Josephson current across a Double Quantum Dot Josephson junction in T-Shape Configuration

Bhupendra Kumar; Sachin Verma; Ajay

doi:10.61343/jcm.v1i02.18

Josephson current across a Double Quantum Dot Josephson junction in T-Shape Configuration

Kumar B^1*, Verma S², Ajay³

DOI:10.61343/jcm.v1i02.18

^1* Bhupendra Kumar, Department Of Physics, Indian Institute Of Technology, Roorkee, Uttarakhand, India 247667.

² Sachin Verma, Department Of Physics, Indian Institute Of Technology, Roorkee, Uttarakhand, India 247667.

³ Ajay, Department Of Physics, Indian Institute Of Technology, Roorkee, Uttarakhand, India 247667.

Keywords: Josephson Current, Double Quantum Dots, Equation of Motion

Corresponding Author	How to Cite this Article	To Browse
Bhupendra Kumar, , Department Of Physics, Indian Institute Of Technology, Roorkee, Uttarakhand, India 247667. Email:	Kumar B, Verma S, Ajay, Josephson current across a Double Quantum Dot Josephson junction in T-Shape Configuration. J.Con.Ma. 2023;1(2):113-117. Available From https://jcm.thecmrs.in/index.php/j/article/view/18

Introduction

Josephson transport through quantum dot junctions has been widely implemented in nanoelectronic devices such as superconducting quantum interference devices (SQUIDS) [1], Cooper pair splitters [2], and superconducting quantum bits [3]. The development of nanofabrication techniques enabled the creation of devices with superconducting leads coupled to quantum dots (QDs). QDs are nanoscopic semiconductor structures that confine electrons to zero dimensions. These QDs, like atoms, have a distinct energy level due to quantum confinement [4]. In the superconductor-quantum dot-superconductor system, Josephson current flows as a result of the creation of subgap states, also known as Andreev Bound states (ABS). The Andreev reflection process happens at the interfaces between the quantum dot and the superconducting leads in a superconductor-quantum dot system. When an electron from one of the leads meets the dot, it can be retro-reflected as a hole, resulting in the creation of a Cooper pair in the superconducting lead. Numerous theoretical [5-9] and experimental [10-12] studies of charge transport in superconductor-quantum dot-superconductor systems have been addressed in recent years.

Additionally, the charge transport properties of Josephson junctions with double quantum dots have been explored in various research papers. These junctions are characterized by the coupling of two quantum dots in a series, parallel, or T-shape arrangement with superconducting leads [13-18]. Because of adjustable characteristics such as quantum dot energy levels, dot-lead coupling strength, and interdot tunneling, studying quantum electronic transport across systems where double quantum dots (DQD’s) are coupled to superconducting leads has been a research area in recent years. Whether the dots are firmly or weakly linked is determined by interdot tunneling. DQD’s controllable adjustable characteristics make it useful for analyzing transport phenomena such as the Coulomb blockade, the Kondo effect, quantum coherence, and so on.

This paper examines the Josephson current in a system with double quantum dots coupled to two superconducting leads in a T-shape configuration. In this setup (S − T DQD − S) (see figure 1), the main quantum dot (QD₁) is coupled directly to the leads,

whereas the side quantum dot (QD₂) is connected to the QD₁ but not to the superconducting leads. In this study, we investigated the transport properties of a

Figure 1: Schematic diagram of T-shaped double quantum dot Josephson Junction

T-shaped double quantum dot Josephson junction by using Keldysh non-equilibrium Green’s equation of motion technique [19-20]. We studied the interdot tunneling and dot-lead coupling dependency of the Josephson current.

Theoretical Formulation

Generalized Anderson and BCS Hamiltonians in second quantization formalism are used to model the double quantum dots system in T-shape configuration as follows:

Where

is the Hamiltonian of BCS superconducting leads. In the first term ∈_k∝ is the energy of superconducting leads (∝ ∈ L, R) and c†_kσ,∝(c_kσ,∝) is the creation (annihilation) operator of electrons with spin σ(↑,↓) and wave vector . The second term denotes the interaction between Cooper pairs where ∆_∝ is the superconducting order parameter and is given as ∆_∝ = |∆₀|e^iφ∝. denotes the Hamiltonian of double quantum dots. The energy of main quantum dot (QD₁) and

side quantum dot (QD₂) is given by ε_di with fermionic operators (d^†_iσ and d_iσ).

is the Hamiltonian for tunneling between both quantum dots and the amplitude of interdot tunneling is symbolized by t.

represents the hamiltonian for tunnelling between QD₁ and superconducting leads where V_k,∝ is the tunneling strength between leads and QD₁.

