Photon-assistant Fano resonance in coupled multiple quantum dots

0.42 MB·English

by  Wanyuan Xie

Photon-assistant Fano resonance in coupled multiple quantum dots
Wanyuan Xie1, Hui Pan1,2, Weidong Chu1, Wei Zhang1 and Suqing Duan1 ∗
1 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
2 Department of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Based on calculations of the electronic structure of coupled multiple quantum dots, we study
systemically the transport properties of the system driven by an ac electric field. We find quali-
tative difference between transport properties of double coupled quantumdots (DQDs)and triple
8
quantum dots. For both symmetrical and asymmetrical configurations of coupled DQDs, the field
0
can induce the photon-assisted Fano resonances in current-AC frequency curve in parallel DQDs,
0
and a symmetric resonance in serial DQDs. For serially coupled triple quantum dots(STQDs), it
2
is found that the Λ-type energy level has remarkable impact on the transport properties. For an

n asymmetric (between left and right dots) configuration, there is a symmetric peak due to resonant
a photon induced mixing between left/right dot and middle dot. In the symmetric configuration, a
J
Fano asymmetric line shape appears with the help of “trapping dark state”. Here the interesting

3 coherenttrappingphenomena,whichusualappearinquantumoptics,playanessentialroleinquan-
2 tumelectronictransport. WeprovideaclearphysicspicturefortheFanoresonanceandconvenient
ways totunethe Fano effects.
]
l
l PACSnumbers: 73.63.Kv,05.60.Gg,25.70.Ef
a
Keywords: coupledmultiplequantum dots,Floquettheory,Fanoresonance, trappingdarkstate
h
-
s
e I. INTRODUCTION 1 1
m L 1 R L 1
at. The electronic transport in quantum dot structures is L t R L t R
2 2
m of great interest and has been the subject of active re- L 2 R 2 R
- search in recent years. The strong spacial confinement Parallel DQDs Serial DQDs
d leads to the discrete energy spectrum. These quantum
n dots(socalled“artificialatoms”)/mutiplecoupledquan-
o tum dots (MCQDs, so called “artificial molecules” ) are L L 1 t 2 t 3 R R
c
very important in understanding fundamental quantum
[
Serial Triple Qantum Dots
phenomena. Many interesting phenomena which appear

1 in real atoms and molecules can be well manifested in FIG. 1: Schematic diagrams of coupled multiple quantum
v thesenanostructures. Tunabilityprovidesmuchmoreop- dots
5
portunities for exploring richer physics which is hard to
1
access in real atoms/molecules. It also leads to various
5
3 applications. For instance, novel quantum logical gates tention has been paid to the photon-assistant transport
. and elementary qubits in quantum computers1,2,3 could inmultiplequantumdotsdrivenbyatime-periodelectric
1
be realized based on these systems. field. Moreover,periodicdriventriplequantumdotswith
0
8 In these nanostructures, quantum coherence and in- therelevantthree-levelstructuremayhavesomeinterest-
0 ing interference phenomena, such as coherent trapping,
terference is one of the most important issues and leads
: like those in quantum optics. In this paper we study the
v to a lot of interesting phenomena such as Aharonov-
photon-assistanttransportinMCQDs(see Fig.1)paying
i Bohm (AB) oscillations4, Kondo effect5,6, coherent
X specialattentiontotheconsequencesofthoseinteresting
trapping7,8, Fano resonance9,10,11 and so on. In re-
r interference phenomena.
a cent years, double and triple quantum dots have at-
tracted much attention due to their rich electric struc- InprevioustheoreticalworkonMCQDs, the quantum
ture and physical phenomena. An AB interferometer levels,whichdependonthe structures ofthe system,are
containing two QDs has been realized12,13,14,15. Some oftenassumedasparameters30,31,andthequantumprop-
experimental groups have been able to fabricate triple erties of MCQDs can not be presented quantitatively.
quantum dots with hight quality16,17, and many rele- Therefore, it is necessary to reveal the quantum behav-
vant theoretical studies have been conducted on these ior of MCQDs based on more realistic model. In our ap-
systems18,19,20,21,22,23,24,25,26,27,28,29. In spite of many proach, we first design DQDs and STQDs (with Λ-type
studiesonthemultiplequantumdots,relativelylittleat- three-level structure) using a two-dimensional confining
modelintheeffectivemassframe. Basedontheobtained
levelstructureandwiththe helpofFloquettheory32, we
studythetransportpropertiesoftheDQDSandSTQDs.
∗Correspondingauthor,Email-address: duan [email protected] Itisfoundforbothsymmetricalandasymmetricalconfig-
2
(a) (b) coupled triple quantum dots as an example. The confin-
ing potential is
E
2
m L E E3 m y V(x,y) = 0.5m∗min{ωl2x(x+d)2+ωl2yy2,ωm2xx2
1 R +ω2 y2,ω2 (x−d)2+ω2 y2}, (1)
my rx ry
x where d is the interdotdistance, m∗ is the effective mass

