Effective level attraction and magnetic flux-induced negative differential conductance in two double quantum dot molecules embedded in an Aharonov-Bohm ring

1.3 MB·

by  M. L. Ladrón de Guevara

Effective level attraction and magnetic flux-induced negative differential conductance
in two double quantum dot molecules embedded in an Aharonov-Bohm ring
M. L. Ladr´on de Guevara1, Gustavo A. Lara2, and P. A. Orellana1
1Departamento de F´ısica, Universidad Cat´olica del Norte, Casilla 1280, Antofagasta, Chile and
2Departamento de F´ısica, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
(Dated: January 19, 2009)
We study transport of non-interacting electrons through two quantum dot molecules embedded
9 in an Aharonov-Bohm interferometer. The system in equilibrium exhibits bound states in the
0 continuum (BIC) and total suppression of transmission. It also shows a magnetic flux-dependent
0 effectivelevelattractionandlinesofperfecttransmissionwhentheintramolecularcouplingisweak.
2 Outofequilibrium,thecurrentdisplaystwokindofnegativedifferentialconductance(NDC)regions,
whichhavedifferentorigins. OneisgeneratedbytheusualmechanismoftheNDCarisinginadouble
n
quantum dot system. The other is induced by the magnetic flux, and it occurs at small voltages
a
and for a well definite range of the intramolecular couplings. We explain this effect in terms of the
J
level attraction displayed bythesystem.
9
1
PACSnumbers: 73.21.La;73.63.Kv;85.35.Ds;85.35.Be

]
l
l I. INTRODUCTION earconductance22. Thiseffectiscausedbytherenormal-
a
h ization of levels by the leads.
- Fromtheapplicabilitypointofview,aninterestingfea-
s Advances in experimental techniques at the nanome-
e terscalehaveallowedtorealizeandmanipulatequantum ture exhibited by the transport through quantum dots
m dotsinacontrolledway1. Thishaspermittedtostudyin is the negative differential conductance (NDC)24. NDC
. thesesystemsawidespectrumofphenomena2,aswellas has been studied in single as well as in double quan-
t
a to takeadvantageofsome propertiesofquantumdots to tum dot systems, and it has applications in amplifiers
m and oscillators in the microwave, mm-wave and Tera-
build nano-devices, such as rectifiers, amplifiers, lasers,
- and others3,4,5,6. For their similitude with atoms, quan- hertz frequency ranges25. In multilevels quantum dots
d NDC can occur when states have different couplings to
tum dots are often viewed as artificial atoms, and two
n the leads26,27,28,29. In a serial DQD, NDC can be pro-
or more coupled quantum dots as artificial molecules.
o
ducedwhenthe biasbreaksthetransmissionchannelex-
c Multiple quantum dot systems, and in particular the
[ double quantum dot (DQD) molecule7, are of particu- tended along the system30. Other theoretical works on
generationof NDC in DQD connected in series are Refs.
lar importance, because they are more controllable than
1
31,32,33,34,35. In a DQD embedded in an Aharonov-
single quantum dots. Moreover, they admit different
v
ring,magnetic-flux-inducedNDCwasfoundinthestrong
5 kind of connections to leads. Initially, most of studies
7 considered serial DQDs8, but posteriorly parallel9,10,11 interdot repulsion regime19. A similar result was found
8 and T-shaped12,13 configurations were also examined, by MourokhandSmirnov in a DQD molecule with three
2 emerging quantum interference effects due to the exis- terminals36. Recently, it was reported NDC induced by
. the electronic correlationin a side-coupled DQD37.
1 tence of different electronic paths. Much attention has
0 received the DQD molecule embedded in an Aharonov- Inthisworkwestudyequilibriumandnon-equilibrium
9 Bohm ring, where the presence of a magnetic flux adds transport through two quantum dot molecules embed-
0 a new tool in the control of the transport properties. ded in an Aharonov-Bohm interferometer. We obtain
:
v Aharonov-Bohm (AB) oscillations and Fano effect in a analytical expressions for the transmission and we cal-
i DQD molecule embedded in a ring have been observed culate numerically the current at zero temperature. In
X
experimentally9,14,15. There is much theoretical work equilibrium, the transmission exhibits Fano resonances,
ar supporting these experiments and exploring new phe- totalreflection, and suppressedpeaks as a manifestation
nomenaintheDQDembeddedinaring. Fanoresonances of bound states in the continuum. Moreover, we find
andthe magnetic-fluxcontrollabilityoftransportareex- for small intramolecular couplings a flux-dependent ef-
amined, for example, in Refs. 10,11,16,17,18,19, both in fective level attraction and lines of perfect transmission.
