Advances in Condensed Matter Physics

Volume 2019, Article ID 5639487, 16 pages

https://doi.org/10.1155/2019/5639487

## Josephson Current through a Quantum Dot Connected with Superconducting Leads

Correspondence should be addressed to Satoshi Kawaguchi; pj.ca.nuf@ihsotas

Received 26 June 2019; Revised 20 August 2019; Accepted 23 August 2019; Published 18 November 2019

Academic Editor: Oleg Derzhko

Copyright © 2019 Satoshi Kawaguchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, we consider the Josephson current in a system composed of a superconductor/quantum dot/superconductor junction. In the model, the Coulomb interaction in the quantum dot is taken into consideration, and the Lacroix approximation is applied to study the electron correlation. We derive Green’s function of the quantum dot by applying the Lacroix truncation. Although the Andreev bound state does not occur in our formulations, the *π*-junction occurs for a restricted parameter range. On comparing the Kondo temperature with that estimated by another method, it is found that our Lacroix approximation does not capture well the Kondo physics in the superconductor/quantum dot/superconductor junction.

#### 1. Introduction

In the last few decades, the Josephson current in a system composed of a superconductor/quantum dot/superconductor (S/QD/S) junction has been extensively studied [1–4]. When identical superconducting leads are separated by a thin layer of insulator, the Josephson current can flow because of the coherent tunneling of Cooper pairs across the insulator in the absence of a potential difference. When the tunneling amplitude across the barrier is small and the spin is conserved, the current depends on the superconductor phase difference *θ* between the left and right leads. The current is expressed as , where denotes the critical current, which is proportional to the normal conductance through the barrier [5]. When *θ* = 0, the Josephson current is zero and the junction is in the ground state. When *θ* = *π*, the current becomes zero; however, in this case, the junction energy is maximum and it is in an unstable state. Very recently, carbon-nanotube Josephson junction systems have been studied intensively [6–8].

When we consider physics at low temperatures under the Coulomb interaction in the QD, the physical behavior of the system depends on the relative magnitude of the Kondo temperature and the BCS gap [9]. When , the Kondo effect is sufficiently strong to break the Cooper pair at the Fermi level, and the localized spin in the QD is screened; it is expected that a Kondo singlet will be formed. This results in a positive critical current (0-junction). By contrast, when , the Cooper pair is strongly coupled, and the Kondo screening is essentially negligible. In this case, the Cooper pair is subjected to a localized magnetic moment in the QD. When the Coulomb repulsion in the QD is large, the ground state of the QD is a magnetic doublet, and the electrons in a Cooper pair can tunnel one by one via virtual processes. The spin ordering of the Cooper pair is reversed, resulting in a *π*-junction [3, 10, 11]. The 0-*π* transition is expected to occur around . The tuning of the *π*-junction in S/QD/S systems has been studied extensively [3, 12–14].

The current density in the S/QD/S system is composed of two parts: continuous and discrete spectrums. The former (latter) arises from outside (inside) the BCS gap. The current from the continuous current density is calculated by the usual numerical integral. By contrast, the current from the discrete current density is calculated by applying the complex function theory. The discrete current density arises from the Andreev bound states (ABSs) inside the BCS gap. The ABSs are determined as poles of the QD Green’s function, and there is a pair of ABSs in the absence of Coulomb interaction. In particular, when the energy level of the QD coincides with the Fermi level , . In this situation, the current jumps discontinuously when the BCS phase difference is [15, 16]. Usually, the current from the discrete current density is much larger than that from the continuous current density.

The Coulomb interaction in the QD is studied by many methods: the numerical renormalization group (NRG) method [17–19], noncrossing approximation [20], quantum Monte Carlo (QMC) method [21–23], and so on. Although the NRG and QMC methods are very precise methods, they are computationally expensive. The simplest method is the Hartree approximation, which corresponds to the zeroth-order approximation. In this method, a two-particle Green’s function is truncated by decoupling at the mean-field level. Beyond the Hartree approximation, the Hartree–Fock (HF) approximation is proposed, where up to the first order of the tunneling amplitude is considered. This method was applied to the level-crossing quantum phase transition between the BCS-singlet and the magnetic doublet states [24–27]. The above two approximations can be applied to describe single-particle physics. The higher-order Green’s functions are not taken into consideration; therefore, these approximations are not sufficient when we consider Kondo physics [28]. To overcome this defect, the Lacroix approximation has been proposed [29]. In this approximation, a greater higher-order correlation effect is included in the QD Green’s function by truncation in the second order. Although the Lacroix approximation suffers from several defects, the mathematical procedures to derive the QD Green’s functions are a simple application of the equation of motion. Although there have been many studies on QD systems that employ the Lacroix approximation [28, 30–34], only a few studies have been conducted on the current in S/QD/S systems [33, 35].

From these standpoints, we examine the current in a system composed of an S/QD/S junction with Coulomb interaction by applying the Lacroix approximation. Under second-order truncation and simplification, Green’s function of the QD is obtained. Using Green’s functions, we calculate the electron occupation number in the QD and the Josephson current. We can observe the *π*-junction in a restricted parameter range, but our Lacroix approximation does not capture well the competition between the Kondo effect and superconductivity.

#### 2. Model and Formulation

We first introduce the setup of the system and give its Hamiltonian. For the system, Green’s functions of the QD and the Josephson current are derived by employing the equation of motion.

##### 2.1. Model and Green’s Functions of the QD

We consider a system composed of an S/QD/S junction, where Coulomb interaction exists in the QD. The geometry of the setup is shown in Figure 1. The total Hamiltonian of the system is written aswherewhere and . represents the superconducting lead *α*; denotes the annihilation operator of an electron with energy , wave number *k*, and spin *σ* in the lead; the order parameter with the BCS gap and the BCS phase ; represents the QD; denotes the annihilation operator of an electron with spin *σ*; and denotes the QD energy level. The occupation number of an electron in the QD with spin *σ* is defined by , and *U* represents the Coulomb repulsion between electrons with up- and downspins. represents the electron tunneling between the leads and the QD. The coupling strength between electrons in the QD and the leads is defined by , where the tunneling amplitude *t* is real and denotes the normal density of states (DOS) at the Fermi level.