Quantum interference and electron correlation in charge transport through triangular quantum dot molecules

0.63 MB·English

by  Chih-Chieh Chen

Quantum interference and electron correlation in charge
transport through triangular quantum dot molecules†
Chih-Chieh Chen,a Yia-chung Chang,∗abc and David M T Kuo∗d
Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
First published on the web Xth XXXXXXXXXX 200X
5
1DOI: 10.1039/b000000x
0
2
We study the charge transport properties of triangular quantum dot molecule (TQDM) connected to metallic elec-

n
trodes, taking into account all correlation functions and relevant charging states. The quantum interference (QI)
a
Jeffect of TQDM resulting from electron coherent tunneling between quantum dots is revealed and well interpreted by
the long distance coherent tunneling mechanism. The spectra of electrical conductance of TQDM with charge filling
0
3from one to six electrons clearly depict the many-body and topological effects. The calculated charge stability dia-
gram for conductance and total occupation numbers match well with the recent experimental measurements. We also

]demonstrate that the destructive QI effect on the tunneling current of TQDM is robust with respect to temperature
l
lvariation, making the single electron QI transistor feasible at higher temperatures.
a
h
-1 Introduction tal studies have been extended to serially coupled triple
s
e quantum dots (SCTQDs) for studying the effect of long
mMoleculetransistors(MTs)provideabrightenedscenario distance coherent tunneling (LDCT) in electron trans-
.of nanoelectronics with low power consumption.1–3 To port.5–7 Triangularquantumdotmolecule(TQDM)pro-
t
adate, the implementation of MTs remains challenging, vides the simplest topological structure with quantum
mand a good theoretical understanding of their charac- interference (QI) phenomena.8–10 The QI effect in the
-teristics is essential for advancing the technology. The coherent tunneling process of TQDM junctions has been
d
current-voltage (I-V) curves of MTs are typically pre- studied experimentally.11,12
n
dicted by calculations based on the density functional
o It was suggested that the tunneling currents through
ctheory(DFT).3However,theDFTapproachcannotfully
benzenemoleculescanalsoshowadestructiveQIbehav-
[capture the correlation effect in the transport behavior
ior.3,9 The tunneling current through a single benzene
of MTs in the Coulomb-blockade regime. A theoretical
1 molecule was theoretically studied by DFT.9 However,
vframework to treat adequately the many-body problem
the influence of the strong correlation on the QI effect
7of a molecular junction remains elusive due to the com-
remainsunclearduetothelimitationofDFT.Manythe-
5plicated quantum nature of such devices. Experimental
6 oretical works have pointed out that electron Coulomb
studies of a artificial molecule with simplified structures
7 interactions have strong influence not only on the elec-
areimportantnotonlyfortheadvancesofnovelnanoelec-
0 tronic structures of TQDM,13,14 but also on the prob-
1.tronics, but also for providing a testing ground of many ability weights of electron transport paths.15–18 When
body theory. For example, the coherent tunneling be-
0 both the intradot and interdot Coulomb interactions in
5tweenseriallycoupleddoublequantumdots(DQDs)was a TQDM are included, the transport behavior involving
1studied and demonstrated for application as a spin filter
multiple electrons becomes quite complicate. The setup
:in the Pauli spin blockade regime.4 Recent experimen-
v for the TQDM junction of interest is depicted in Fig. 1.
i Herewepresentafullmany-bodysolutiontothetunnel-
X
†ElectronicSupplementaryInformation(ESI)available. SeeDOI:
ingcurrentofTQDM,whichcanwellillustratethePauli
r10.1039/b000000x/
aa Department of Physics, University of Illinois at Urbana- spin blockade effect of DQDs,4 LDCT of SCTQDs5−7
Champaign, Urbana, Illinois 61801, USA. andQIofTQDMs11 forbothequilibriumandnonequilib-
b Research Center for Applied Sciences, Academic Sinica, Taipei, riumcases. Thus,ourtheoreticalworkcanprovideuseful
11529 Taiwan. E-mail: [email protected]
c Department of Physics, National Cheng-Kung University, guidelines for the design of future molecular electronics
Tainan, 70101 Taiwan. and the realization of large scale quantum registers built
d Department of Electrical Engineering and Department of by multiple QDs.19
Physics, National Central University, Chungli, 320 Taiwan. E-
We adopt the equation of motion method (EOM),
mail: [email protected]
1–7 |1
Hubbard Hamiltonian for multiple QDs.
