as:



H=

Hα + H D + H T ,

(1)

α=L, R

Hα =

†

εα ,k aα ,kσ aα ,kσ

k,σ

+



 † † Δe i ϕα aα ,k↓ aα ,−k↑ + Δe −i ϕα aα ,−k↑ aα ,k↓ ,

k

HD =

2 

ε j d†jσ d jσ +

σ , j =1

HT =





εα a†α ,σ aα ,σ + Δ˜ e iϕα aα ,↓aα ,↑ + Δ˜ e−iϕα a†α ,↑a†α ,↓ ,

σ

HT =





V˜ L a L ,σ d1σ + V˜ R a R ,σ d1σ + H.c. . †

†

(3)

σ

This approach, usually referred to as the zero bandwidth model (ZBWM), has been discussed for some previous studies [18,19]. One can see that the ZBWM can give qualitatively correct results and can grasp the ground state properties in this kind of systems in the approximate range of Γ  Δ, where Γ is the standard tunneling rate to the leads. We should mention that the Hilbert space of this S/TDQD/S system within the ZBWM is restricted to 44 states and the z component of the total spin S is a good quantum number. Thus, the eigenstates can be characterized in terms of S z and the eigenenergies can be obtained by block diagonalization of the Hamiltonian matrix. In the superconducting case we distinguish four different ground states, i.e., the pure 0 and π states (for these states, the ground state energy as a function of the superconducting phase difference ϕ = ϕ L − ϕ R has a global minimum at the points of ϕ = 0 and π ), and two intermediate 0 and π  phases (they are both local minima and dependent on whether ϕ = 0 or ϕ = π is global minimum [11]). The Josephson current flowing through the S/TDQD/S system at zero temperature can be obtained by deriving the ground state energy with respect to the phase difference, i.e.,

IJ =

2e ∂ E (ϕ ) h¯

∂ϕ

.

(4)

3. Results and discussion

with



Hα =



†

σ †



t c d2σ d1σ + H.c. + †



2 

U j n j↑n j↓ ,

j =1

V L ,k a L ,kσ d1σ + V R ,k a R ,kσ d1σ + H.c. .

(2)

k,σ

In the above equations, H α (α = L , R ) is the standard BCS meanfield Hamiltonian for the superconducting leads with phase ϕα and energy gap Δ. The chemical potentials of both leads are set as zero. H D models the T-shaped DQDs. And, H T denotes the tunneling part of the Hamiltonian between lead-L ( R ) and QD-1. t c is the in†

†

terdot coupling coefficient. On the other hand, aα ,kσ and d j σ (aα ,kσ and d j σ ) are operators to create (annihilate) an electron with momentum k and spin orientation σ = ↑ (↓) in lead-α and in the jth QD, respectively. εα ,k and ε j denote the corresponding energy levels. U j indicates the strength of intradot Coulomb repulsion in the corresponding QD. V α ,k denotes the coupling between lead-α and QD-1. Note that the determination of the ground state properties of this model is a formidable task, which requires some approximation scheme. A great simplification can be made by integrating

By using the formulas developed in Section 2, we perform a numerical calculation to investigate the characteristics of the Josephson currents in the S/TDQD/S structure. In this Letter, we consider that temperature is zero and Δ˜ = 1, i.e., all the energy quantities are scaled by Δ˜ . In principle, the effective parameters Δ˜ and V˜ α in this approach have to be determined from the bare parameters Δ and V α by means of a self-consistency condition and using a renormalization group analysis, as discussed by Affleck et al. in a previous paper [21]. However, we shall adopt here the simplified assumption that Δ˜ = Δ = 1 and V˜ L = V˜ R = 1 without an attempt to fine tune them within the range of parameters considered. This is a reasonable choice as far as we are interested in the qualitative trends rather than in the detailed quantitative results. To get the main physical insight, we set the Fermi energy of leads to be zero. And only the case of symmetric QD-lead coupling is considered with V˜ L = V˜ R = 1. Furthermore, we set ϕ L = −ϕ R = ϕ /2, ε j = ε0 and t c = 6 unless otherwise specified. Fig. 2(a) shows a (ε0 , U ) phase diagram for the 0–π transition of the Josephson current in this S/TDQD/S structure. For comparison, the 0–π transition phase diagram of the serial DQDs system with the same parameters is also shown in the inset. In this figure, one can clearly see that there are three 0–π transition regions for the T-shaped structure but only two transition regions for the serial one. In order to further investigate the mechanism of phase transition, we consider the case of U = 15 and plot the total mean charge per spin, namely N e↑ and N e↓ as functions of QD level ε0 . The numerical results are shown in Fig. 2(b). The total

