Solid State Communications 144 (2007) 37–41www.elsevier.com/locate/ssc

Quantum supercurrent in a quantum dot Aharonov–Bohm interferometer

Hui Pana,∗, Rong Lub, Cong Wanga

a Department of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinab Center for Advanced Study, Tsinghua University, Beijing 100084, China

Received 1 May 2007; received in revised form 17 June 2007; accepted 23 July 2007 by S. MiyashitaAvailable online 31 July 2007

Abstract

The quantum supercurrent through an Aharonov–Bohm interferometer containing four quantum dots is investigated theoretically. Thedependence of the supercurrent on the arrangement of the four quantum dot energy levels is obtained. For various quantum dot energy levels,the supercurrent shows different numbers of resonant peaks. By tuning the magnetic flux, the sign of the supercurrent can be changed, whichresults in the π -junction transition. Whether the magnetic flux can reverse the supercurrent depends distinctly on the four quantum dot energylevels. This opens the possibility to manipulate the quantum supercurrent by varying the magnetic field and the gate voltage.c© 2007 Elsevier Ltd. All rights reserved.

PACS: 74.45.+c; 73.23.-b; 73.63.Kv

Keywords: A. Superconductors; D. Electronic transport

Quantum coherence phenomena in the resonant tunnelingprocesses of the quantum dot (QD) systems have attractedmuch attention recently [1]. Interference experiments aboutAharonov–Bohm (AB) effects [2] have been done in an ABinterferometer containing one or two quantum dots in order toprobe the coherence of an electron through a QD [3,4]. Theobserved magnetic oscillation of the current indicates coherenttransport through the QD [4–11]. It is possible to manipulateeach of the quantum dots separately, which enlarges thedimension of the parameter space for the transport properties ascompared to a single-dot AB interferometer. On the other hand,the superconductor coupled mesoscopic hybrid systems havealso attracted much attention in recent years because of boththe fundamental interest and the potential applications in futurenanoelectronics [12–15]. The Andreev reflection happens atthe normal–metal/superconductor (N/S) interface [16]. Anotherimportant and interesting transport characteristic of S/QD/Sdevices is the Josephson effect that gives rise to a dcsupercurrent at zero bias. The supercurrent originates from theAndreev reflection at the interface between the superconductingleads and the central region. One of the most intriguing

∗ Corresponding author.E-mail address: hpan@buaa.edu.cn (H. Pan).

0038-1098/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2007.07.025

experimental results on mesoscopic superconductivity hasbeen the controlling of the supercurrent through a Josephsonjunction [17].

For the S/FQD/S system, the four QDs are coupled andtheir energy levels are mixed, which is quite different fromthe S/QD/S system. This square structure enclosing a magneticflux is the basic unit of two-dimensional quantum dot arrays.Compared with single QD systems, it is expected that thisFQD system makes the quantum transport phenomena richand varied. Motivated by this, we investigate how to controlthe supercurrent in the S/FQD/S system in this work. Theschematic diagram of the device is depicted in Fig. 1, wherefour quantum dots are used to form an AB interferometer. TheQDs 2 and 3 are embedded into the opposite arms of the ABring. The left and right superconducting lead are coupled to theQDs 1 and 4, respectively. By using the nonequilibrium Green’sfunction (NGF) techniques [18,19], we have analyzed quantumtransport properties of the S/FQD/S system. It is found that thesupercurrent is very sensitive to the systematic parameters, suchas the QD energy levels and the magnetic flux. This providesa mechanism for a very sensitively controlled supercurrent.The AB oscillations of the supercurrent can change not onlythe current amplitude but also the current sign. Therefore,the typical Josephson relation between the supercurrent and

38 H. Pan et al. / Solid State Communications 144 (2007) 37–41

Fig. 1. A schematic diagram of a parallel four quantum dot ring systemconnected with two superconducting leads.

the macroscopic phase difference between the superconductorsusually given by Is = Ic sin(ϕ) changes to Is = −Ic sin(ϕ) =

Ic sin(ϕ +π) (with Ic denoting the critical current). This can beused to realize the π -junction transition.