(cid:88) (cid:88)
H = E n + U n n (1)
QDs (cid:96) (cid:96),σ (cid:96) (cid:96),σ (cid:96),σ¯
(cid:96),σ (cid:96)
(cid:88) (cid:88)
Electrode Gate + U(cid:96)jn(cid:96),σnj,σ(cid:48) + t(cid:96)jd†(cid:96),σdj,σ,
Electrode (cid:96)<j,σ,σ(cid:48) (cid:96)(cid:54)=j,σ
whereE isthespin-independentQDenergylevel,n =
(cid:96) (cid:96),σ
d† d , U and U ((cid:96) < j) denote the intradot and
(cid:96),σ (cid:96),σ (cid:96) (cid:96)j
interdot Coulomb interactions, respectively and t de-
Substrate (cid:96)j
Substrate scribes the electron interdot coupling. The interdot
Coulomb interactions as well as intradot Coulomb inter-
actions are important for nanoscale semiconductor QDs
and molecules. Therefore, U cannot be ignored.
(cid:96),j
Fig. 1 Illustration of the TQDM junction system of UsingtheKeldysh-Green’sfunctiontechnique15,21,the
interest.
electricalcurrentfromreservoirαtotheTQDMjunction
is calculated according to the Meir-Wingreen formula
which is a powerful tool for studying electron transport, ie(cid:90) (cid:88)
taking into account electron Coulomb interactions.15–18 J = d(cid:15) Γα[G<((cid:15))+f ((cid:15))(Gr ((cid:15))−Ga ((cid:15)))],
α h j jσ α jσ jσ
This method has been applied to reveal the transport jσ
behaviors of a single QD with multiple energy levels16 (2)
and DQDs.17,18 For a TQDM with one level per QD, where Γαj((cid:15))=(cid:80)k|Vk,α,j|2δ((cid:15)−(cid:15)k) is the tunneling rate
there are 64 configurations for electrons to transport be- between the α-th reservoir and the j-th QD. Through-
tween electrodes.20 Previous theoretical works have ig- out the paper, for two-terminal devices we assume that
noredthehigh-orderGreenfunctionsresultingfromelec- the left (right) lead is only coupled to the left (right)
tron Coulomb interactions to simplify the calculation.20 QD with tunneling rate ΓL (ΓR), while there is no cou-
To have a full solution becomes crucial for depicting the pling between the center QD and the two leads. For
charge transport involving a few electrons. We solve the three-terminal devices, the coupling between the cen-
EOM of Green functions up to six electrons, taking into ter QD and a third gate is described by the tunnel-
account all correlations caused by electron Coulomb in- ing rate ΓC. fα((cid:15)) = 1/{exp[((cid:15) − µα)/kBT] + 1} de-
teractions and electron hopping between TQDM. This notes the Fermi distribution function for the α-th elec-
involves solving 4752 Green’s functions and 923 correla- trode, where µα is the chemical potential and T is
tion functions self-consistently. the temperature of the system. e, h, and kB de-
note the electron charge, the Planck’s constant, and
the Boltzmann constant, respectively. G<((cid:15)), Gr ((cid:15)),
2 Model jσ jσ
and Ga ((cid:15)) are the frequency domain representations of
jσ
the one-particle lessor, retarded, and advanced Green’s
We consider an artificial molecule made of nanoscale
functions G<(t,t(cid:48)) = i(cid:104)d† (t(cid:48))d (t)(cid:105), Gr (t,t(cid:48)) =
QDs, in which the energy level separations are much jσ j,σ j,σ jσ
larger than the on-site Coulomb interactions and ther- −iθ(t − t(cid:48))(cid:104){dj,σ(t),d†j,σ(t(cid:48))}(cid:105), and Gajσ(t,t(cid:48)) = iθ(t(cid:48) −
mal energies. Thus, only one energy level for each quan- t)(cid:104){d (t),d† (t(cid:48))}(cid:105), respectively. These one-particle
j,σ j,σ
tum dot is included. The extended Hubbard-Anderson Green’sfunctionsarerelatedrecursivelytootherGreen’s
model is employed to simulate the TQDM junction with functions and correlators via the many-body equation
Hamiltonian given by H = H + H + H , where of motion,16–18 which we solve via an iterative numer-
0 T QDs
H = (cid:80) (cid:15) c† c is the Hamiltonian for free ical procedure to obtain all n-particle Green’s func-
0 k,σ,α k k,σ,α k,σ,α
electrons in the electrodes. c† (c ) creates (de- tions (n = 1,··· ,6) and correlators for the TQDM.