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Fig. 2. (Color online.) (a) ε0 –U phase diagram of the S/TDQD/S structure in the ZBWM, and the 0–π transition phase diagram of a serial DQD system with the same parameters in its inset. (b) Total charge per spin N e↑(↓) in the T-shaped DQDs region for U = 15. (c) The Josephson current as function of QD level ε0 for U = 15 with a magnified plot for the central π phase region. (d) The spin correlations as function of QD level ε0 for U = 15. The solid and dashed lines in Fig. 2(d) correspond to the cases of superconducting and normal leads, respectively. The other structure parameters are taken to be the following values: Δ˜ = V˜ L = V˜ R = 1, t c = 6, ϕ = π /2.

electron numbers N corresponding to the left, right, and central π phase regions are N = 3, N = 1, and N = 2 (half-filled), respectively. Obviously, in the former two cases, the result of the magnetic symmetry breaking comes into being with N e↑ = N e↓ , which is identical with the previous results [18,19]. However, the last case is counterintuitive at the first glance because of the zero magnetic moment in this π -phase region. In order to explain the inconsistency, we plot the curves of the Josephson currents and the spin–spin correlations as functions of QD level ε0 in Fig. 2(c)–(d). First, in Fig. 2(c)–(d) we observe that each positive peak in Josephson current curve corresponds to the negative peak in the curve of S L S 1 . This means that the Kondo antiferromagnetic correlation between the spins in lead-L and QD-1 gives rise to the enhancement of the Josephson effect. Similar result has already been observed in a carbon nanotube superconducting quantum interference device (nanoSQUID) [22]. In such an experiment, it was seen that the maximum of Josephson current coincides with the Kondo ridge [23]. On the other hand, we find that I J is negative in the corresponding π -phase regions. And, in the left and right ones, the spin–spin correlation S L S 1  is strongly suppressed by the presence of superconducting leads (blue solid line), which is opposite to the case of normal metallic leads (blue dashed line). In the previous works, this result was observed in a one-QD structure, and it was associated with the disappearance of the Kondo effect and the appearance of an unscreened magnetic moment. But in this structure, the situation is more complicated due to the interplay among the Kondo effect, the interdot antiferromagnetic coupling, and the pairing correlations. We use S 1 S 2  to denote the interdot spin correlation between the QDs and plot the corresponding curves vs. QD level ε0 , as shown in Fig. 2(d). In the central π phase

region, S 1 S 2  curves both have an antiferromagnetic character in the case of normal and superconducting leads. Furthermore, the superconductivity of the leads can notably strengthen the interdot antiferromagnetic correlations in this region. Thus, one can understand that the central 0–π transition region is due to the antiferromagnetic and pairing correlations. As we will show below, the large intradot Coulomb interaction only allows the electrons in a Cooper pair to tunnel one by one via virtual processes in which the spin ordering of the Cooper pair is reversed. As a result, a negative Josephson current (π -junction) comes up. In Fig. 3(a)–(g) we illustrate the lowest order (left panel) and one of higher-order tunneling processes (right panel) to further understand the physics mechanisms in this structure. Here we only focus on the half-filled case because the mechanism of the other π -phase regions has been analyzed in Ref. [24]. We consider that the initial state of the electrons in the DQDs is in a certain antiferromagnetic order. As for the spin order of the Cooper pair in lead-L, we assume the spin-up electron is in the left. In the left panel, the sequence of tunnel processes is indicated by the number j ( j = 1–4). First, the electron in QD-1 must tunnel out to the right superconducting leads, and then the other tunnel events occur sequentially. Finally, the remnant electron in the left superconducting leads tunnels into QD-1. After these processes, the antiferromagnetic order of the electrons in the DQDs is still restored, as shown in Fig. 3(c). Similarly, we present the higher-order tunneling processes in Fig. 3(d)–(g). It is understandable that all the higher-order co-tunnelling events result in the spin-ordering reversal of the Cooper pair in the final state, namely, the Cooper pair in lead-R is created in the order |↑, |↓ while the pair in lead-L is annihilated in the order |↓, |↑. This spin reversal causes a sign change of the

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Fig. 4. (Color online.) Schematic of the tunneling processes for the DQD coupled in series when the electron state is half-filled.

