Aharonov-Bohm Effect for Parallel and T-shaped Double Quantum Dots

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by  Yoichi Tanaka

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Aharonov-Bohm Effect for Parallel and T-shaped Double Quantum Dots
Yoichi Tanaka and Norio Kawakami
6 Department of Applied Physics, Osaka University, Suita, Osaka 565-0871
0
(ReceivedFebruary2,2008)
0
2 We investigate the Aharonov-Bohm (AB) effect for the double quantum dots in the Kondo
regimeusingtheslave-bosonmean-fieldapproximation.Incontrasttothenon-interactingcase,
n
wheretheABoscillationgenerallyhastheperiodof4πwhenthetwo-subringstructureisformed
a
J viatheinterdottunnelingtc,wefindthattheABoscillationhastheperiodof2π intheKondo
regime. Sucheffectsappearfor thedoublequantumdotsclose totheT-shaped geometry even
9
1 in the charge-fluctuation regime. These results follow from the fact that the Kondo resonance
is always fixed to theFermi level irrespective of thedetailed structure of thebare dot-levels.

]
l KEYWORDS: Aharonov-Bohm effect, Kondo effect, DoubleQuantumDots
l
a
h
-
s Recent rapid progress in the nanotechnology makes it fold degenerate Anderson Hamiltonian
e possible to fabricate double quantum dot (DQD) sys-
m tems.1 One of the remarkable features in the DQD sys- H = Xεkαc†kασckασ+Xεdfm†σfmσ
. temsisthe interferenceeffectinducedviamultiple paths kα,σ m,σ
t
ma oexfteelrencatrlomnapgrnoeptaicgafltiuoxn,, gwihviicnhg irsisveertyostehnesitAivheartoontohve- + Ntc X(f1†σb1b†2f2σ+h.c)
- Bohm (AB) effect.2,3 The AB effect has been theoret- σ
d
on iicnatrllaydosttuCdoieudlofmorbtinhteerDacQtDionw.iAths4n–8otoedr iwnitthhoeultit9e–1r3attuhree + √1N X (Vαmc†kασb†mfmσ+h.c.) (1)
kα,m,σ
c recently,10–13 the interdot tunneling between two dots,
[ where c is the annihilation operator of an electron
which forms two-subring structure, generates a new AB kασ
with spin σ in the lead α (α = L,R), and the an-
2 oscillation period of 4π instead of the normal period
v of 2π. However, the arguments are based on the non- nihilation operator of an electron in the dot m (m =
3 interacting electron model, and the influence of electron 1,2) with the energy εd is represented as b†mfmσ with
5 the constraint, f† f + b† b = 1, where b (f )
correlationsonthis new AB oscillationhasnotbeendis- mσ mσ m m m mσ
2
is the slave-boson (pseudo-fermion) annihilation opera-
0 cussed yet.
tor for an empty state (singly occupied state). We use
1 In this note, we address this question about the AB
5 effect by explicitly taking into account the Kondo effect the mean-field approximation at zero temperature (see
0 ref.14). The effect of the magnetic flux Φ is symmet-
in the DQD shown in Fig. 1. We will show that if the
/ rically incorporated in the tunneling between the dot
t
a m and the lead α in eq. (1), V = W e−iφ/4,
m dot1 L1(R2) L1(R2)
V =W eiφ/4, where φ=2πΦ/Φ (Φ =h/e).
- WL1e-iφ /4 WR1ei φ /4 TLh2i(sRm1)eans Lth2(aRt1)the magnetic flux equall0y pi0erces the
d
n Φ two subrings formed by the interdot tunneling tc. We
o lead L ⊗ tc lead R systematically change the arrangement of the DQD by
c modifyingtheresonancewidthΓα =π W 2ρ(ρ
: 1,1(2,2) | α1(2)|
v W eiφ /4 W e-iφ /4 isthedensityofstatesforleadsattheFermienergy):the
L2 R2
Xi dot2 parallel DQD is given by Γα1,1 ∼ Γα2,2, and the T-shaped
DQD by Γα 0 with fixed Γα .15
r 2,2 ∼ 1,1
a Fig. 1. DQDsystemwiththemagneticfluxΦ. Figure 2 shows the conductance as a function of the
magnetic flux φ for severalvalues of Γα . We have stud-
2,2
iedtwotypicalcasesintheKondoregimeandthecharge-
Kondo effect is taken into account in the parallel DQD,
fluctuationregime.Thereareseveralremarkableproper-
the AB period should be practically regarded as 2π in-
ties in the conductance. (i) As seen in Fig. 2(a), the AB
stead of 4π. This tendency becomes more remarkable in
period of the conductance seems to be always 2π in the
the DQD close to the T-shaped geometry, which is an-
Kondoregime(ε = 3.5)evenifwechangethearrange-
d
other typical DQD system sensitive to the interference −
ment of the DQD from the parallel to T-shaped geom-
effect.
etry. This is in contrast to the conclusion for the non-
In the following discussions, the intradot Coulomb in-
interacting case: the period should be 4π generally,10–13
teractionisassumedtobesufficientlylarge,sothatdou-
which follows from the two-subringstructure formed via
bleoccupancyateachQDisforbidden.Thisassumption
the interdot tunneling t . (ii) On the other hand, in the
c
which allows us to use a slave-boson representation of
charge-fluctuation regime (ε = 2.4) in Fig. 2(b), the
d
correlated electrons in the dots. In this representation, −
AB oscillation for the parallel DQD (Γα = 0.9) indeed
the DQD system in Fig. 1 is modeled with the N(=2) 2,2