k,σ,α k,σ,α (See supplemental materials.) Our procedure is valid
stroys) an electron of momentum k and spin σ with en-
ergy(cid:15) intheαelectrode. H =(cid:80) (V c† d + in the Coulomb blockade regime, but not the Kondo
k T k,(cid:96),α k,α,(cid:96) k,σ,α (cid:96),σ regime.22,23 Throughout this paper, we assume the on-
V∗ d† c ). V describes the coupling between
k,α,(cid:96) (cid:96),σ k,σ,α k,α,(cid:96) site Coulomb interaction U(cid:96) = U0 = 100Γ0 for all three
the α electrode and the (cid:96)-th QD. d† (d ) creates (de- QDs and the same tunneling rates at all leads, Γα = Γ
(cid:96),σ (cid:96),σ j
stroys)anelectroninthe(cid:96)-thdot. H istheextended with j labeling the QD directly connected to lead α.
QDs
2| 1–7
G is vanishingly small at the value of ∆ where |t |
e C eff

crosses t as indicated by dashed lines in Fig. 2(a). For
0.20 LR
2(())(()) /hG2ee0000....00114826 (atL)C=tCR=3GGL0EtLLCD CEt1LRC tCERRG R D C2 tttttLLLLeRRRRff====0000...0125GG 00G 0 iT|titslolLhutRaesolt|sprvo(aoaCtlnosiooihgsmnoihc,wipantanlhgreeeffiGnceduecaFtrs,ovihgtec-hc.dfueoo2rret(slta|eet)acde.ttfrfwol|o(niS=wt-hehee|ortdlLesa∆ChssChotyCremtdRw-md|i/ctaeu(hstUrhrvyCienedRdsc)or+.ceeuas∆DsrnviCnuoege)t)
0.00
0.08 (b) tl,j -tl,j holdfortheenergyspectrumofTQDM.13 WhenTQDM
))))h 0.06 has identical QD energy levels (E(cid:96) = E0) and homoge-
2(((( /G2ee00..0024 EL EC ER+U0 nl(cid:15)eo=vueslE(cid:15)e0l−=ecttErfoo0nr+thh2oetpcpaanisndegoosfntUere(cid:96)dn=oguthUbsl(cid:96)y,jt(cid:96)d=,jeg0=e.nUetr,nalwtiekeelehtvhaeevlecwaosintehes
0.00 of DQDs and SCTQDs, the lowest energy level depends
-30 0 30 60 90 120
D DDD cG/GGG 0 onthesignoft(cid:96),j. Thisisamanifestingresultofelectron-
hole asymmetrical behavior of TQDM. Therefore, it will
be interesting to examine the sign effect of t on the QI
(cid:96),j
Fig. 2 Electrical conductance (G ) of TQDM as a function
e behavior. Physically,thesignoft dependsonthesym-
(cid:96),j
of central QD energy for different t strengths in the Pauli
LR metrypropertiesoforbitals,whichcanchangeindifferent
spin blockade configuration with E =E and
L F configurations. We can replace t by −t to examine
E =E −U . (a) t =t =3Γ , and (b) t replaced (cid:96),j (cid:96),j
R F R LC CR 0 (cid:96),j the QI effect with respect to the electron-hole symmetry.
