Transport through side-coupled double quantum dots: from weak to strong interdot coupling
ABSTRACT We report low-temperature transport measurements through a double quantum dot
device in a configuration where one of the quantum dots is coupled directly to
the source and drain electrodes, and a second (side-coupled) quantum dot
interacts electrostatically and via tunneling to the first one. As the interdot
coupling increases, a crossover from weak to strong interdot tunneling is
observed in the charge stability diagrams that present a complex pattern with
mergings and apparent crossings of Coulomb blockade peaks. While the weak
coupling regime can be understood by considering a single level on each dot, in
the intermediate and strong coupling regimes, the multi-level nature of the
quantum dots needs to be taken into account. Surprisingly, both in the strong
and weak coupling regimes, the double quantum dot states are mainly localized
on each dot for most values of the parameters. Only in an intermediate coupling
regime the device presents a single dot-like molecular behavior as the
molecular wavefunctions weight is evenly distributed between the quantum dots.
At temperatures larger than the interdot coupling energy scale, a loss of
coherence of the molecular states is observed.
- SourceAvailable from: Pablo S. Cornaglia[Show abstract] [Hide abstract]
ABSTRACT: We analyze the transport properties of a double quantum dot device in the side-coupled configuration. A small quantum dot (QD), having a single relevant electronic level, is coupled to source and drain electrodes. A larger QD, whose multilevel nature is considered, is tunnel-coupled to the small QD. A Fermi liquid analysis shows that the low temperature conductance of the device is determined by the total electronic occupation of the double QD. When the small dot is in the Kondo regime, an even number of electrons in the large dot leads to a conductance that reaches the unitary limit, while for an odd number of electrons a two stage Kondo effect is observed and the conductance is strongly suppressed. The Kondo temperature of the second stage Kondo effect is strongly affected by the multilevel structure of the large QD. For increasing level spacing, a crossover from a large Kondo temperature regime to a small Kondo temperature regime is obtained when the level spacing becomes of the order of the large Kondo temperature.Physical Review B 01/2014; 89(11). · 3.66 Impact Factor
arXiv:1202.1580v1 [cond-mat.mes-hall] 8 Feb 2012
Transport through side-coupled double quantum dots: from weak to strong interdot
D. Y. Baines,1T. Meunier,1D. Mailly,2A. D. Wieck,3C. B¨ auerle,1L. Saminadayar,1
Pablo S. Cornaglia,4,5Gonzalo Usaj,4,5C. A. Balseiro,4,5and D. Feinberg1
1Institut N´ eel, CNRS and Universit´ e Joseph Fourier, 38042 Grenoble, France
2Laboratoire de Photonique et Nanostructures, CNRS, route de Nozay, 91460 Marcoussis, France
3Lehrstuhl f¨ ur Angewandte Festk¨ orperphysik, Ruhr-Universit¨ at,
Universit¨ atsstraße 150, 44780 Bochum, Germany
4Centro At´ omico Bariloche and Instituto Balseiro, CNEA, 8400 Bariloche, Argentina
5Consejo Nacional de Investigaciones Cient´ ıficas y T´ ecnicas (CONICET), Argentina
(Dated: February 9, 2012)
We report low-temperature transport measurements through a double quantum dot device in a
configuration where one of the quantum dots is coupled directly to the source and drain electrodes,
and a second (side-coupled) quantum dot interacts electrostatically and via tunneling to the first one.
As the interdot coupling increases, a crossover from weak to strong interdot tunneling is observed in
the charge stability diagrams that present a complex pattern with mergings and apparent crossings
of Coulomb blockade peaks. While the weak coupling regime can be understood by considering a
single level on each dot, in the intermediate and strong coupling regimes, the multi-level nature of the
quantum dots needs to be taken into account. Surprisingly, both in the strong and weak coupling
regimes, the double quantum dot states are mainly localized on each dot for most values of the
parameters. Only in an intermediate coupling regime the device presents a single dot-like molecular
behavior as the molecular wavefunctions weight is evenly distributed between the quantum dots.
At temperatures larger than the interdot coupling energy scale, a loss of coherence of the molecular
states is observed.