To compute the spectral and transport properties of S − T DQD − S system, the single-particle retarded Green’s function of main quantum dot QD1 is required. We have resolved the above-mentioned Hamiltonian (Eq. 1) using Green’s equation of motion approach (EOM).

The following equation of motion should be satisfied by the Fourier transform of the above retarded Green’s function.

(2)

The Green’s function EOM technique is used to get the coupled set of equations (Eq. 2), and when the closed set of coupled equations is solved, the expression for the single particle retarded Green’s function of QD₁ may be written as follows:

(3)

In the Green’s function shown above, I₁, I₂ and I₃ refer to the diagonal and off-diagonal parts of self-energy, respectively, which correspond to the pairing that is produced as a result of the connection between the quantum dot and superconducting leads. The expressions for I₁, I₂ and I₃ are written as follows:

The expression of Josephson current can be written as [7, 21]. In these references, authors gave a full derivation for the expression of Josephson current.

where Γ_∝ is the main dot-lead coupling strength, ∆_∝ is the superconducting gap parameter, f(ω) is the Fermi-distribution function, and A(ω) is the denominator of the single particle retarded Green function (Eq.3).

Results And Discussion

In this section, on the basis of the theoretical formulation discussed above, we compute the Josephson current as a function of QD₁ energy level for distinct values of interdot hopping (t) and dot-lead coupling (Γ_∝).

∆₀, which is in meV, is taken as the energy unit. For

Figure 2: Josephson current (I_J) vs QD₁ energy level for different interdot hopping (t) for various dot-lead coupling strengths. The other parameters are T = 0.2∆₀, . Inset (a1) is the magnified view of the Γ = 0.1∆₀ case for different interdot hopping.

simplicity, we assume both the superconducting leads are identical i.e. ∆_L = ∆_R = ∆ and also considered symmetric tunneling between QD′s and superconducting leads i.e. Γ_L = Γ_R = Γ.

In figure 2 (a-d), Josehsonn current shows a symmetric resonant peak centered at ε_d = 0, and the peak height suppresses with the increasing interdot hopping (t). This behavior of Josephson current can be explained by the interference of two paths. In the absence of QD₂ i.e. t=0, electrons directly transport from the one superconducting lead to QD₁ and then the other superconducting lead, inducing a larger peak (blue solid line in Figure 2). When QD₂ is connected to the QD₁ i.e. t > 0, electrons tend to tunnel into QD₂ and thus Josephson current suppresses.

Further in the same figure (a-d), we show the impact of dot-lead coupling strength Γ on Josephson current.

The enhancement in the amplitude of Josephson current with increasing Γ is obvious and shown in figure 2 (a-d) for different dot-lead coupling strengths. As only QD₁ is connected to superconducting leads so on increasing the dot-lead coupling strength the electrons tunnel directly from one superconducting lead to the other superconducting lead and thus enhances the amplitude of Josephson current.

Conclusion

We have addressed the Josephson current versus QD₁ energy level across the double quantum dot Josephson junction in a T-shape configuration. It is observed that interdot tunneling and dot-lead coupling strengths of quantum dots perform a key role in the tunability of Josephson current. It is observed that Josephson current exhibits a symmetric resonant peak centered at ε_d = 0, and the peak height suppresses with increasing interdot tunneling. We also show that Josephson current enhances with increasing dot-lead coupling strengths. In future superconducting devices, the double quantum dot-based Josephson junction might provide tunable supercurrents and new noise features. This research may be expanded to include multi-dot and multi-terminal Josephson junctions.

Acknowledgments

Research scholar Bhupendra Kumar acknowledges the support provided by the Ministry of Education (MoE), India, with a Ph.D. fellowship.

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Manuscript Received	Review Round 1	Review Round 2	Review Round 3	Accepted
2023-11-02	2023-11-07	2023-11-14	2023-11-21	2023-12-01
Conflict of Interest	Funding	Ethical Approval	Plagiarism X-checker	Note
None	Nil	Yes	19.65

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