by −t . Other physical parameters are k T =1Γ ,
(cid:96),j B 0
The results are shown in Fig. 2(b). We find that G is
Γ=0.3Γ , U =U =30Γ , and U =0. e
0 LC CR 0 LR
enhanced with increasing t , which is attributed to the
LR
constructive QI effect, in contrast to the destructive QI
3 Results and discussion effect shown in Fig. 2(a).
To gain deeper understanding of the destructive and
constructive QI shown in Fig. 2, we compare our
Although the QI of TQDM was theoretically investi-
gated previously24,25, the effect of intradot and inter- full calculation with the weak interdot-coupling the-
ory,18,26 which allows simple closed-form expression for
dot Coulomb interactions was not considered. Here, we
the electrical conductance of TQDM. We obtain G =
utilize the LDCT effect to tune the effective hopping e
2e2/h(cid:82) d(cid:15)T((cid:15))[∂f((cid:15))/∂E ] ≈ (2e2/h)T(E ) at low-
strength between outer QDs to achieve the destructive F F
temperature limit, where the transmission coefficient
andconstructiveQIsinthepresenceofelectronCoulomb
T((cid:15)) with 64 configurations is approximately given by
interaction. We consider a TQDM junction in the Pauli
spin blockade (PSB) configuration4,18 with E = E ,
L F T ((cid:15)) (3)
PSB
E = E −U , t = t = 3Γ , Γ = Γ = 0.3Γ ,
R F 0 LC CR 0 L R 0 4Γ Γ P F
ULC = UCR = 30Γ0, and ULR = 0. Fig. 2(a) shows = |µ µ µ −t2 µ −tL2 Rµ P−SBt2 QµI −2t t t |2,
the electrical conductance (Ge) as a function of central 1 2 3 CR 1 LC 3 LR 2 LR LC CR
QD energy level (∆ = E −E ) for t varying from
C C F LR where F = µ2(t t /µ +t )2 is a factor related
0 to 0.2Γ at k T = 1Γ (in weak interdot coupling QI 2 LC CR 2 LR
0 B 0 to QI. µ = (cid:15) − E + iΓ , µ = (cid:15) − E − U and
1 L L 2 C RC
regime). For ∆ less than −15Γ , G is not sensitive
C 0 e µ = (cid:15)−E −U +iΓ . P denotes the probability
3 R R R PSB
to the variation of t , indicating that the transport is
LR weightinthePSBconfiguration.18 FromEq.(3),wehave
mainly through the upper path involving the center QD
asshownintheinsetofFig.2(a). Inthecaseoft =0,
LR
Ge canbewellexplainedbytheLDCTeffectwhenEC is G = 2e2 PPSB4Γ2(teff +tLR)2 ,
farawayfromE .26ThecentralQDprovidesaninterme- e h (Γ2+2t t +t2 )2+Γ2 (t +t )2
F eff LR LR h1 h2
diated state for electrons in the outer QDs. Through the (4)
upperpath,TQDMbehaveslikeadoubleQDwithanef- wheret =−t t /(U +∆ ),t =−t2 /(U +
eff LC CR CR C h1 LC CR
fectivehoppingt =−t t /(U +∆ ), whichcan ∆ ), and t = −t2 /(U +∆ ) with Γ = Γ ≡ Γ.
eff LC CR CR C C h2 CR CR C L R
alsobeunderstoodbythesecondorderperturbationthe- For t = t = t = 0, Eq. (4) reduces to the con-
LC CR c
ory.26 Once t (cid:54)= 0 (the lower path turns on), electron ductance of DQD,17 while for t = 0, it reduces to the
LR LR
transport through the two paths with t and t lead G of SCTQD.26 At ∆ = 15Γ and 60Γ , which sat-
eff LR e C 0 0
to a destructive QI. Note that G for the case of t (cid:54)=0 isfy the condition of t + t = 0 for t = 0.2Γ
e LR eff LR LR 0
isreducedcomparedtothecaseoft =0. Inparticular, and t = 0.1Γ , respectively, and we see G vanishes
LR LR 0 e
1–7 |3
there. This well illustrates the destructive QI seen in
Fig. 2(a). Once we make the substitution t → −t
(cid:96),j (cid:96),j 0.8 6
in Eq. (4), we can reveal the constructive QI in Ge as 0.6 (a) NG, ,k kBTT==00 ? e 5
s(theofwf/nΓin(cid:28)Fi1g).,2t(hbe).prNoobtaebtihliatytiwnetihgehtwPeaPkSBcooufpElinq.g(r4e)gicmale- 2))))2e/h 0.4 e Gee, keBBT2=1G 0 e e e 5 6 34
culatedaccordingtotheproceduresinRef. 18,wherethe (((( e0.2 1 3 4 2
interdot two-particle correlation functions are factorized G 1
as the product of single occupation numbers, is consis- 00..08 (b) kBT=0 <<nnLs sss>> 0
tent with the full calculation, but not for t /Γ (cid:29) 1. s Cssss
eff n <n n >
Atrwonaycofrrroemlattihoenswbeaeckocmoeupimlinpgorrteagnitm.eT,otheexpinlitceirtdlyotreevleeac-l relatio 00..46 <<nnLLLssssssssssssnnCRCsssssss-sssssn>Rs-sss>
theimportanceofelectroncorrelationeffects,weplotthe Cor 0.2 <nLssssnCs-sssnRssss>
curve of t =0.2Γ (with triangle marks) calculated by 0.0
LR 0 0 50 100 150 200 250
the procedure of Ref. 18 (including 64 configurations) D DDD gG/GGG 0
in Fig. 2(a). Comparison between the full solution and
the approximation considered in Ref. 18, we find that
Fig. 3 (a) Electrical conductance of TQDMs as a function
the electron correlation effects become very crucial when
of gate voltage ∆ with E =E =E =E +10Γ −∆
t /Γ(cid:29)1. Once electron transport involves more elec- g L R C F 0 g
eff and Γ=1.6Γ .(b) Correlation functions for k T =0Γ .