Double quantum dot (DQD) devices have been the
object of numerous experimental1–4and theoretical5–7
studies due to their potential applications in both classi-
cal and quantum computing,5,6,8–10and also because of
their usefulness as model systems to study the physics of
strongly correlated electrons.4,11–15
The transport signatures of these devices depend
strongly on their topology and on the geometry of the
quantum dots, which determines their energy level spac-
ings and charging energies.16–21In the so-called side-
coupled configuration, were only one of the quantum dots
is coupled to the electrodes, a rich variety of correlated
phenomena has been predicted,7,13,22–25yet few experi-
ments are available.4,26Signatures of two-channel Kondo
physics have been measured for a device with a large side-
coupled quantum dot,4while a two-stage Kondo effect
has been proposed for a small (single-level) side-coupled
dot in the Kondo regime.13Furthermore, a device with
an intermediate size of the side-coupled dot has been pre-
dicted to be a realization of the Kondo box problem.27–34
As we will show in what follows, in the weak quantum
dot-electrodes tunneling regime, where the Kondo effect
is exponentially suppressed, this type of device allows for
a controlled study of the interplay between the interdot
tunnel coupling, the temperature and the multiple levels
of the quantum dots.
In this work we measure the electronic transport
through a DQD in the side-coupled configuration and
characterize the effect of the interdot tunnel coupling and
the temperature. For sufficiently weak interdot tunnel-
ing coupling, the charge stability diagrams can be under-
stood within the usual two-level representation.2As the
tunneling coupling increases, however, the device enters
a molecular regime where the multi-level nature of the
quantum dots needs to be taken into account in order to
capture the physics of the low temperature regime.
Numerical simulations based on a simplified multi-level
double dot Hamiltonian enable to calculate the conduc-
tance through the system in the sequential tunneling
regime. We show that a qualitative understanding of the
transport properties at low and high temperature can be
reached by comparing experimental and numerical data
in the energy range where the experiment is carried out.
The rest of the paper is organized as follows: In Sec.
II we describe the experimental setup. In Sec. III we
present transport measurements illustrating the weak to
strong interdot tunneling crossover and the effect of tem-
perature. In Sec. IV we present the model and methods
for the calculation of the conductance. In Sec. V we
present numerical results that reproduce the main fea-
tures observed experimentally in the crossover from weak
to strong interdot tunneling. In Sec. VI we characterize
numerically and analytically the different interdot tun-
neling and temperature regimes. Finally, in Sec. VII we
present our concluding remarks.
Our device consists of a double quantum dot de-
fined in a two-dimensional electron gas formed in a
GaAs/AlxGa1−xAs heterostructure (density 2.4×1011
cm−2, mobility 1×106cm2V−1s−1). The quantum dots
are designed following a side-coupled (or T-shape) config-
uration where a small quantum dot (500 nm) is connected
to electron reservoirs and is side-coupled to a large quan-
tum dot (1500 nm) (Fig. 1, A). The bare charging energy
and mean level spacing of each dot extracted from non
linear measurements in the weak coupling regime are:
Ud=700 µeV, ∆d=150 µeV and UD=250 µeV, ∆D=20
µeV, for small dot (d) and large dot (D) respectively.
The differential conductance dI/dV of the double dot
system is measured by applying a DC and AC (11 Hz,
2 µV) voltage excitation on the top small quantum dot
lead, as indicated by Vbiasin Fig. 1 (A). In this particular
electrical set-up, transport occurs only through the small
quantum dot. The side-coupled large quantum dot influ-
ences the transport mechanisms via the interdot tunnel
coupling that is controlled with the voltages applied on
the middle gates. All measurements are performed in a
dilution refrigerator with a base electron temperature of
30 mK. The electronic temperature for different fridge
temperatures has been calibrated from weak localization
measurement realized in earlier experiments.35
III. TRANSPORT MEASUREMENTS
The regime of interest is a strong interdot tunnel cou-
pling regime compared to the weak coupling limit widely
studied in lateral quantum dots.36In order to empha-
size the influence of the increase of the interdot tun-
nel coupling in our system we make a comparison by
means of Fig. 1 (B, C and D) that shows two experi-
ments performed on the same double dot system at low
temperature (30 mK) and with different interdot hop-
ping strengths. Figure 1 (B) shows the weak coupling
limit. The conductance pattern follows a honeycomb lat-
tice. Note that due to the side-coupled configuration,
conductance occurs mainly at the degeneracy points of
the small dot with the Fermi energy.18Nevertheless, de-
tection of current on the degeneracy line of the large dot
with the reservoir indicates finite though weak interdot
hopping. The position of the charge states of each dot
can therefore be identified through the modulation of the
conductance on the different degeneracy lines. In other
words, a large conductance peak indicates a charge state
holding an important weight of the small dot wavefunc-
tion and vice versa. Moreover, the small dot-leads hy-
bridization (Γ) results in very thin degeneracy lines and