of electron, ω , ω and ω are confining trap
3.6 lx(y) mx(y) rx(y)
(c) frequencies of the left, middle and right dots in the x(y)
directionrespectively. ThemodelHamiltonianofanelec-
3.4 2.7meV 3.0meV
tron in the coupled quantum dots can be written as
V)
me 3.2
y ( H = P2 +V(r), (2)
nerg 3.0 2m∗
E
where ~r = (x,y). In our calculation, the material pa-
2.8
rameter of GaAs QDs is used as m∗ = 0.067m and the
3.0meV 2.7meV e
valueofdistakentoresembleexperimentalsystems. We
2.7 2.8 2.9 3.0 3.1 3.2 3.3
use the eigenstates ϕ ,ϕ ,ϕ for each dot as the ba-
hw r ( meV) sis of the Hilbert spailce.imConirsidering the nonorthogo-
nality of the basis states, we obtain the eigenstates of
FIG. 2: (a) Expected Λ-type three-level scheme for STQDs. Eq. (2) by solving the generalized eigenvalue of the sys-
(b) Confining potential model of STQDs. (c)The three solid tem. Here, we investigate the levels of the STQDs by
lines are levels for a STQD as functions of ~ωr for STQDs varying ~ω = ~ω = ~ω with constant parameters
of ~ωl = 3.00 meV, ~ωm = 3.52 meV, and d = 77.70 nm. of ~ω = r~xω = r~yω = 3r.00 meV, ~ω = ~ω =
Thedotted twolines are levels for DQDsof ~ωl=3.00 meV, ~ω =lx 3.52 mlyeV, andld = 77.70 nm. Fomrxsuch pamryam-
d = 77.70 nm. Insert: The eigenwavefunctions for STQDs m
(up) and DQDs (bottom) at ~ωr =3.00 meV and 2.70 meV eters, the tunneling energy between the left and middle
dot is about 51 µeV. The lowest three energy levels of
respectively.
STQDs form a Λ-type structure as shown in Fig. 2(a)
(c). Experimentally, the levels of each dot are tuned
by changing the voltage of the corresponding electrode
urationsof coupledDQDs, there is photon-assistedFano
gate. Such change of voltage corresponds to the vari-
resonancesinparallelDQDs,andasymmetricresonance
ation of confining strength of the dot in our model. As
in serial DQDs. The electronic transport is very differ-
thevalueof~ω increases,therightlevelincreases. When
ent from that in STQDs with Λ-type energy level: the r
~ω = ~ω = 3.00 meV, the energies of |1i and |3i anti-
photon-assistanttunneling leads tothe symmetricBreit- r l
Wigner36 resonance when the system is asymmetric (be- cross. The corresponding eigenwavefunctions are shown
in the insert. It is clear that a pair of delocalized bond-
tween left and right quantum dots). When the system
ing and antibonding states are formed, while the other
is in a symmetric (between left and right quantum dots)
state is localized in the middle quantum dot. When ~ω
configuration,the formationof“trapping darkstate” re- r
is away from 3.00 meV, the three levels are nearly en-
sultsintheinterestingasymmetricFanoresonanceunder
ergies of the ground states of the left, middle and right
resonant condition (photon energy equals to the energy
dots respectively, and the eigenstates are all localized in
difference between left/right quantum dot and the mid-
each quantum dots. The continuous manifolds on the
dle dot.) Our studies show that the transportproperties
two sides of the Fig. 2(a) correspondto electronic states
arequite sensitiveto the number ofquantum dots inthe
withchemicalpotentialµ andµ andthe twosidedots
coupled systems. L R
are coupled to the leads with the dots-lead hopping rate
The organization of the paper is as follows. In sec-
Γ and Γ .
tionIIwedescribethemodelandcalculatetheelectronic L R
Using the same method, we investigate the levels of
structure of the MCQDs. In section III we present the
the DQDs by varying ~ω with constant parameters of
photon-assistant transport properties of the system and r
~ω = 3.00 meV, and d = 77.70 nm. The lowest two
discuss the results. A brief summary is given at the end l
energy levels and the corresponding eigenwavefunctions
of the paper.
withdifferentvaluesof~ω ofDQDsasshowninFig.2(c).
r
Similarly, when ~ω = ~ω = 3.00 meV, the energies of
r l
|1i and |2i anticross and a pair of delocalized bonding
II. MODEL AND ELECTRONIC STRUCTURE and antibonding states are formed. When ~ω is away
r
from 3.00 meV, the two levels are nearly energies of the