trons, the high-order (beyond two-particle) Green func- 0 B 0
Other physical parameters are t =3Γ and
tions and correlation functions should be included (see (cid:96),j 0
U =U =U =30Γ .
LR LC CR 0
the results of Fig. 3). The difference between the con-
ventionalmean-fieldtheoryofRef.3withthefullsolution
is even larger. The comparison between mean-field the-
trum, labeled by (cid:15) , n = 1,··· ,6 and some secondary
n
ory and the procedure of Ref. 18 has been discussed in
peaks. At higher temperature, the six main peaks are
the appendix of Ref. 18.
suppressedandbroadenedasshownbythedashedcurve,
According to Eq. (4), constructive and destructive QI andthesecondarypeaksarewashedout. Wealsoplotthe
effects depend on the sign of t(cid:96),j. Therefore, if the wave- total occupation number, N =(cid:80)σ(NL,σ+NR,σ+NC,σ)
function of the center dot has opposite parity (say, an as the black solid curve, which shows a stair-case behav-
x-like state) with respect to the wavefunctions in two ior with plateaus at N = 1,··· ,6, corresponding to the
outer dots (assumed to be s-like), then tLC will have op- fillingofTQDMwith1to6electrons. Itcanbeseenthat
posite sign compared with tCR and tLR, and the sign of thesixmainpeaksoccurat∆g whereN isincreasedby1.
teff will be flipped. Consequently, the destructive QI Thus, the peak positions (cid:15)n correspond to the chemical
shown in Fig. 2(a) will become constructive QI. Thus, potential of electrons in TQDM, i.e. the energy needed
for a center QD with an s-like ground state and an ex- to add an electron to the system. The main peak posi-
cited p-like state, it is possible to see the change of QI tions can be approximately obtained by the calculation
between destructive and constructive by tuning the gate of chemical potential of TQDM without considering the
voltage, which sweeps through different resonance ener- coupling with leads as done in Ref. 13. For example,
gies of the center QD in addition to the change of sign of (cid:15) = E −2|t |, (cid:15) = E +U −8t2/(U −U ) and
1 L c 2 L LC 0 LC
teff when the Fermi level goes from below the resonance (cid:15)3 = EL+2U(cid:96),j −3Jex/2+2|tc|+16t2c/(U0−ULC) un-
level to above the resonance level. From the results of der the condition U > U (cid:29) t , where J ≡ E (S =
0 LC c ex 0
Fig. 2, QI effect can be electrically controlled by the en- 3/2)−E (S =1/2)isthedifferenceinenergybetweenthe
0
ergy level Ec. This advantage of TQDM may be useful spin-3/2 and spin-1/2 configuration.13 However, the rel-
for improving the spin filtering of DQDs4. ativestrengthsofpeaksintheconductancespectrumcan
In addition to QI effects, spin frustration and topo- only be obtained by solving the full Anderson-Hubbard
logical effects (due to electron-hole asymmetry) on the model self-consistently.