in a very low conductance in the Coulomb blockade val-
leys (10−4e2/h). Such a regime of weak tunneling (to
the leads and between the dots) can be accounted for in
a standard two-level representation.2The stronger cou-
pling situation is met in Fig. 1 (C and D). One can notice
that the degeneracy lines seen in the stability diagram
still appear as thin conductance lines indicating a rather
weak coupling to the leads. This point can be confirmed
by monitoring the conductance in the Coulomb blockade
valleys which is of the order of 10−3e2/h which is still
FIG. 1. (Color online) A: SEM image of the device. The red
gates are pushed far in the pinch-off regime. The interdot
tunnel coupling is controlled via the yellow gates. All ohmic
contacts are set to the same potential (V0) except the top
small dot lead where the bias voltage Vbias is applied. The
energies and occupancies of each dot are changed with the use
of the green plunger gates. B: Weak coupling stability dia-
gram (30 mK). The differential conductance (color scale) is
plotted versus the small dot and large dot plunger gate volt-
ages, Vgd and VgD respectively. C and D: Low temperature
(30 mK) stability diagrams (2D and 3D) with stronger cou-
pling between the dots. At such low temperatures, a complex
conductance modulation pattern is found.
far from the strong coupling limit (∼0.1 e2/h).37We will
consider therefore the tunnel coupling to the leads as a
weak perturbation and concentrate mainly on the effect
of the interdot hopping. Concerning the conductance
pattern, we observe that it deviates from a honeycomb
lattice. From Fig. 1 (C) one can argue that the effect
of enhanced tunneling between the dots is an effective
smoothing of the honeycomb structure which one can
expect from earlier literature.2However the nonperiodic
modulation of the conductance in each direction of the
voltage gate space depicted in Fig. 1 (D) shows unusual
transport features such as the apparent merging and the
crossing of peak structures in the VgD direction (black
lines). Such conductance features cannot be accounted
for in the common two-level representation. To be able
to capture the low temperature physics, the multi-level
structure of the large dot on the energy range of the in-
terdot tunnel coupling, tdD> ∆D, has to be considered.
In order to justify the use of a multi-level representa-
tion of the hybridization between both quantum dots we
make use of Fig. 2 (A). The depicted low-temperature
stability diagram represents a larger gate voltage scan
centered around the same region of parameters used
in the previous low temperature measurements (white
dashed rectangle). From Fig. 2 (A), it appears that the
area scanned in Fig. 1 (C, D) corresponds to a crossover
region between two different regimes. On the bottom left
side of Fig. 2 (A) one can identify smoothed honeycomb
cells (red cells). By studying the deviation of these cells
from pure honeycomb cells, we can extract a rough es-
timation of the interdot tunnel coupling, that is to say
tdD∼ 30µeV . In this limit, tdD∼ ∆D, the experimental
data point towards a two-level system behavior, hence
the apparent honeycomb cells. However by depolarising
both plunger gates one can note through the straighten-
ing of the degeneracy lines (red lines) that the hopping
term increases. This feature can be easily understood
due to crosstalk between the different gates. Once the
regime tdD> ∆Dis reached, hybridization involves mul-
tiple energy levels in dot D. As a result the molecular
addition spectrum38gains in complexity which we expect
to be reflected in the conductance through the device. A
proper calculation will be presented below to illustrate
Before going into the core of the multi-level double
dot model, it is interesting to address the question of the
evolution of the transport properties as temperature is
increased. Experimental data shown in Fig. 2 (B and C)
indicate that at high temperature, 500 mK, the strong
irregularities found at low temperature are completely
washed out and a periodic pattern is recovered (Fig. 2
B, black lines). Whereas at low temperature one can-
not interpret the conductance via the filling of both dots
independently, in the high temperature limit the stan-
dard picture can be applied. Periodic oscillations of the
conductance as a function of both plunger gate voltages
enable to keep track of the addition of electrons in each
quantum dot separately. Therefore by heating the device,
the coherence of the molecular eigenstates is broken and a
suitable description is found by simply thinking in terms
of the occupancy states in dots d and D. More precisely,
by bringing thermal energy to the system, the Fermi dis-
tribution of the metallic leads is broadened which leads
to a larger effective conduction window. We argue that
once kBT > tdD, the conductance measured through the
double quantum dot represents an average over several
molecular levels lying in the conduction window which
results in the loss of coherence of molecular states and
gram (30 mK) corresponding to a larger scan than previously
shown. The white dashed rectangle corresponds to the gate
scan seen in Fig. 1 B,C. Due to cross-talk between the plunger
gates and the middle gates defining the interdot tunneling
two different regimes can be identified in the diagram. As
tunneling increases we go from a two-level system behavior
to a multi-level system behavior. B: Color plot of the high
temperature stability diagram monitored at 500 mK in the
same plunger gate voltages range made initially (Fig 1 B,C).