The S/FQD/S system under consideration is described by thefollowing Hamiltonian

H =

∑α=L ,R

Hα + Hdot + HT , (1)

with

Hα =

∑k,σ

εα,kaĎα,kσ aα,kσ

+

∑k

[1e−iϕα aĎα,k↑

aĎα,−k↓

+ H.c.], (2)

HD =

4∑σ,i=1

(εi − Vgi )dĎiσ diσ (3)

+

∑σ

[t12eiφ/4dĎ1σ d2σ + t13e−iφ/4dĎ

1σ d3σ

+ t24e−iφ/4dĎ4σ d2σ + t34eiφ/4dĎ

4σ d3σ + H.c.], (4)

HT =

∑kσ

(tLkaĎLkσ d1σ + tRkaĎ

Rkσ d4σ + H.c.), (5)

where Hα (α = L/R) is the standard BCS mean-fieldHamiltonian for the α superconducting leads with phase ϕα andthe energy gap 1. The chemical potential of both leads are keptas zero. The Hamiltonian HD models the AB interferometerconsisting of four QDs with magnetic flux φ. The quantum dotshave the discrete energy levels εi with i = 1, 2, 3, and 4, whichcan be tuned experimentally by modulating the gate voltage ofeach quantum dot. The Hamiltonian HT denotes the tunnelingpart of the Hamiltonian between the leads L(R) and QD 1(4),and ti j is the hopping matrix.

The supercurrent can be calculated from the standard NGFtechniques, and can be expressed in terms of the QD Green’sfunctions as [19]

Iα(t) =2eh

Re∫

dt1Tr{σz[G<(t, t1)Σaα(t, t1)

+ Gr (t, t1)Σ<α (t, t1)]}, (6)

where σz is a 8 × 8 matrix with the Pauli matrix σz asits diagonal components. The 8 × 8 Nambu representationis used to include the physics of the Andreev reflection.The retarded and lesser Green’s functions are defined as

Gr (t, t ′) = −iθ(t − t ′)〈{Ψ(t),ΨĎ(t ′)}〉 and G<(t, t ′) =

i〈ΨĎ(t ′)Ψ(t)〉, respectively, with the operator ΨĎ=

(dĎ1↑

, d1↓, dĎ2↑

, d2↓, dĎ3↑

, d3↓, dĎ4↓

, d4↑). By taking the Fouriertransformation, the current formula becomes

Iα =2eh

∫dε

2πTr{σz Re[G<(ε)Σa

α(ε) + Gr (ε)Σ<α (ε)]}. (7)

In the steady transport, the total current is given by thefollowing expression

I =12(IL − IR) =

eh

∫dε

2πTr{σz Re[G<(Σa

L − ΣaR)

+ Gr (Σ<L − Σ<

R )]}. (8)

Now we need to obtain the retarded Green’s function Gr (ε) andthe lesser Green’s function G<(ε) of the system. First, Gr (ε)

can be given from Dyson’s equation

Gr (ε) = [gr (ε)−1− Σ r (ε)]−1, (9)

where Σ r (ε) is the self-energy and gr (ε) is the QD Green’sfunction without the coupling between the four QDs andleads. In the Nambu representation, gr (ε) can be written asthe equation given in Box I. Under the wide-bandwidthapproximation, the linewidth functions Γα are independentof the energy variable. Thus the retarded self-energy can bederived as

Σ r (ε) = Σ rL(ε) + Σ r

R(ε), (10)

with

Σ rL(ε) = −

i2ΓLρ(ε)

1 −

1

εe−iϕL

−1

εeiϕL 1

0

0 0

, (11)

and

Σ rR(ε) = −

i2ΓRρ(ε)

0 0

0

1 −1

εe−iϕR

−1

εeiϕR 1

. (12)

The factor ρ(ε) is defined as

ρ(ε) =

|ε|√

(ε2 − 12)|ε| > 1

ε

i√

(12 − ε2)|ε| < 1.