measured quantities (electrical conductance or current) Unlike the G spectrum of DQDs,17 the G spectrum
e e
are also interesting issues.11 Fig. 3(a) shows G as a ofTQDMdoesnotshowtheelectron-holesymmetrydue
e
function of gate voltage, ∆ for E = E +10Γ −∆ ; to topological effect. Note that N = 4 and N = 5 cor-
g (cid:96) F 0 g
(cid:96) = L,C,R at two different temperatures, k T = 0 and respond to two-hole and one-hole configurations, respec-
B
1Γ . Here, we consider the homogenous configuration tively. A large Coulomb blockade separation between (cid:15)
0 3
with U =U =U =30Γ and t =t =3Γ . At and (cid:15) is given by ∆ = U +3J −4t −8t2/(U −
LR LC CR 0 (cid:96),j c 0 4 34 0 ex c c 0
lowtemperature,therearesixmainpeaksintheG spec- U ). Here, (cid:15) corresponds to the two hole ground state
e LC 4
4| 1–7
with spin triplet instead of singlet. The magnitude of
G is smaller than the quantum conductance 2e2/h for
e
t /Γ(cid:29)1asaresultofelectronCoulombinteractions.17
LR
The mechanism for understanding the unusual G be-
havior in nanostructure junction systems is a subjeect of Ge (e2/h) N
-100 -100
high interest.27 Due to electron Coulomb interactions,
0.6 5
the magnitudes of peaks are related to the probability -50 0.5 -50 4
wpaerigtihctlse oofccquupaanttiounmnpuamthbse,rswahnicdhmaarenyr-eplaatretidcletocosirnregllae-- ΓV/g20 0 000...234 ΓV/g20 0 23
tion functions.18 50 0.1 50 1
0 0
To reveal the configurations for each main peak, the
100 100
one-particle occupation number N(cid:96),σ ≡ (cid:104)n(cid:96),σ(cid:105), inter- -100 -50 0 50 100 -100 -50 0 50 100
dot two particle correlation functions (cid:104)n n (cid:105) , and Vg1/Γ0 Vg1/Γ0
(cid:96),σ j,σ¯
three particle correlation functions ((cid:104)n n n (cid:105),
L,σ C,−σ R,σ
and (cid:104)n n n (cid:105)) are plotted in Fig. 3(b). We al-
L,σ C,−σ R,σ
ways have the relation N = N (cid:54)= N , because
L,σ R,σ C,σ
theouterQDsaredirectlycoupledtoelectrodes, butnot
thecentralQD.Sucharelationalsoholdsfortwo-particle Fig. 4 Charge stability diagram of TQDM. Γ =Γ ,
(cid:96) 0
and three-particle correlation functions. The six main kBT =1.5Γ0, ULC =UCR =ULR =12Γ0, t(cid:96),j =3Γ0. The
peaks in Fig. 3(a) indicate the filling of TQDM up to energy levels are shifted according to
the n-electron ground state for n = 1,··· ,6. For ex- E(cid:96) =EF −ULR−(cid:80)2m=1β(cid:96),meVgm, where the gate coupling
constants are β =0.5, β =1, β =0.5, β =1,
ample, (cid:15) indicates the formation of two-electron state L,1 C,1 R,1 L,2
2
β =0.5, and β =1.
with spin singlet, while (cid:15) indicates the formation of C,2 R,2
3
three-particle state with total spin S = 1/2, which can
be described as the spin-frustration state11,13,28. Be-
cause the on-site Coulomb interaction favors homoge- arenotsymmetricalwithrespecttogatevoltage. InRef.
neous distribution of three electrons in TQDM, whereas 11, a capacitive interaction model was employed to plot
the interdot Coulomb repulsion favors the charge fluc- the diagram of N. In their model, the electron hopping
tuation. As seen in Fig. 3(b), for ∆g ≤ (cid:15)3 ( 78Γ0) N(cid:96) strength t(cid:96),j was ignored. Consequently, the charge sta-
in each dot clearly displays the charge fluctuation be- bility diagram of Ge cannot be obtained. The charge
havior. When TQDM goes into a three-particle state stability diagram of Ge obtained by our full calculation
(∆ > (cid:15) ), the charge fluctuation is suppressed, and [as shown in Fig. 4(a)] bears close resemblance to the
g 3
each QD is filled with one particle (with N =N = experimental results as shown in Fig. 2 of Ref. 11.