presence and the absence of electronic correlations. The This effect occurs no matter how weaklycoupled are the
existenceofboundstatesinthecontinuum(BICs)isdis- quantum dots forming each molecule.
cussed in Refs. 10,16. The interplay between quantum In the non-equilibriumregime,we identify twokind of
interference and Kondo physics in the parallel-coupled NDC regionsin the I-V characteristics,occurringat dif-
DQD has been also explored20,21. A closely related sys- ferent scales and of different origin. One is generated by
tem which has received attention is the AB interferome- the usual mechanism of the NDC in a double quantum
ter with two embedded quantum dots22,23. Kubala and dot systems, and it is independent of the magnetic flux.
K¨onigstudiedequilibriumtransportinthissystem,find- The current will increase or decrease with voltage, de-
ing aneffective flux-dependent level attraction in the lin- pending on whether the voltage makes the levels of the
2
different quantum dots become aligned or not aligned.
The secondNDC regionisinduced bythe magneticflux,
and it occurs only for a definite range of intramolecular
couplings. Anabruptriseofcurrentoccursforsmallbias
voltages as consequence of the effective level attraction
of the hybridized levels produced by the magnetic flux.
Thedecreaseofcurrentisresultofthedestructionofthis
effect when the bias voltage is increased.
FIG. 1: Two double quantum dot molecules embedded in
parallel in an Aharonov-Bohm interferometer
II. MODEL
with VA(B), the tunneling coupling connecting the left
α
The system under consideration is shown in Fig. 1.
(right)dotoftheα-thmoleculewiththeleft(right)lead.
Two equal double quantum dot molecules are embed-
Werestricttothecaseinwhichthereisnotmagneticfield
ded in an Aharonov-Bohm ring, which is attached to
actingdirectlyontheelectrons,sothatthesituationwill
large contacts through one-dimensional leads. Equilib-
beidenticalforthetwovaluesofspin. Thereforeinwhat
riumtransportinasimilarconfigurationwithadditional
follows we omit the spin index.
connectionsbetweendotswasstudiedbyLiet al.38. The
In presence of a magnetic flux threading the ring,
left and right contacts are in thermodynamic equilib-
and using gaugeinvariance,we add the Aharonov-Bohm
rium with thermodynamical potentials µ and µ , re-
L R phase φ=2πΦ/Φ around the ring by the replacement
0
spectively. The leads are assumed to be ballistic con-
ductors. We assume that a bias voltage V/e is applied VA = t e−iφ/6, t =t e−iφ/6 VB =t e−iφ/6,
+ A + c + B
betweensourceanddrainsuchthatthesiteenergyisV/2
VA = t eiφ/6, t =t eiφ/6 VB =t eiφ/6, (3)
for the left lead and V/2 for the right lead. We con- − A − c − B
−
sideronlyonelevelrelevantineachofthequantumdots.