(13)

Second, we solve the lesser Green’s function G<(ε). From thefluctuation–dissipation theorem, G<(ε) can be obtained as

G<(ε) = f (ε)[Ga(ε) − Gr (ε)], (14)

where Ga(ε) = (Gr (ε))Ď and f (ε) = 1/(eβε+ 1) is

the Fermi distribution function. Under the wide-bandwidthapproximation, the lesser self-energy can be derived as

Σ<(ε) = Σ<L (ε) + Σ<

R (ε), (15)

H. Pan et al. / Solid State Communications 144 (2007) 37–41 39

Fig. 2. The supercurrent I (ϕ =π2 ) at φ = 0 vs gate voltage Vg for various QD levels (a) ε1 = ε4 = 0, ε2 = ε3 = 0.4 (solid line), ε1 = ε4 = 0.4, ε2 = ε3 = 0

(dashed line), (b) ε1 = ε2 = ε3 = ε4 = 0 (solid line), ε1 = ε2 = ε4 = 0, ε3 = 0.4 (dashed line), and (c) ε1 = ε2 = 0, ε3 = ε4 = 0.4 (solid line),ε1 = 0.1, ε2 = 0.2, ε3 = 0.3, ε4 = 0.4 (dashed line).

[gr (ε)]−1=

ε − ε1 + i0+ 0 −t12eiφ/4 0 −t13e−iφ/4 0 0 00 ε + ε1 + i0+ 0 t12e−iφ/4 0 t13eiφ/4 0 0

−t12e−iφ/4 0 ε − ε2 + i0+ 0 0 0 −t24eiφ/4 00 t12eiφ/4 0 ε + ε2 + i0+ 0 0 0 t24e−iφ/4

−t13eiφ/4 0 0 0 ε − ε3 + i0+ 0 −t34e−iφ/4 00 t13e−iφ/4 0 0 0 ε + ε3 + i0+ 0 t34eiφ/4

0 0 −t24e−iφ/4 0 −t34eiφ/4 0 ε − ε4 + i0+ 00 0 0 t24eiφ/4 0 t34e−iφ/4 0 ε + ε4 + i0+

.

Box I.

with

Σ<α (ε) = f (ε)[Σa

α(ε) − Σ rα(ε)], (16)

where Σaα(ε) = (Σ r

α(ε))Ď. After obtaining the self-energy andthe Green’s functions, the supercurrent can be expressed as

I =eh

∫dε

2πf (ε)Tr{σz Re[Ga(Σa

L − ΣaR)

− Gr (Σ rL − Σ r

R)]}. (17)

We perform the calculations at zero temperature in units ofh = e = 1. The energy gap of the superconductor is fixed as1 = 1. All the energy quantities in the calculations are scaledby 1. The couplings between the four QDs are set as ti j = 0.1.Furthermore, we set ϕL = −ϕR = ϕ/2 and ΓL = ΓR = Γ withsmall values 0.1 for the symmetric and weak-coupling case.

In the following, the numerical results of the supercurrentsare discussed in detail with various QD energy levels. Fig. 2presents the dependence of I (ϕ =

π2 ) on Vg (= Vg1 =

Vg2 = Vg3 = Vg4) with different QD energy levels. To gaina more clear physical insight into the dependence of the su-percurrent of the system on the QD energy levels, some maincases are considered in the following. The different numberand height of supercurrent peaks are related to the eigenval-ues and corresponding eigenfunctions of the Hamiltonian HD .As shown in Fig. 2(a), the supercurrent shows two asymmet-ric peaks at Vg = −0.08, 0.48 for the case of (ε1 = ε4 =

0, ε2 = ε3 = 0.4). When the Hamiltonian HD is diagonalized,four different eigenvalues are obtained as (ε1 = −0.08, ε2 =