C: Three-dimensional representation of the above diagram.
At high temperature, the conductance follows a periodic pat-
tern. One can clearly identify the addition of electrons in one
quantum dot or the other.
(Color online) A: Low temperature stability dia-
leads to a regular stability diagram. The picture at low
temperature becomes clearer now. As we will show be-
low, for tdD > ∆D > kBT and at the degeneracy with
the leads, the addition of an electron in the system can
only be done via a single molecular energy state. In this
regime, Fig. 1 (C, D) represents in a sense a spectroscopy
of single molecular levels. The irregular stability diagram
therefore reflects the complexity of the molecular addi-
tion spectrum of the system as already mentioned.
In what follows we present a theoretical analysis of
the experimental results using a simplified model that
captures qualitatively the main features observed in the
IV. MODEL AND METHODS
The double quantum dot (DQD) system is modeled in
the constant interaction model by the Hamiltonian2(for
related models, see e.g. Ref.  and references therein)
H = HC+ Ht+ He+ HV + Hel. (1)
Here HCdescribes the electrostatic interactions
+ UdD(ˆ ND− ND)(ˆ Nd− Nd)
2(ˆ Nℓ− Nℓ)2
dot ℓ with its corresponding gate electrode, Uℓ is the
charging energy and UdD is given by the QDs mutual
= CgℓVgℓ/Uℓ, Cgℓ is the capacitance of
describes the tunneling coupling between the different
levels on the two dots with effective single electron energy
levels ˜ ǫℓα:
describes the coupling between QD d and the left (L)
and right (R) electrodes, which are modeled by two non-
interacting Fermi gases:
We follow Refs. [18, 40, and 41] to calculate the con-
ductance through the system,
Here A(ω) is the QD spectral density and we have as-
sumed proportional (ΓL∝ ΓR) and energy independent
dot–leads hybridization functions:
[ΓL(R))]ℓ,ℓ′ = 2πρL(R)(EF)V∗
where EF = 0 is the Fermi energy of the electrodes,
ρL(R)(ǫ) is the electronic density of states of the left
(right) electrode, and VL(R),ℓ(ǫ) equals VkL(R),ℓfor ǫ =
To lowest order in Γ/kBT we replace in Eq. (7) the
exact spectral density A(ǫ) of the isolated DQD:
× ?Ψi|dmσ|Ψj?δ[ǫ − (Ej− Ei)], (9)
where |Ψi? and Ei are the exact eigenfunctions and
eigenenergies of the DQD, and Z =?
ie−βEiis the par-
tition function. We get:
where Pi= e−βEi/Z.
In the experimental setup, Γ is nonzero only for the
small dot d, and we choose it to be level independent:
?Γδn,m if n ∈ d
0 if n ∈ D
We finally have:
(Pi+ Pj)f(Ei− Ej)f(Ej− Ei)
valid for Γ ≪ kBT.
It is clear from this formula that the conductance is
suppressed for |Ei− Ej| ≫ kBT, i.e. the conductance
is low, unless two states with N + 1 and N electrons
in the DQD are nearly degenerate allowing the charge
in the molecule to fluctuate.18This is not a sufficient
condition, for a charge fluctuating in and out of a state
whose weight is mainly located in the large dot, the ma-
The calculation of the conductance maps then reduces
to obtaining the eigenenergies and eigenfunctions for the
isolated DQD molecule. This can be done by exact di-
agonalization for a limited number of states on each dot,
due to the exponential increase of the size of the Fock
space with the number of levels.