Fig. 3. (Color online.) Schematic of the lowest-order (left panels) and one of higherorder tunneling processes (right panels) for the S/TDQD/S structure in the case of half-filled are plotted in (a)–(c) and (d)–(g), which (from top to bottom) represent initial, intermediate and final states, respectively. The number j ( j = 1–6) denotes the sequence of tunnel events, and the red (blue) line corresponds to the tunneling of the spin-up (spin-down) electron.

√

Cooper√pair singlet state, i.e., from (|↑↓ − |↓↑)/ 2 to e iπ (|↑↓ − |↓↑)/ 2, which exactly leads to a π -shift in the Josephson relation and a negative supercurrent [24]. For comparison, in Fig. 4(a)– (c), we illustrate the analogous tunneling processes, which result in a 0-junction for the serial DQDs system in the half-filled case. By the above discussion, the difference between the 0–π transition phase diagrams of the T-shaped and serial DQDs system can be well explained. Recently, some theoretical and experimental works reported that the Josephson junction made of a multilevel QD can behave as a π -junction due to the permutation of tunnel events and the parity of orbital wavefunctions, even when the QD is nonmagnetic without localized spin [24,25]. But in this structure, the 0–π transition mechanism can be caused by the interplay between the interdot antiferromagnetic coupling and electron correlations.

In order to further investigate the difference in the two 0–π phase transition mechanism, in Fig. 5 and Fig. 6, we present the curves of Josephson current I J , eigenenergies of system and N e↑(↓) as the function of superconducting phase difference ϕ . Fig. 5(a)– (d) correspond to the one of phase transition process in the left π phase region in Fig. 2(a). We can see that this system undergoes a 0 → 0 → π  → π quantum phase transition process when the QD level ε0 changes and U = 15. Correspondingly, the Josephson currents (green dashed-dotted lines) in Fig. 5(a) and (d) have opposite directions. In the intermediate phases 0 and π  , the energy levels of ground (blue solid lines) and first excited states (red dashed lines) are cross and the currents show a kink-like result [11,13], as shown in Fig. 5(b) and (c). Furthermore, in the right panels of Fig. 5, we find that the magnetic symmetry breaking comes into being, namely N e↑ = N e↓ when the two lowest eigenenergy levels of system are crossed. This means that the ground state of system transitions form Kondo singlet with total spin S = 0 to magnetic doublet [13,26]. When we consider the central 0–π transition region corresponding to the half-filled case, the ground state of this system is always in the state of total spin S = 0, namely N e↑ = N e↓ , as shown in Fig. 6(a)–(d). In this case, the localized moments on the dots form a local spin singlet. Another difference with the 0–π transition process in Fig. 5(a)–(d) is that the intermediate 0 and π  phases are not present here. It can also be obtained from the phase diagrams in Fig. 2(a). As discussed above, we can conclude that the 0–π transition phenomena in this system can be induced by the two different mechanisms. The former is caused by the competition between the Kondo effect and superconductivity and the latter arises from the interplay of interdot antiferromagnetic coupling and electron correlations.

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Fig. 5. (Color online.) Energy levels, Josephson currents (left panels) and N e↑(↓) (right panels) as functions of superconducting phase difference ϕ for the S/TDQD/S structure in the ZBWM. From top to bottom: ε0 = −21, −20.3, −20, −19.7. In left panels, solid (blue), dashed (red) and dashed-dotted lines (green) correspond to the ground, first excited states and Josephson currents, respectively. The structure parameters are Δ˜ = V˜ L = V˜ R = 1, U = 15, t c = 6.

Fig. 6. (Color online.) Ground-state levels, Josephson currents (left panels) and N e↑(↓) (right panels) as functions of ϕ for the S/TDQD/S structure in the ZBWM. From top to bottom: ε0 = 0, −7.5. In left panels, solid (blue) and dashed-dotted lines (green) correspond to the ground-state energies and Josephson currents, respectively. Other parameters are the same as Fig. 5.