0, ε3 = 0.4, ε4 = 0.48), and the corresponding eigenvec-tors are [A1 = (−0.6533, 0.2706, 0.2706, −0.6533), A2 =

(0.7071, 0, 0, −0.7071), A3 = (0, 0.7071, −0.7071, 0), A4 =

(−0.2706, −0.6533, −0.6533, −0.2706)]. Since the eigenvec-tors corresponding to ε2 = 0.0 and ε3 = 0.4 have zero com-ponents, the two states are localized and contribute little to thecurrent. This qualitatively explains that only two resonant cur-rent peaks appear in this case. In addition, converting the val-ues of ε1,4 and ε2,3 exchanges the positions of the two currentpeaks. This swap effects can be used to design a switch. Asshown in Fig. 2(b), the supercurrent shows three resonant peaksin the case of (ε1 = ε2 = ε3 = ε4 = 0). When the Hamilto-nian HD is diagonalized, three different eigenvalues can be ob-tained. Since each component of the corresponding eigenvec-tors has nonzero values, the three states spread throughout thering. This explains the appearance of the three resonant peaks.As shown in Fig. 2(c), the supercurrent shows four resonantpeaks in the case of (ε1 = ε2 = 0, ε3 = ε4 = 0.4). Whenthe Hamiltonian HD is diagonalized, four different eigenval-ues are obtained and the corresponding states can also spreadthroughout the ring, which results in the four resonant currentpeaks. Similarly, other cases in Fig. 2 can also be qualitativelyexplained from the eigenvalues and corresponding eigenvectorsof Hamiltonian HD .

It is seen that the supercurrents are quite different in the twocases: 1. ε12 = ε1 = ε2 and ε34 = ε3 = ε4; 2. ε14 = ε1 = ε4and ε23 = ε2 = ε3. To gain a full picture on their influence onthe supercurrent, the images of the supercurrent I (ϕ =

π2 ) as

40 H. Pan et al. / Solid State Communications 144 (2007) 37–41

Fig. 3. Images of the supercurrent I (ϕ =π2 ) at Vg = 0 as a function of the

QD levels: (a) ε12 = ε1 = ε2 and ε34 = ε3 = ε4, and (b) ε14 = ε1 = ε4 andε23 = ε2 = ε3.

a function of ε12 and ε34 or ε14 and ε23 are plotted in Fig. 3.The bright regions correspond to positive current and the grayregions to zero current. At zero magnetic flux φ = 0, tuningthe energy levels influences the magnitude of the supercurrentsignificantly, but leaves the sign of the current unchanged. Inthe first case, the current is symmetrical about the line ofε12 = ε34 and ε12 = −ε34 in the diagram. When the energylevels in the four QDs are aligned to the Fermi energy, themaximal supercurrent can flow through the device. When theenergy levels of two dots (ε12 or ε34) are tuned away from theFermi energy, the supercurrent reduces distinctly. In the secondcase, when ε14 moves away from the Fermi energy but ε23is kept at the Fermi energy, the supercurrent reduces rapidly.When ε23 moves away from the Fermi energy but ε14 is keptat the Fermi energy, the supercurrent reduces slowly, since bothenergy levels of QDs 1 and 4 connected with the leads still lineup with the Fermi energy of the leads.

It is interesting to discuss the AB oscillations of thesupercurrent as a function of the magnetic flux. Figure 4presents the dependence of I (ϕ =

π2 ) on φ for different QD

energy levels. In the hybrid system, the interference channelpaths contain two parts due to the Andreev reflection. One isthe incident electron from the left superconductor lead to theright one, and the other is the reflected hole from the rightsuperconductor lead to the left one. In the cases of (ε1 = ε4 =

0, ε2 = ε3 = 0.4), (ε1 = ε4 = 0.4, ε2 = ε3 = 0) and(ε1 = ε2 = ε3 = ε4 = 0) as shown in Fig. 4(a) and (b),the sign of the supercurrent does not change. At φ = 2nπ , thesupercurrent peak for (ε1 = ε4 = 0, ε2 = ε3 = 0.4) is muchhigher than that for (ε1 = ε4 = 0.4, ε2 = ε3 = 0), and the largecurrent peaks for the former case correspond to the small onesfor the latter case. However, in the case of (ε1 = ε2 = 0, ε3 =