withΦ =h/ethe flux quantum. We lookforthe steady
The system is modeled by a non-interacting Anderson 0
states ψ of the whole Hamiltonian H. The Hamil-
Hamiltonian, which can be written as | ki
tonian describing the leads, H , corresponds to a free-
0
H =H +H +H , (1) particle Hamiltonian on a lattice, the eigenfunctions be-
M 0 I
ing Bloch functions
where H describes the dynamics of the isolate
M
molecules, k = eikβj j , β =L,R, (4)
β
| i | i
HM = εid†iαdiα Xj
α=X+,−i=XA,B where kβ isthemomentumeigenstateand j aWannier
| i | i
+ (t d† d +t∗d† d ), (2) state localized at the j-th site. The corresponding dis-
α A,α B,α α Bα Aα
persionrelationsareε=V/2 2vcosk ,fortheelectrons
α=X+,− originated in the left contact−, and ε =L V/2 2vcosk
R
whereεA(B)isthelevelenergyoftheleft(right)quantum forthoseoriginatedinthe rightcontact−. The−eigenstates
dot in the molecule α (α=+, ); d (d† ) annihilates of the entire Hamiltonian can be written as
− i,α i,α
(creates)anelectronindotiinthe moleculeα,andt is
α −1 N
the intramolecular tunneling hopping. H0 is the Hamil- ψ = ak j + bk i,α + ak j (5)
tonian for the noninteracting electrons in the leads, | ki j| i i,α| i j| i
j=−N α=+,−i=A,B j=1
X X X X
−1 −1
H = V c†c +v (c†c +c† c ) We assume electrons as described by a plane wave inci-
0 2 i i i i−1 i−1 i dent fromthe far left (right)with unit amplitude, reflec-
i=−N i=−N
X X tion amplitude r (r′), and transmission amplitude t (t′).
N N
V Thus, for electrons incident from the left,
c†c +v (c†c +c† c )
− 2 i i i i+1 i+1 i
Xi=1 Xi=1 eikLj +re−ikLj j 1
akL = ≤− (6)
whereci (c†i)istheannihilation(creation)operatorofan j (teikRj, j ≥1
electron in the site i-th of the leads, and v the hopping
betweensites inthe leads. The termH accountsforthe and for electrons incident from the right,
I
tunneling between molecules and leads,
e−ikRj +r′eikRj j 1
HI =− (VαAd†Aαc−1+VαA∗c†−1dAα) akjR =(t′e−ikLj, j ≥ 1 (7)
α=+,− ≤−
X
(VBd† c +VB∗c†d ) where the two contributions are independent. Inserting
− α Bα 1 α 1 Bα Eqs. (5),(6)and(7)intheSchr¨odingerequationH ψ =
α=+,− k
X | i
3
E ψ , we can solve for ak and bk . We are interested action of the magnetic flux in the conductance G =
inkt|hekitransmission and thje curreni,tαthrough the system (2e2/h)T(0). This is given by
for an applied voltage V/e between contacts. We center
2e2 4cos2(πΦ/Φ )(t /Γ)2
in the the symmetrical configuration, that is, equal left G= 0 c . (12)
and right dot-lead couplings, tA = tB. Additionally, the h [cos2(πΦ/Φ0)+(tc/Γ)2]2
energies of the quantum dots are ε =µ and ε =µ .
A L B R WedistinguishinEq. (12)twodifferentbehaviors. When
The current in the leads is given by
t Γ there is always a value of Φ for which G=2e2/h,
c
≤
2iev while when t > Γ this never occurs. These behaviors
I = c† c c†c , (8) c
¯h h j+1 ji−h j j+1i are illustrated in Fig. 2, which shows the conductance
(cid:16) (cid:17) versus the magnetic flux for two different values of the
where
t . For t = 0.25Γ, the conductance reaches the max-
c c
1 1
c†c = f(ε µ )akα∗akα. (9)
h i ji 2 N kα − α i j
α=X{L,R} Xkα
Weassumethatthevoltagedropoccursonlybetweenthe
dotAandthedotB,sothattheenergiesofthequantum
dots are equal to the site energies of the adjacent leads,
ε = V/2 and ε = V/2. At zero temperature, the
A B
−
states that contribute to the net current are the states
of the left lead with energies between µ = V/2 and
R
−
µ =V/2. Evaluating (8)-(9), we arrive to the following
L
expression for the current
2e V/2
I(V)= T(ε)dε, (10)
h
Z−V/2
FIG. 2: (Color online) Conductance versus magnetic flux for
where T(ε) is the transmission. We have assumed that
tc =0.25Γ (solid line) and tc =1.5Γ (dash line).
the Fermi level in equilibrium is equal to 0.