We use a two-dimensional model to describe multiple ground states of the left and right dots respectively, and
coupled quantum dots laterally. Here, we give serially the eigenstates are localized in each quantum dots.
3
III. PHOTON-ASSISTANT TRANSPORT G<(t,t′) = ihΨ†(t′)Ψ(t)i, respectively, with the opera-
PROPERTIES tor Ψ† =(d†,d†).
1 2
The total self-energy is Σr = Σr +Σr, in which
α α c
A. Formulism for photon-assistant transport in Σr and Σr are caused by the α lead and the interdot
α c
coupled QDs system couplings, respectively. Under tPhe wide-band approxi-
mation, the retarded self-energy caused by the α lead is
We first construct our formulism based on parallelly defined as
coupledDQDs(seeFig.1)withanexternaltime-varying
i Γα ΓαΓα
field. This system can be described by the following Σr(t,t′)=− δ(t−t′) 1 1 2 , (8)
α 2 ΓαΓα Γα
Hamiltonian: (cid:18) 1 2 p 2 (cid:19)
where Γα is the linewidth fpunction defined by Γα =
H = H +H +H , (3) i i
α D T 2πρ t∗ t withρ beingthe densityofstatesofthe cor-
α=L,R α αi αi α
X respondinglead,describing the coupling betweentheith
QD and the α lead. The advanced self-energy can be
with
obtained from Σa(t,t′) = (Σr(t,t′))†. The self-energy
α α
H = ǫ a† a , (4) caused by the interdot coupling is
α α,k α,k α,k
k
X 0 −t
Σr(t,t′) =δ(t−t′) c . (9)
c −t 0
c
(cid:18) (cid:19)
H = E (t)d†d −(t d†d +H.c.), (5)
D i i i c 1 2 The lesser self-energy is Σ<(t ,t )= Σ<(t ,t ) and
i=1,2 1 2 α α 1 2
X
Σ<(t,t′)=i dǫf (ǫ)e−iǫ(t−t′) PΓα1 Γα1Γα2 (.10)
HT = tαid†iaα,k+H.c.. (6) α Z 2π α (cid:18) Γα1Γα2 pΓα2 (cid:19)
α,kX,i=1,2 where fα(ǫ) = 1/(e(ǫ−µα)/kBT + 1p) denotes the Fermi
distribution function of electrons in the α lead.
H (α=L,R) describes the left and right normal metal
α
leads. H models the parallel-coupled DQD where d† The Green’s function of uncoupled DQD without the
D i couplings to the two leads can be easily obtained as
(d )representsthecreation(annihilation)operatorofthe
i
electronwithenergyE inthedoti(i=1,2). t denotes
the interdot coupling istrength, which can becobtained gr(t,t′)=−iθ(t−t′) e−iRtt′E1(t1)dt1 0 (11.)
from the calculations of the electronic structure. Under 0 e−iRtt′E2(t1)dt1 !
theadiabaticapproximation,theexternalacelectricfield
After taking the Fourier transformations, the Dyson
canbe reflectedin the single-electronenergieswhich can
equation and Keldysh equation become
be separated into two parts as E (t) = E +∆ (t) and
1 1 0
E (t) = E −∆ (t) for the central conductor. E is the
2 2 0 i Gr (ǫ)=gr (ǫ)+ Gr (ǫ)Σr(ǫ)gr (ǫ), (12)
time-independent single-electronenergies without the ac mn mn ml ll ln
field, and ∆0(t) is a time-dependent part from the ac Xl
field, which can be written as ∆ (t) = edAcosΩt. H
0 T and
represents the tunneling coupling between the DQD and
leads where tαi is the hopping strength between the ith G<mn(ǫ)= Grml(ǫ)Σ<ll(ǫ)Galn(ǫ), (13)
QD and the α lead. To capture the essential physics
l
X
of photon-assisted Fano resonance, we consider the sim-
plest case with noninteracting electrons in two single- respectively. With these Green’s functions, the time-
level QDs. dependent current can be expressed as
ThecurrentI (t)fromtheαleadtothecentralregion
α 2e
can be calculated from standard NGF techniques, and I (t)= Re eilΩt{ dǫTr[Gr(ǫ)Σ<(ǫ)+G<(ǫ)Σa(ǫ)] }.
α h α α l0
can be expressed in terms of the dot’s Green function l Z
X
as33,34,35 (14)
Then, the averagecurrent is
2e
I (t)= Re dt′Tr{[Gr(t,t′)Σ<(t′,t)+G<(t,t′)Σa(t′,t)]},
α ~ α α 2e
Z (7) I =hIα(t)i= h Re{ dǫTr[Gr(ǫ)Σ<α(ǫ)+G<(ǫ)Σaα(ǫ)]00}.
Here, the Green’s function Gr,< and the self-energy Z (15)
Σa,< are all two-dimensional matrices for the DQD sys- Here we give another formulism suitable for the seri-
tem. The bold-faced letters are used to denote ma- ally coupled QDs system. Making use of the Floquet
trices. The retarded and lesser Green functions are decomposition, the physical picture is clearerin this for-
defined as Gr(t,t′) = −iθ(t − t′)h{Ψ(t),Ψ†(t′)}i and mulism. As seen in Fig. 1, we should have the serially
4