dnσ|Ψi?|2are small and suppress
V. TUNNELING CROSSOVER
In this section we present numerical results for the
conductance maps that reproduce qualitatively the main
experimental observations, as a crossover from weak to
strong interdot tunneling. We exactly solve a model of an
isolated DQD with three levels on each quantum dot and
use Eq. (12) to calculate the conductance in the weak
dot-leads coupling regime (Γ/kBT ≪ 1)
We use the experimentally obtained values for the pa-
rameters: Ud=700 µeV, ∆d=150 µeV and UD=250 µeV,
∆D=20 µeV, and UdD= 100µeV. We consider fixed in-
tradot level splittings and the same interdot tunneling
dD= tdDfor all levels α,β. To model the tun-
neling crossover observed in the experiments we include
a linear crosstalk of the gate that determines the tunnel-
ing coupling between the dots with the gate voltages Vgd
tdD∝ Vgt+ αDVgD+ αdVgd
Figure 3 shows a three-dimensional representation of
the calculated conductance as a function of the gate volt-
ages including a crosstalk with the interdot tunneling
amplitude.42The hopping amplitude increases linearly
from a minimum at the lower left corner of the figure
to its maximum at the top right corner.
see in the next section, the system goes from a low-tdD
regime for small ND,Nd, to a high-tdD regime for high
ND,Nd, both characterized by a regular array of con-
ductance peaks (honeycomb diagram), associated to the
charging of the small QD. In the intermediate tunnel-
ing regime, the electronic wavefunctions are highly de-
localized between the two QDs and the DQD enters a
single-dot molecular regime where the conductance maps
present diagonal lines of high and relatively uniform con-
ductance. The result is an apparent merging and crossing
of peaks with increasing NDas observed experimentally.
Eventually, the emergence of a regular pattern of peaks
at high tdDimplies that the wavefunctions are localized
on each dot as in the low-tdD case. As we will show in
the next section this is a consequence of the structure of
the wavefunctions in the regime ∆d∼ tdD≫ ∆D.
We now focus on the effect of the temperature on the
conductance maps. Figure 4 presents conductance maps
calculated with the same parameters as in Fig. 3 for
different values of the temperature. Each panel is calcu-
lated using a different temperature but the parameters
are otherwise equal. Increasing the temperature allows
us to observe its effect on regions with different values of
the hopping amplitude and investigate the regularization
of the patterns observed experimentally.
For temperature regimes where kBT is much smaller
than the DQD’s energy level spacings, the main effect of
increasing the temperature is to increase the width and
reduce the height of the Coulomb blockade peaks. This
can be readily seen from Eq. (12) assuming that a single
state from each charge sector contributes to the conduc-
tance at a given peak. When the temperature becomes
As we will
conductance for a double quantum dot with 3 × 3 levels, in-
cluding a gate voltage dependence of tdD = 0.015 meV+0.01
meV(ND+ Nd/3) and kBT = 0.0075 meV.
Three-dimensional representation of the calculated
of the order or larger than the level spacing in a given
charge sector, several states may contribute to the con-
ductance producing in some cases a qualitative change
in the conductance maps.
in the present DQD geometry whenever the states that
contribute to a single CB peak have a markedly different
weight on each dot. In that case, the intensity of the con-
ductance peak at low temperatures will be very different
depending on the nature of the state that dominates the
charging: a small conductance if the large dot is being
charged and a large conductance if the small dot is being
charged. At temperatures larger than the level spacing,
however, several states with a different weight on each
dot may statistically contribute to the charging resulting
in an intermediate value of the conductance.
This type of behavior is observed in Fig. 4, where an
increase in the temperature produces a broadening of the
charge degeneracy lines and leads to a more homogeneous
intensity of the conductance along them.
At the highest temperature shown in the lowest panel
of Fig. 4, the conductance pattern presents a regular
lattice of maxima.This is the expected behavior for
kBT ? tdD with the position of the conductance max-
ima given by the charging energies as in the tdD → 0
Such changes are expected
In the previous section we showed that the main fea-
tures of the measured conductance maps can be repro-
duced numerically and that the observed merging and
apparent crossing of peaks are associated to a crossover
from weak to strong interdot tunneling regimes. In this
section we characterize the different tunneling regimes.
We start with the case of uniform charging energies
(UdD = Ud = UD = U) that is obtained in the regime
of large interdot capacitance. This case allows for an
FIG. 4. Calculated conductance maps for a double quantum
dot with 3 × 3 levels, including a gate voltage dependence of
tdD = 0.015meV + 0.01meV(ND + Nd/3) and different val-
ues of the temperature: a) kBT = 0.0075 meV, b) kBT =
0.02meV, c) kBT = 0.04meV, and d) kBT = 0.05meV.