imum G = 2e2/h in the interval Φ = 0 to Φ = Φ /2,
0
being symmetric around Φ /2. Such a maximum occurs
III. RESULTS 0
at ε = arccos(t /Γ)Φ /π. Similar features are found for
c 0
any value of t Γ, no matter how small it is. This is a
Thetransmissionprobabilitycanbewritteninthesim- c ≤
remarkableresult,takingintoaccountthatforamolecule
ple form
inseriestheconductance,proportionalat2,getsprogres-
c
T(ε)= 4t2cΓ2(∆ε)4cos2φ/2 (11) sivelysmallerastc decreases7. Incontrast,fortc >Γthe
[(∆ε)4+Γ2(ε q)2][(∆ε)4+Γ2(ε+q)2] conductancedecaysmonotonouslywithΦinthesamein-
− terval, never reaching the quantum limit.
where (∆ε)2 = ε2 p2, with p2 = (V/2)2 + t2 and Theoccurrenceofmaximumconductanceforsmallval-
− c
q2 = (V/2)2 +t2cos2(φ/2), where Γ = 4πt2ρ(0) is the uesoft isassociatedtoaneffectivelevelattractionofthe
c A c
characteristic line width, with ρ(0) the density of states hybridized levels, similar to the discussed in Ref. 22 in
in the leads atthe Fermi level. It is evident from the ex- an Aharonov-Bohm interferometer with two single-level
pression(11)thatthetransmissionhasaperiod∆φ=2π quantum dots. This effect is illustrated in Fig. 3, which
(∆Φ=Φ0). On the other hand, it is totally suppressed, shows transmission versus energy for tc = 0.25Γ. In the
both in and out of equilibrium, when φ is an odd mul- absenceof magnetic flux, the transmissionbehavesqual-
tiple of π (Φ = nΦ0/2, n odd). This fully destructive itatively as that of the DQD in series. In Fig. 3a (dash
interferenceeffect forthis value ofφis expected, since in line), t is small and a single and flat peak is observed.
c
the absence of magnetic flux the upper and lower paths The hybridized states are not resolved yet, which is the
are equivalent, then the magnetic flux introduces in the usualforamoleculeinserieswithsmallt . However,two
c
wavefunctionaphase π/2alongonearmandπ/2along BICsareoccurring,similarlytowhathappenfora DQD
−
the other. The total suppression of transmission was embedded in anAB ring10,16. In other words,two of the
discussedinan Aharonov-Bohminterferometerwith two hybridized states are localized and do not participate of
quantum dots, in equilibrium22. transmission. This situation repeats whenever Φ = nΦ
0
(n integer). When the flux is on, the BICs are replaced
by two Fano resonances, as seen in the rest of Figs. 3.
A. Equilibrium transport The Fano peaks reach T =1.
In the sequence from Fig. 3a to Fig. 3d it can be
The equilibrium transmission is obtained from Eqs. observedthe flux-controlled level attraction between hy-
(11)-(13) by making V = 0. Let us first analyze the bridized states. As Φ is increased from 0 to Φ /2 the
0
4
FIG. 5: (Color online) Positions of the T = 1 peaks for dif-
FIG. 3: Transmission versus energy for tc = 0.25Γ and a) ferent valuesof tc Γ/√2.
Φ = 0 (dash line) and Φ = 0.1Φ0 (solid line), b) Φ = 0.2Φ0, ≤
c)Φ=0.3Φ0,d)Φ=0.4Φ0 (solidline)andΦ=0.46Φ0 (dash
line).
the band as Φ increases. The peaks meet at ε = 0 at
Φ = arccos(t /Γ)Φ /π. The situation slightly changes
c 0
Fano peaks get progressively closer to each other until when Γ/√2 < tc Γ, where two new real solutions of
≤
overlapping completely, the transmission decaying when Eq. (13) arise in the interval arccos(tc/Γ)Φ0/π < Φ
riΦnegai.cshTceldhoeswevhateolnueΦth20/ee22t/w(hdoianpshetahkleisnceoovniendrluFacpitga.tnoc3teadl)sl,yh.ouwInntniclaovbnaotnrviaesshits-, iafnorrcFcaoilgsl.(Φ16−≤foΓarr2c/tc2cots2c=)(Φ100−./82ΓΓπ2a,/n2wdth2c)itlΦce0=ε/+12Γπa..nTdThhεi−1es isbsoeliucllotuimsotnersarteεea≤+1dl
for tc = 1.5Γ the level attraction is not significant, as and ε+1 never meet, but ε+2 and ε−2 do so at ε=0, being
observed in Fig. 4. responsible of the conductance G = 2e2/h for these val-
ues of t . When t >Γ there are four T =1 peaks when
c c
Φ arccos(1 Γ2/2t2)Φ /2π. These peaks never reach
≤ − c 0
the center of the band, as shown for t =1.5Γ.