2 J.Phys.Soc.Jpn. FullPaper AuthorName
Γα =0.9 the self-energy shift inherent in the Kondo effect prop-
2,2
h) 1 (a) εd=−3.5 ΓΓ22αα,,22==00..36 1 (b) εd=−2.4 eerlelycttruonnesnuthmebpeorsiintiotnheofdtohtesrceosnonstaanncte.sTsohiassmtoakneye-pbotdhye
22e/0.8 0.8 effect gives rise to the effective period of 2π, in contrast
ce (0.6 0.6 to the non-interacting case.10–13
n
cta0.4 0.4
u
nd0.2 0.2
co 0 0 1(a) φ=0 (4π) (b) φ=π (3π) (c) φ=2π
0 1 2 3 4 0 1 2 3 4
φ/π φ/π 0.8
ε)0.6
T(0.4
Fig. 2. Conductance as a function of the magnetic flux φ: (a) 0.2
Kondoregime,(b)charge-fluctuation regime.Wechoosethein- 0
terdotcouplingtc=1.5intheunitofΓα1,1 (=1). −0.05 0ε 0.05 −0.05 0ε 0.05 −0.05 0ε 0.05
Fig. 4. Transmission Probability T(ε) for Γα2,2 = 0.3 in the
has the period of 4π. Even in this case, however, as the charge-fluctuation regime(εd=−2.4).
DQDsystemischangedfromtheparalleltoT-shapedge-
ometry,theperiodagainapproaches2π(seeΓα =0.3in
2,2 In the charge-fluctuation regime (ε = 2.4), the res-
d
Fig.2(b)).Theseresultsimply thatelectroncorrelations −
onances are both lifted slightly above the Fermi energy.
play a crucial role for the AB oscillation in the DQD.
The resulting asymmetric nature in T(ε) naturally gives
theABeffectwiththeperiodof4π,asseenforΓα =0.9
2,2
in Fig. 2(b). However, even in this charge-fluctuation
(a) φ=0 (4π) (b) φ=0.5π (3.5π)
1 regime, we encounter a remarkable fact that when the
0.8 systemapproachestheT-shapedDQD(Γα =0.3inFig.
εT ()00..46 (c) φ=π (3π) 2(b)), the effective period becomes 2π ag2,a2in. To under-
standthistendency,weshouldrecallthatwhilethewhole
0.2
0 DQDsystemisinthecharge-fluctuationregime,thedot-
0.8 2 itself falls into the Kondo regime because it is almost
ε)0.6 decoupled from the leads in the T-shaped geometry. We
T (00..24(e)φ=2π (d)φ =(21..55ππ) −0.004 0ε 0.004 cdaipnsstereucthtuartetahtetbhreoaFderrmesioennaenrgcey,owfhthicehdisotc-a1udseedvebloyptshae
0
−0.004 0ε 0.004−0.004 0ε 0.004 sharpKondoresonanceformedinthedot-2.Inthiscase,
the DOS of the dot-1 mainly controls the transmission
probability T(ε) shown in Fig. 4. When the magnetic
Fig. 3. Transmission probability T(ε) around the Fermi energy
flux φ is turned on, the dip in T(ε) is slightly modified
(ε =0) in the Kondo regime (εd =−3.5). We set tc =1.5 and
Γα2,2=0.9. with the resonance-width almost unchanged. Therefore,
although the current flows mainly through the dot-1 in
the T-shaped geometry, the sharp dip structure caused
by the Kondo effect in the dot-2 essentially controls the
The above characteristic properties result from the
linear conductance in the system. This gives rise to 2π
Kondo resonances affected by the interference effect. In
fortheeffectiveABperiodeveninthecharge-fluctuation
Fig.3thetransmissionprobabilityT(ε)isshownaround
regime, as observed in Fig. 2(b).
the Fermienergy (ε=0) in the caseclose to the parallel
geometry (Γα = 0.9). At φ = 0, the Kondo resonance Finally,wewouldliketocommentontheABperiodin
2,2
more general cases for which the magnetic flux piercing
splits into two distinct resonances due to the interdot
thetwosubringsisnotequal:theratiooffluxisassumed
coupling t : they consist of a sharp bonding Kondo res-
c
tobe anirreduciblefractionm/n.Inthe non-interacting
onance at the Fermi energy and a broad anti-bonding
Kondo resonance.14 The interference between these two case, the AB period of the conductance is 2π(m+n). In
theKondoregime,however,theABperiodbecomeshalf,
resonances produces a dip in T(ε) around ε=0, as seen
π(m+n), provided that both m and n are odd, while it
in Fig. 3(a). As the magnetic flux φ increases, the in-
remains 2π(m+n) if either m or n is even.
terference effect is smeared (Fig. 3(c)), and the Kondo
In summary, we have shown that in spite of the two-
resonances in T(ε) become almost symmetric at φ = π.
subring structure in our DQD system, the AB oscil-
At φ=2π, the shape of T(ε) is inverted with respect to
lation practically has the period of 2π in the Kondo
the Fermi energy as shown in Fig. 3(e), and then T(ε)
regime. Such effects appear for the DQD close to the T-
returns to the original shape with another increase by
shaped geometry even in the charge-fluctuation regime.
2π. We thus see that T(ε) changes with the period of
This conclusion, which is contrasted to the noninteract-
4π, as should be expected. However, it should be noted
ingcase,10–13 arisesfromtheself-energyshiftinherentin
thatthevalueofT(ε=0)oscillateswiththeperiodvery
the Kondoeffect:the Kondoresonanceisalwaysfixedto
close to 2π, which characterizes the linear conductance
showninFig.2(a)forΓα =0.9.Thisnew periodicityis the Fermi level irrespective of the detailed structure of
2,2
the bare dot-levels.
due to the constraintimposedonthe Kondo resonances:
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