15 in Fig.3 (a) and (b) respectively. We find that the cur-
(a) 16.5 (b)
rentcurvehasasymmetricBreit-Wignerlineshapewhen
12
pA) 9 16.0 the two dots are serial, and the current has an asym-
I ( 1 5.5 metric Fano line shape when the dots are parallel. Here
6
15.0 the Breit-Wigner line shape is a consequence of photon-
03 14.5 assistant resonant, i.e., ~Ω = (E1−E2)2+4t2c. The
20 30 40 50 60 20 30 40 50 60 new photon-assistant Fano lineshape comes from the
p
60 interference between photon-modified bound-antibound
50 (c) (d)
50 channels. Then we apply the same confining potentials
pA) 40 40 to the two dots and research the electron transport of a
I ( 30 30 symmetricsystem. The resultswith ~ω =3.00meV are
20 20 r
displayed in Fig.3 (c) and (d). Similarly, we find that
10 10
a Breit-Wigner resonance appears in the serially DQDs
0 0
10 15 20 25 30 35 10 15 20 25 30 35
andaFanoresonanceoccursintheparallelDQDs. From
(GHz) (GHz)
above discussion, we conclude that the ac electric field
FIG.3: (a)AveragecurrentIoftheserialDQDs. (b)Average
caninduce the photon-assistedFano resonancesfor both
currentIoftheparallelDQDswhere~ωr =3.10meV,kBT = symmetrical and asymmetrical parallel configurations of
0. (c) (d) are the same as (a) (b) respectively except that
~ωr =3.00 meV. DQDs,butcannotinduce Fanoresonanceintheserially
DQDs, whether the system is symmetric or not, which
means that the energy level of DQDs does not have re-
coupled DQD, if we set Γ1 = Γ2 = 0. In this case, the markable impact on the transport properties.
R L
self-energies are simplified. It can be shown that
e +∞ C. Transport properties of STQDs
(k) (k)
I = dε{T (ε)f (ε)−T (ε)f (ε)}, (16)
h LR R RL L
k=X−∞Z As one observes in the last section that there is no
Fano effect in a serially coupled DQD due the lack of
where
apparent interference channels. One may expect that
T(k)(ε)=Γ Γ |h1|G(k)(ε)|Ni|2 (17) thereisalsonoFanoresonanceinaSTQD.However,our
LR L R
studies shows the transport in STQD is quite different
from that in a DQD, and we will give a quite different
T(k)(ε)=Γ Γ |hN|G(k)(ε)|1i|2 (18) story in this section.
RL R L
Based on the results of electronic structure in section
(N=2)denotethetransmissionprobabilitiesforelectrons II, we study the transport properties through the sys-
from the right lead and from the left lead, respectively. temwith Λ-typethree-levelstructure(Fig.2 )under the
fℓ(ε) = (1 + exp[(ε − µℓ)/kBT])−1 denotes the Fermi action of an ac driving field with a frequency Ω.
function and In our numerical calculations, we set the applied volt-
ageµ −µ =120µV,andtheotherparameterswehave
G(k)(ε)= |uβ,k′+kihu+β,k′| (19) choseLn areRthe same as the ones for DQDs.
ε−(ǫ +k′~Ω−iγ )
β,k′ β β
X
isthe Fouriercoefficientsofthe retardedGreenfunction,
where |u i are the Fourier coefficients of the Floquet Wefirststudythetransportpropertiesofanasymmet-
β,k
state|u (t)i,ǫ ,γ (β =1,2)aretherealandimaginary ric system by applying different confining potentials to
β β β
parts of quasi energies respectively. This formulism is the left and right dots. The averagecurrent I as a func-
valid for general N serially coulped QDs system.32 tion of the driving frequency Ω with ~ω = 3.10 meV
r
is presented in Fig.4 (a). We find the current curve
has a symmetric Breit-Wigner line shape around Ω =
B. Transport properties of DQDs 129.48 GHz, suggesting that there is a resonance for
electrons in the system in this case (~Ω = E − E ).
2 1
In our numerical calculations, we assume k T = 0 The time evolution of the probabilities for an electron
B
and set the energy independent dots-lead hopping rate in left, middle, right dot are shown in Fig.4 (b). Here
Γ =Γ =9µeV,theappliedvoltageµ −µ =15µV, we have performed our calculation in a closed system
L R L R
and the ac field magnitude A = 1.54 V/cm. First we (i.e.,without interaction with the leads) and have used
study the transport properties of an asymmetric system the initial condition PL = 1 and the resonant condi-
by applying different confining potentials to the left and tion Ω = 129.48 GHz. Fig. 4(c) shows the time av-
right dots. The averagecurrent-frequencycurves of seri- erage of the probability distribution of Floquet states,
ally and parallel DQDs with ~ω =3.10 meV are shown PF = 1 T |u (t)|2dt,β =1,2,3. Itis clearthatthere is
r β T 0 β
R
5