L,σ R,σ
NC,σ =0.5), while(cid:104)nL,σnC,σnR,−σ(cid:105)=(cid:104)nL,σnC,−σnR,σ(cid:105)= So far, the results shown in Figs. 2-4 are all related
(cid:104)nL,−σnC,σnR,−σ(cid:105). This also demonstrates the spin frus- to the linear response. To further clarify the QI effect
tration condition as depicted in the inset of Fig. 3(a). at finite bias (V = 10Γ ) for different temperatures, we
a 0
Figure 4 shows the charge stability diagram for zero- plot in Fig. 5(a) the tunneling current as a function of
bias electrical conductance (G ) and total occupation center QD energy, E = E + ∆ for the configura-
e C F c
number (N) as functions of gate voltages exerted on any tion shown in the inset of Fig. 5(b). For ∆ ≥ 2Γ ,
C 0
twoQDs(labeledbyV andV )foraTQDMconnected the tunneling current is suppressed as temperature in-
g1 g2
tothreeterminals. ThemagnitudesofG andN areindi- creases. Thisisattributedtoareductionofelectronpop-
e
catedbydifferentcolors. ItisnoticedthatG isenhanced ulation in the electrodes for electrons with energy near
e
on the borders that separate domains of different values E +10Γ . For ∆ ≤ 2Γ , we notice that N quickly
F 0 C 0 c,σ
of occupation number (N) with larger G occurring at jumps to 0.4, indicating that the central QD is filled
e
V = V . This is a result of higher degeneracy and with charge, which causes an interdot Coulomb block-
g1 g2
charge-fluctuationinthestate. ThelargestG forN ≤3 ade for electrons entering the left QD. (See the reduc-
e
occursatthejunctionbetweenN =1andN =2domains tion of N in Fig. 5(b)). This explains the sharp dip
L,σ
when V =V . This feature corresponds to the (cid:15) peak of G for ∆ ≤ 2Γ in Fig. 5(a). As temperature in-
g1 g2 2 e C 0
of Fig. 3(a). The diagram Fig. 4(a) is simply a collection creases, such a dip in tunneling current is smeared out.
of curves displayed in Fig. 3(a) at different values of V At ∆ =10Γ +t t /t ≡∆ , the tunneling cur-
g2 C 0 LC CR LR QI
that shifts the QD energy levels. We note that in the rent vanishes for all temperatures considered due to the
domains of N = 1 and N = 2, the areas with stripes QIeffect. SucharobustdestructiveQIeffectwithrespect
1–7 |5
ments very well. Finally, we demonstrated that the QI

effect in TQDM is robust against temperature variation
))))h 12..60 (a) JMax kkBBTT==00, Uij=0 and it can be utilized to control the charge storage.
/0 1.2 kBT=2G GGG 0 Acknowledgments
GGGGe 0.8 kT=4G GGG
((((J 0.4 D DDD QI B 0 ThisworkwassupportedinpartbytheNationalScience
Council, Taiwan under Contract Nos. NSC 101-2112-M-
0.0
0.4 (b) kT=0 001-024-MY3 and NSC 103-2112-M-008-009-MY3.
B
0.3 E
C
E E N References
Nssssl0.2 EF+Va L R E NLssss
F Cssss
0.1 NRssss 1 M. A. Reed, C. Zhou, C. J. Muller, T. P. Burgin and J. M.
0.0 Tour,Science,1997,278,252–254.
0 20 40 60 80 100 120 140 2 C. Joachim, J. Gimzewski and A. Aviram, Nature, 2000, 408,
D DDD cG/GGG 0 541–548.
3 J.BergfieldandC.Stafford,Phys. Rev. B,2009,79,245125.
4 R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha and
Fig. 5 (a) Tunneling current as a function of L.M.K.Vandersypen,Rev. Mod. Phys.,2007,79,1217–1265.
∆C =EC−EF at finite bias Va =10Γ0 for various 5 M. Busl, G. Granger, L. Gaudreau, R. S´anchez, A. Kam,
temperatures. (b) Occupation numbers as a function of ∆ M. Pioro-Ladri`ere, S. A. Studenikin, P. Zawadzki, Z. R.
C
at k T =0. The other physical parameters used are Wasilewski, A.S.SachrajdaandG.Platero, Nature nanotech-
B
nology,2013,8,261–265.
E =E =E +10Γ , U =U =30Γ , U =10Γ ,
L R F 0 LC CR 0 LR 0
6 F. R. Braakman, P. Barthelemy, C. Reichl, W. Wegscheider
t =t =3Γ , t =0.4Γ , and Γ=Γ .
LC CR 0 LR 0 0 and L. M. K. Vandersypen, Nature nanotechnology, 2013, 8,
432–437.