ε4 = 0.4) as shown in Fig. 4(c), the sign of the supercurrentchanges from positive to negative, resulting in the π -junctiontransition. The sign change caused by the AB oscillations isrelated to the difference between the energy levels of QDs 2 and3, which results in ε2↑ 6= ε3↓ (or ε2↓ 6= ε3↑). Since the currentcarrying density depends sensitively on the configuration ofε↑ and ε↓ of the middle conductor, the supercurrent sign canbe changed when the difference between ε↑ and ε↓ is largeenough [20]. Thus, both the magnitude and the sign of thesupercurrent can be controlled by tuning the total magnetic fluxand the QD energy levels. The superconducting phase ϕ andthe magnetic phase φ together can lead to a complex picturefor the supercurrent. In Fig. 5, the images of the supercurrentI versus ϕ and φ for (ε1 = ε2 = 0, ε3 = ε4 = 0.4) and(ε1 = ε4 = 0, ε2 = ε3 = 0.4) are plotted, respectively.The bright regions correspond to positive current and the darkregions to negative current. In the case of (ε1 = ε2 = 0, ε3 =

ε4 = 0.4), tuning φ at some certain ϕ can change not onlythe magnitude but also the sign of the supercurrent. However,in the case of (ε1 = ε4 = 0, ε2 = ε3 = 0.4), tuning φ

only changes the magnitude of the supercurrent and leaves thesign unchanged. Whether the AB oscillations can reverse thesupercurrent depends on the four QD energy levels distinctly.The system can be used as magnetic flux-controlled π junctionsat some proper gate voltages.

In summary, we have studied the quantum supercurrentthrough an Aharonov–Bohm interferometer containing fourquantum dots in terms of the nonequilibrium Green’s functionmethod. The detailed dependence of the supercurrent on thearrangement of the four quantum dot energy levels (ε1, ε2, ε3,ε4) is obtained. For various quantum dot energy levels, thesupercurrent shows different numbers of resonant peaks. Byexchanging the quantum dot energy levels ε1 = ε4 and ε2 = ε3,

Fig. 4. The periodic oscillation of the supercurrent I (ϕ =π2 ) at Vg = 0 vs the magnetic flux φ for various QD levels; (a) ε1 = ε4 = 0, ε2 = ε3 = 0.4 (solid line),

ε1 = ε4 = 0.4, ε2 = ε3 = 0 (dashed line), (b) ε1 = ε2 = ε3 = ε4 = 0, and (c) ε1 = ε2 = 0, ε3 = ε4 = 0.4.

H. Pan et al. / Solid State Communications 144 (2007) 37–41 41

Fig. 5. Images of the supercurrent I at Vg = 0 as a function of thesuperconducting phase ϕ and magnetic flux φ for dot levels: (a) ε1 = ε2 = 0,ε3 = ε4 = 0.4, and (b) ε1 = ε4 = 0, ε2 = ε3 = 0.4.

the two resonant supercurrent peaks can be exchanged, showinga swap effect. The sign of the supercurrent does not changeby only varying the quantum dot energy levels. However, thesupercurrent can oscillate from positive to negative by tuningthe magnetic flux φ when the difference between the middletwo quantum dot energy levels ε2 and ε3 is large enough,resulting in the π -junction transition. Whether the supercurrentcan be reversed by the magnetic flux depends distinctly onthe four quantum dot energy levels. This opens the possibilityto manipulate quantum supercurrent in a nontrivial way byadjusting the magnetic flux and the gate voltage.

Acknowledgments

H.P. would like to thank Prof. Tsung-Han Lin for manyhelpful discussions. This work was supported by the Programfor New Century Excellent Talents in University (NCET).

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