c
FIG. 4: Transmission versus energy for tc = 1.5Γ and a)
Φ = 0, b) Φ = 0.03Φ0, c) Φ = 0.1Φ0, d) Φ = 0.2Φ0 (solid
line) and Φ=0.4Φ0 (dash line). FIG.6: Positions of themaxima of transmission for different
values of tc > Γ/√2. The solid lines correspond to ε+1 and
It follows from Eq. 11 that perfect transmission takes ε−,and thedash lines to ε+ and ε−.
1 2 2
place at energies obeying the following equation
We show below that he flux-dependent level attraction
ε4 ε2(2t2 Γ2)+t2[t2 Γ2cos2(φ/2)]=0, (13)
− c − c c − and the complete overlapping of resonances present for
small values of t strongly influence the behavior of the
with two pair of solutions c
non-equilibrium transport.
ε± = [A+B1/2]1/2, ε± = [A B1/2]1/2 (14)
1 ± 2 ± −
with A = t2 Γ2/2 and B = Γ2 4t2sin2(φ/2). Fig. B. Non-equilibrium transport
c − − c
5 shows the positions of the T = 1 peaks for different
values of t Γ/√2. These correspond to ε+ and ε−, Let us now consider a voltage V applied between con-
c ≤ 1 1
the only real solutions of Eq. (13). In the three cases tacts and let us study the current in the leads. Fig. 7
the peaks positions shift progressively to the center of (upper panel) shows the current-voltage characteristics
5
for Φ = 0 and different values of intramolecular cou-
plings.
Inallcasesthe current-voltagecharacteristicsdisplays
a peak, with the corresponding region of NDC. This
feature occurs analogously to the observed in the serial
DQD30. The current increases when the bias allows a
transmissionchannelexists along the left and rightsides
ofthesystem. Ifthebiascontinuestoincrease,thechan-
nel is destroyed resulting in the drop of current. On the
other hand, as occurs for quantum dots connected in se-
ries,ingenerallargervaluesoft givelargervaluesofthe
c
current for the same voltage.
The existence of a magnetic flux produces changes in
the I-V characteristics which become important when
tc < Γ/√2 and Φ is within an interval close to Φ0/2, as FIG. 8: (Color online) I-V characteristics for tc = 0.1Γ and
illustrated in Fig. 7 (lower panel), where Φ = 0.46Φ0. Φ=0.46Φ0 (left panel). Transmission spectrum for different
We observe sharper current peaks at lower voltages as bias voltages for the same parameters (right panel).
compared to the case Φ=0. The abrupt increase of the
eters of Fig. 8. In equilibrium (V =0) a molecular state
is formed. For the cases 2 and 3 the coherence is still
preservedbutforhighervoltages(cases4-5)the physical
picture changes. In these cases, the coherence between
dots is lost, the electron is localized at the left quantum
dots and the molecular bridge is broken.
FIG. 7: (Color online) I-V characteristics for Φ = 0 (upper
panel) and Φ = 0.46Φ0 (lower panel), and tc = 0.1Γ (solid
line), tc = 0.25Γ (dash line), tc = 0.5Γ (dotted line), tc = FIG. 9: (Color online) Densities of states of the electrons
0.75Γ (dash-dotted line), tc =Γ (short dash line). coming from the left at the quantum dots A (left panel) and
B (right panel), for tc =0.1Γ and Φ=0.46Φ0.