7 S. Amaha, W. Izumida, T. Hatano, S. Teraoka, S. Tarucha,
to temperature provides a remarkable advantage for the J. A. Gupta and D. G. Austing, Phys. Rev. Lett., 2013, 110,
realization of single electron QI transistors at room tem- 016803.
perature.12 Tounderstandtheinterdotcorrelationeffect, 8 A.NitzanandM.A.Ratner,Science,2003,300,1384–1389.
9 C.A.Stafford,D.M.CardamoneandS.Mazumdar,Nanotech-
we also plot the case without interdot Coulomb interac-
nology,2007,18,424014.
tion (U = 0) for k T = 0 in Fig. 5(a). (See solid
(cid:96),j B 10 C. Po¨ltl, C. Emary and T. Brandes, Phys. Rev. B, 2013, 87,
curve) We notice that the QI effect remains qualitatively 045416.
the same, except that the tunneling current is slightly 11 M. Seo, H. K. Choi, S.-Y. Lee, N. Kim, Y. Chung, H.-S. Sim,
enhanced with interdot Coulomb interaction turned off. V. Umansky and D. Mahalu, Phys. Rev. Lett., 2013, 110,
046803.
As the QI effect suppresses the current flow, the charge
12 C. M. Gu´edon, H. Valkenier, T. Markussen, K. S. Thygesen,
will accumulate in the left dot. Thus, N reaches the
L,σ J.C.HummelenandS.J.vanderMolen,Naturenanotechnol-
maximum at ∆C =∆QI while NR reaches the minimum ogy,2012,7,304–309.
as seen in Fig. 5(b). This implies that the QI effect can 13 M. Korkusinski, I. P. Gimenez, P. Hawrylak, L. Gaudreau,
be utilized to control charge storage in TQDM. S.A.StudenikinandA.S.Sachrajda,Phys. Rev. B,2007,75,
115301.
14 C.-Y.Hsieh,Y.-P.ShimandP.Hawrylak,Phys. Rev. B,2012,
4 Conclusions 85,085309.
15 Y.MeirandN.S.Wingreen,Phys.Rev.Lett.,1992,68,2512–
2515.
In summary we have obtained full solution to the charge
16 D. M.-T. Kuo and Y.-C. Chang, Phys. Rev. Lett., 2007, 99,
transport through TQDM junction in the presence of
086803.
electron Coulomb interactions, which includes all n- 17 B.R.Bu(cid:32)lkaandT.Kostyrko,Phys. Rev. B,2004,70,205333.
electron (n=1,··· ,6) Green’s functions and correlation 18 D. M.-T. Kuo, S.-Y. Shiau and Y.-C. Chang, Phys. Rev. B,
functions. ThedestructiveandconstructiveQIbehaviors 2011,84,245303.
19 D.LossandD.P.DiVincenzo,Phys.Rev.A,1998,57,120–126.
ofTQDMareclarifiedbyconsideringtheLDCTeffecton
20 I.Weymann,B.Bu(cid:32)lkaandJ.Barna´s,Phys. Rev. B,2011, 83,
theconductancespectrum. Theconductancespectrumof
195302.
TQDM with total occupation number varying from one 21 H. Haug and A.-P. Jauho, Quantum kinetics in transport and
to six directly reveals the electron-hole asymmetry due optics of semiconductors,Springer,2007.
to topological effect. The calculated correlation func- 22 D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-
Magder,U.MeiravandM.A.Kastner,Nature,1998,391,156–
tions also illustrate the charge fluctuation and spin frus-
159.
tration behaviors of TQDM. Our numerical results for
23 T. Numata, Y. Nisikawa, A. Oguri and A. C. Hewson, Phys.
charge stability diagram match experimental measure- Rev. B,2009,80,155330.
6| 1–7
24 B. Michaelis, C. Emary and C. Beenakker, EPL (Europhysics
Letters),2006,73,677.
25 C.Emary,Phys. Rev. B,2007,76,245319.
26 D.M.T.KuoandY.-C.Chang,Phys.Rev.B,2014,89,115416.
27 F. Bauer, J. Heyder, E. Schubert, D. Borowsky, D. Taubert,
B. Bruognolo, D. Schuh, W. Wegscheider, J. von Delft and
S.Ludwig,Nature,2013,501,73–78.
28 S.Andergassen,Nature,2013,495,321–322.
1–7 |7