Finally, we investigate how the Josephson current influenced by the competition among the Kondo effect, the interdot antiferromagnetic coupling, and the superconductivity. In Fig. 7(a) and (b), the images of the Josephson current I J versus the QD energy levels and the interdot coupling parameter t c are shown for U = 15 and U = 5, respectively. With the increasing of t c , one can find that the changes of I J have various features when the values of (ε0 , U ) are

(−7.5, 15), (2.5, 15), (−2.5, 5) and (5, 5), as shown in Fig. 7(c) and (d). These values of (ε0 , U ) are corresponding to the point marked by “A”, “B”, “C” and “D” in Fig. 2(a), respectively. (i) In the case of

(ε0 , U ) = (2.5, 15), the curve of I J has two positive current peaks. But, in the region of 3 < t c < 8, I J changes from positive to negative, which means that the S/TDQD/S system changes from 0 to

π -junction. In this region, the Kondo antiferromagnetic correlation

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Fig. 7. (Color online.) (a)–(b): The images of the Josephson current as function of QD level ε0 and interdot coupling parameter t c . (c)–(h): The Josephson currents and the spin correlations as function of t c for the various U and ε0 . Left panels: U = 15. Right panels: U = 5.

S L S 1  is strongly suppressed by superconductivity. This can be explained as follows. In this system, the normal state Kondo temperature T K can be expressed at zero temperature as [27]:

T K = 0.182U







2Γ J K (t c , ε , U )/π exp −π /2Γ J K (t c , ε , U ) , (5)

where J K (t c , ε , U ) depends on the values of t c , ε , and U . With the increase of t c , T K first decreases and then increases and the relation between the two competitive energy scales is Δ  T K in this π phase region. So, this 0–π –0 transition process is caused by the competition between the Kondo effect and superconductivity. (ii) In the case of (ε0 , U ) = (−7.5, 15), one can see in Fig. 7(e) that the amplitude of S L S 1  is almost a small constant as a function of t c . The DQDs is half-filled and the interdot antiferromagnetic coupling is dominant. The curve of I J is always negative and the system is in the π phase which arises from the interplay between interdot antiferromagnetic coupling and electron correlations. (iii) In the case of (ε0 , U ) = (5, 5), one can see in Fig. 7(d) and (f) that the curve of I J has a positive current peak corresponding to the negative peak in the curve S L S 1 . It means that Kondo effect is dominant in this case and the Kondo resonance gives rise to the enhancement of the Josephson effect [22,23]. (iv) For (ε0 , U ) = (−2.5, 5) case, the situation is more complicated. One

can see that I J suddenly changes from positive to negative with increasing t c . The antiferromagnetic correlation S L S 1  decreases but S 1 S 2  increases at the same time. It can be explained as follows. When the value of t c is very small and U = 5, the interdot exchange interaction J ≈ 4t 2 /U is also very small. So, the sidecoupled QD becomes irrelevant and this structure can be viewed as a single QD connected to the superconducting leads. At this time, the Kondo antiferromagnetic correlation between QD-1 and the leads is dominant. With the increasing of t c , the interdot antiferromagnetic coupling is dominant and this system entrance into the π phase region once J  T K . But in the case of U = 15, Δ  T K will hold even if the value of t c is very small due to the increase of U can reduce the Kondo temperature T K rapidly [27]. 4. Summary and conclusions In conclusion, we have theoretically investigated the Josephson effect in a S/TDQD/S system by exact diagonalization and ZBWM. When we taken into account the effect of electron correlations, the Josephson current I J can flow through the S/TDQD/S system in the presence of various electron correlations. In the half-filled case, we found that there is a novel 0–π transition behavior arising from

G. Yi et al. / Physics Letters A 377 (2013) 1127–1133

the interplay of interdot antiferromagnetic coupling and electronic correlations. By adjusting the relevant parameters in such a structure, we observed a rich phase transition of the Josephson current I J due to the competition among the Kondo effect, the interdot exchange interaction, and pairing correlations. Finally, it should be emphasized that the ZBWM can only give qualitatively correct results for the interplay among the Kondo effect, the interdot antiferromagnetic coupling, and superconducting pairing correlations. In order to obtain the good quantitative results about the competition among the three different electron correlations, more sophisticated methods like the NRG approach are required [20]. Acknowledgements Our numerical results are obtained via an exact diagonalization program based on the SNEG library. This work was financially supported by Fundamental Research Funds for the Central Universities (Grant Nos. N100305002 and N110405010), the National Natural Science Foundation of China (Grant No. 10904010), and Science and Technology Projects of Heilongjiang Provincial Education Department (Grant No. 12521389). References [1] Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601.

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