currentat low voltagesfor small t is consequence of the
c
level attraction discussed for zero bias. To visualize this Fig. 10showsthecurrentversusbiasvoltageandmag-
we have plotted in Fig. 8 the I-V characteristics for a netic flux for t = 0.1Γ. It can be identified clearly the
c
fixed t and Φ (left panel), and the transmission spectra tworegionsofmaximalcurrentatdifferentscales. Inthe
c
associated to the bias voltages indicated in the current upper panel, there is a broad peak centered in Φ = 0
curve (right panel). As observed, for small bias (cases and V 1.2Γ. The behavior of the current versus volt-
≈
1-3) the transmission keeps large in all the transport re- age in this region was already discussed for zero flux in
gion ( V/2 < ε < V/2), due to the existence of two Fig. 7. The current maximum in this case decreases
−
overlapped resonances close to each other. Larger bias monotonously with the magnetic flux, remaining its po-
voltages make the heights of the resonances fall, so that sition almost unchanged. In the same panel it is high-
the transmission in all the window of transport becomes lighted a region close to Φ /2 and small voltages, which
0
smaller, occurring the observed decrease in the current. is plotted in the lower panel. The observed peak corre-
A further insight of this is obtained through the density sponds tothe enhancementofcurrentwiththe magnetic
ofstatesoftheleftandrightquantumdots. Fig. 9shows flux taking place for weak intramolecular couplings, de-
theleftandrightquantumdotsDOSforthesameparam- scribed in Fig. 7, lower panel. The maximum of current
6
now occurs around Φ = 0.44Φ and V = 0.029Γ. Also, not break the negative differential conductance. A work
0
this peak is clearly defined when t Γ/√2, when the in this direction is under progress.
c
≤
flux produces a total level attraction in equilibrium and
it is not present when t Γ or larger. It is important
c
∼
IV. SUMMARY
We studied the transmission and the I-V characteris-
tics for two double quantum dot molecules embedded in
an Aharonov-Bohm ring. We showed that for t Γ,
c
≤
the magnetic flux can be used to controltotally the con-
ductance, allowing this to take any value between 0 and
2e2/h. When t Γ/√2 the flux produces an effective
c
≤
levelattractionandlinesofperfecttransmission,allowing
the levels to meet at the center of the band at a deter-
minate value of the flux. On the other hand, the system
displays BICs when Φ=nΦ (n integer), and total sup-
0
pression of transmission when Φ=nΦ /2 n odd. In the
0
non-equilibriumregime,we identify twokindofNDC re-
gions in the I-V characteristics, occurring at different
scales and of different origin. A first current peak exists
FIG.10: (Coloronline)Currentversusbiasvoltageandmag-
atvoltagesofthe orderofthe characteristiclinewidth Γ,
netic fluxfor tc =0.1Γ.
and it is independent of the magnetic flux. The role of
the flux inthis caseis to controlthe heightofthe overall
to note that the latter feature does not exist if the mag-
current. The drop of current with the increase of bias
netic flux is absent, so that in this case we can properly
has analogous explanation to the NDC region in a se-
talk of magnetic flux-induced NDC. Similar results are
rialDQD.AsecondpeakintheI-V characteristicstakes
discussed in a parallel DQD molecule embedded in an
place only when t < Γ, at voltages V Γ/10 and it
Aharonov-Bohmring19 and in a molecule in a three ter- c ∼
is strongly dependent on the magnetic flux. In fact, it
minalsconfiguration36,inbothcasesNDCoccuredinthe
does not exist if the flux is absent. The current suffers
strong interdot repulsion limit.
an abrupt rise for small bias voltages, as consequence of
We expect that the above picture remains valid even
an effective level attraction of the hybridized levels pro-
if the electron-electron interaction is taken into account.
duced by the flux. The decrease of current is result of
In fact, in embedded QD arrays, the main effect of the
the destruction of this effect when the bias is increased.
electron-electron interaction is to shift and to split the
? ?
resonance positions . This occurs because the on-site
Coulomb repulsion energy U introduces a renormaliza-
Acknowledgments
tion of the site energies. In analogy with QD arrays in
series,we expectthatdepending onthe relationbetween
theinterdotcouplingandtheon-siteCoulombinteraction The authors acknowledge financial support from
different regimes arise. For t /U 1, the resonances FONDECYT, under grant 1080660. M. L. L. de G.
c
≪
and antiresonances would split into two distinct mini- thanks financial support from Milenio ICM P06-067-F,
bands separated by the on-site Coulomb energy, while and P. A. O. and G. A. L. from CONICYT/Programa
for t /U 1, the resonances and antiresonances would Bicentenario de Ciencia y Tecnolog´ıa (CENAVA, grant
c
≫
occur in pairs. We think that the above behavior would ACT27).
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