0.6
16 (a) 1.0 (b) (c) F
P3
14 y F F
A) 12 abilit 0.8 P1 0.4 P1 P2
p b 0.6
I ( 10 Pro
0.4
0.2
8 P2
0.2
6 P3
0.0 0.0
100 120 140 160 0 100 200 300 400 -150 -100 -50 0 50 100 150

0.6
608 (d) 1.0 (e) (f) P2F
y
A)606 bilit 0.8 P1 0.4
I (p604 Proba 0.6 P1F&P3F
0.4
0.2
P3
602 0.2
P2
600 0.0 0.0
100 120 140 160 0 20 40 60 80 -150 -100 -50 0 50 100 150
(GHz) t / T x (nm)
FIG.4: (a)AveragecurrentIasafunctionofthedrivingfrequencyΩ. (b)Time-dependentoccupationprobabilitiesforelectron
inleft,middle,rightdots. (c)TimeaverageofelectronprobabilitydistributionsoftheFloquetstatesatΩ=129.48GHzwhere
~ωr =3.10meV,kBT =0. (d)(e)(f)arethesameas(a)(b)(c)respectivelyexceptthatΩ=130.57GHzand~ωr =3.00meV.
a photon assistant mixing between left and middle dot.

This photonassistantchargetransfer leads to the occur-
0.6
rence of the current resonance phenomenon.
0.4 1
Thenweconsiderthecasethatthesystemissymmetric
0.2
by applying the same confining potential to the two side )
dfroetqsu.eTnchyeΩavewriatghe~cωurr=ent3.I00asmaeVfunisctdioisnploafytehdeindrFiviign.4g Phase ( -00..02 23
r
(d). We find that the current curve has an asymmetric -0.4
Fano line shape aroundΩ=130.57GHz andthe current -0.6
amplitude is much larger than that in the asymmetric -0.8
case. Inordertounderstandtheintriguingphenomenon, -1.0
100 120 140 160
we did similar calculation of the time evolution of the
(GHz)
occupation probabilities for an electron in left, middle,
right dot and the time average of probability distribu-
FIG. 5: Phases as functions of thedriving frequency Ω.
tion of Floquet states. As shown in Fig.4 (e) and (f),
we find that two delocalized Floquet states ( which is
related to bonding and antibonding states ) are formed
and another Floquet state is still localized in the middle
dot in this case, indicating that the middle dot medi- channels. The localized Floquet state |u i has small oc-
2
ates the super-exchangeinteractionbetween the left and cupation probability in left and right dots and is weakly
right dots. The states |1i,|3i behave like “trapping dark coupled to the leads. It has very narrow width and can
state” in atomic system. Though state |2i has a very be viewed as resonantdiscrete channel. The interference
small occupation probability in the left and right dots, between strongly coupled channels and weakly coupled
it plays an importantrole in the occurrenceof the Fano- channel leads to the observed Fano resonance37. While
type resonance, which can be presented in the following in the asymmetric situation, the Floquet state |u i is
2
discussion. alsoeffectivelycoupledtotheleadsviaaphoton-assistant
AsseenfromEq. (19),threeFloquetstatesformthree mixingbetweenleftandmiddledot. Thusallthreestates
conducting channels. Due to the interaction with leads, form strongly coupled channels, so no Fano effect is ob-
the three states may have different width. In the sym- served. Herewediscussindetailtheinterferencebetween
metric situation, the bonding and antibonding Floquet different channels. From Eq. (19), one can see that the
states |u i, |u i have large occupation probability in left average current I is mainly decided by |hN|G(k)(ε)|1i|2.
1 3