Other parameters are Ud= 0.7meV, UD = 0.25meV, UdD =
0.1meV, ∆d= 0.15meV, and ∆D = 0.02meV.
analytical solution and already contains the underlying
structure of the general case.
A. Large interdot capacitance
For UdD = Ud = UD = U the isolated DQD can be
readily solved as the interaction term only depends on the
total number of electrons in the molecule N = Nd+ ND
which is a good quantum number. The energy of the
system is given by EN= EN
the eigenenergies of Htb= Ht+ He, and EN
N)2, where N = Nd+ND. The charge degeneracy points
(Ej− Ei= 0) satisfy the equation
αεα, where the εαare
Nd+ ND= N +1
that determines a series of parallel lines in the (Nd,ND)
plane. The intensity associated to these lines in the con-
ductance map is proportional to the weight of the addi-
tional electron’s wavefunction in dot d. In the case of
uncoupled dots (tdD = 0), only the lines associated to
the charging of dot d present a maximum in the conduc-
tance. In the general case (tdD?= 0) the wavefunctions of
the DQD are delocalized between the dots and the inten-
sity of the conductance lines is modulated accordingly.
Surprisingly, for large tdD the wavefunctions are again
mainly localized on each dot as in the small tdDregime.
To show this latter point, we further simplify the model
by considering ˜ εℓα= ˜ εℓ(i.e. ∆d= ∆D= 0) and tαβ
where 1 is the identity matrix and
tdD tdD ··· tdD
tdD ··· ··· tdD
tdD ··· ··· tdD
The Hamiltonian matrix Htbcan be diagonalized exactly
for its eigenvectors. Among all wavefunctions, only two
are strongly delocalized between the dots, having half of
the weight on each dot, the rest of the states are either
fully localized at dot d and have energy ˜ εd or are fully
localized at dot D and have energy ˜ εD (assuming ˜ εd?=
˜ εD). For a finite level spacing on each dot ∆D,∆d≪ tdD,
the states remain localized in one of the dots, with only
a small weight (≪ 1) on the other dot. This structure
of eigenstates persists even if one of the level spacings
becomes of the order or even larger than the hopping
amplitude, e.g. ∆d? tdD≫ ∆D.
B.Experimental situation: Ud> UD > UdD
In the experimental situation, there is a hierarchy of
interactions: Ud > UD > UdD and it is generally no
longer possible to solve the interaction and tight-binding
parts of the Hamiltonian independently as it was done
in the previous section. However, as we shall see, the
analysis presented above serves as a guide when tackling
the general case.
We first focus on the limit of small tdD where the
charge in each dot is well defined. The charging con-
ditions for dots d and D are given by
(Nd− Nd), (17)
respectively, where ˜ εℓNℓ+1is the energy of the concerned
level on dot ℓ, and determine two sets of parallel straight
lines in the (Nd,ND) plane. The charging of the DQD
tum dot with 3 × 3 levels, for different values of the interdot
hopping tdD a)0meV, b)0.02meV, c)0.04meV, d)0.05meV,
and e)0.08meV. Other parameters are Ud = 0.7meV,
UD = 0.25meV, UdD = 0.1meV, ∆d = 0.15meV, and
∆D = 0.02meV. The high-tdD and low-tdD with segments
of high conductance are clearly observed.
Calculated conductance maps for a double quan-
and the conductance maps are determined by these equa-
tions and the result, for UdD > 0, is a series of high
conductance segments with slope −Ud/UdD associated
to the charging of the small dot [see Fig. 5(a)]. The end-
points of these segments are given by the intersections
of the charge degeneracy lines (CDL) of the small dot
by the CDLs of the large dot, i.e by triple degeneracy
points: E(Nd,ND) = E(Nd,ND+ 1) = E(Nd+ 1,ND)
and E(Nd,ND) = E(Nd+ 1,ND− 1) = E(Nd+ 1,ND).
The separation between two consecutive segments, for a
fixed ND, is associated to the extra energy required to
add an electron and depends on the parity of electron
number on dot d: it is Ud, when a second electron is
added to a partially occupied level (odd electron valley)
and Ud+ ∆d when it is added to an empty level (even
A finite but small tdD< ∆D,∆dproduces a distortion
of the high conductance segments, and a small peak in
the conductance associated to the charging of the large
dot, due to interdot mixing. The main features in this
regime can be understood within a simplified model with
a single level on each dot [see Fig. 5(b)].