and right dots, thus are strongly coupled to the leads. Theinterferenceeffectscanbeseenfromfollowingcalcu-
They have wide width and can be viewed as continuous lation (here we show term with k = 0, ε = 60 µV as an
6
example) dependence of the ac current on the electronic structure
of the system. It is found that the two-level structure
hN|G(0)(ε)|1i= hN|uβ,k′ihu+β,k′|1i = A eiφβ. does not influence the transport properties of DQDs re-
ε−(ǫ +k′~Ω−iγ ) β markably. For both symmetric and asymmetric configu-
Xβ,k′ β β Xβ rations,thereis FanoresonanceinparallelDQDsandno
(20)
FanoresonanceinserialDQDs. However,itisadifferent
In Fig.5, we show the phases φ as the func-
β(β=1,2,3) case for STQDs. The Λ-type three-level structure has
tions of the driving frequency Ω. Here, the phases φ
1 great impact on electron transport: when the system is
andφ arephasesoftwostronglycoupledstates|u iand
3 1 asymmetric, the symmetric Breit-Wigner resonance ap-
|u i,andφ isthephaseoftheweaklycoupledstate|u i.
3 2 2 pears due to phonon assistant tunneling; When the sys-
From Fig.4, we find that φ has a phase shift that varies
2 tem is symmetric, the interesting asymmetric Fano res-
from0toπasthefieldfrequencyΩismovedthroughthe
onance occurs under resonant condition. In this case,
resonancefrequency,whichprovidesresonantchannelfor
quantuminterferenceresultsintheformationoftrapping
electronsto transportthe system. φ , φ are nearlycon-
1 3 dark states. These trapping dark states are the delocal-
stantandcanbe assumedto be independent ofthe driv-
ized bonding and antibonding states which serve as the
ing frequency, which can be considered as background
continuous channels. Here the middle dot plays a dual
or nonresonant channels. Therefore, the interference be-
role: (1) It mediates the super-exchange interaction be-
tween the strongly coupled channels and weakly coupled
tween the left and the right dots; (2) The localized state
channel results in the Fano-type resonance. Finally, we
within it serves as the resonant discrete channel. Our
would like to point out that the above picture also ex-
work presents the unique quantum interference features
plains why there is no Fano resonance in a serially cou-
ofthemultiplequantumdotssystemandisusefulforthe
pled double dot. As a matter of fact, in that situation,
design of novel nanodevices. In short, for transport in a
thereareonlystronglycoupledbondingandantibonding
systemwith Nquantum dots,the situationwith N=1,is
states. Therefore, there is no apparent Fano effect.
different from that with N=2 and the case with N=2 is
In summary, we have studied the transport properties
different from that with N=3. We do not expect quali-
of DQDs and STQDs under the action of an ac electric
tative difference for system with N >3.
field. Using a two-dimensionalconfining potentialmodel
in the effective mass frame, the two-level structure and
Λ-type three-level structure are obtained through solv- ThisworkissupportedinpartbytheNationalNatural
ing the generalized eigenvalue of the system. Based on Science of China under No. 10574017, 10774016 and a
thelevelstructureandFloquettheory,weinvestigatethe grantoftheChinaAcademyofEngineeringandPhysics.
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