For intermediate values of tdDthe states are strongly
mixed between the two dots and the charge on each dot
is no longer well defined. In this case, the conductance
maps are not expected to have a regular pattern of seg-
ments with high conductance. Instead, as is shown in
Figs. 5(c) and 5(d), the situation is similar to the high
interdot capacitance case (see previous section) with the
conductance map showing diagonal lines of high conduc-
tance separated by an effective interaction.
In the regime of large tdD, however, the state wavefunc-
tions are, as in the uncoupled case, mainly localized on
each dot. Equations (17) determine the conductance map
patterns in the limit tdD→ 0 but also for tdD→ ∞ and
give their qualitative shapes in the regimes tdD ≪ ∆D
and ∆d∼ tdD≫ ∆D.
An important difference between the large and small
tdDlimits is that while in the small tdDlimit, all states
have a well defined number of electrons on each dot
(excluding specific regions in the parameter space), in
the large tdD limit there are only two states which are
strongly delocalized between the dots.
One of these states, which is fully symmetric between
dots, is the ground state and is the first to be charged
as the gate voltages are swept. The complete charging of
this state adds a single electron to each dot and there-
fore changes the parity for the subsequent charging of
the dots. This parity change alters the sequence of dis-
tances between high conductance peaks as the gate volt-
age of the small dot is swept: the separation between two
consecutive segments, for a fixed ND, is now: Udwhen
adding an odd electron on dot d and Ud+∆dfor an even
The above-mentioned parity effect persists even for
tdD? ∆d, and the pattern of peaks in the conductance
follows qualitatively what is expected in the high tdD
limit. A qualitative understanding of the peaks positions
and shapes can be obtained considering that for a finite
but large value of tdDthe states are not fully localized on
each dot. A small interdot mixing of the states reduces
the effective charging energy of dot d and increases the
effective interdot interaction. This leads to a reduction of
the slope of the high conductance segments [see Fig. 5e)]
which is given by −Ud/UdDin the tdD→ 0 and tdD→ ∞
VII. SUMMARY AND CONCLUSIONS
We have studied the transport through a double-
quantum-dot system in a side-coupled configuration as
a function of temperature and interdot tunneling cou-
pling. We have focused on the weak QD-electrodes cou-
pling regime and analyzed the structure of the DQDs
molecular wavefunctions. The topology of the device al-
lows us to study via transport measurements, how the
wavefunction weight is distributed between the QDs for
each molecular state. The geometry of the device makes
it possible to explore different tunneling coupling and
temperature regimes. We have constructed and solved a
simplified model that reproduces the experimentally ob-
For a weak interdot coupling (tdD ≪ ∆D,∆d) the
molecular states can be accurately described considering
a model with two-levels, one from each QD, coupled by
a hopping term tdD. The resulting molecular wavefunc-
tions are essentially localized on one of the quantum dots
for most values of the plunger gate voltages. The con-
ductance in the (Vgd,VgD) plane reflects this structure
of molecular eigenstates and shows a series of CB peaks
associated to the charging of the QD directly coupled to
the electrodes. Much weaker CB peaks are obtained as
the side-coupled QD is charged due to a small mixing
between the QDs.
When the interdot coupling is increased, more le-
vels from each QD are involved in the formation of a
given molecular state. An intermediate tunneling regime
(tdD ? ∆D) can be reached where the molecular wave-
functions are strongly delocalized between the QDs. In
this situation, the conductance maps present a series of
lines of high and relatively uniform conductance in the
(Vgd,VgD) plane and resemble those expected for a single
quantum dot coupled to two plunger gates.
For large enough interdot coupling,(tdD∼ ∆d≫ ∆D),
the nature of the eigenfunctions changes and most molec-
ular states become increasingly localized on each QD as
in the weak tdD limit. There is, however, an important
difference between these two regimes due to the emer-
gence, in the high tdD limit, of two states of a different
nature that result from a mixing of several levels from
each dot. These states are a symmetric and antisymmet-
ric combination between the states of the two QDs, and
have (for tdD> 0), a lower and higher energy than their
component states, respectively. The charging of the low-
est lying of these states involves adding a single electron
to each QD and alters the even-odd sequence of CB peaks
producing a shift of the high conductance peaks in the
In the experiments there is a crosstalk between the
plunger gates of each QD and the gate controlling the
tunneling barrier between the QD. As a consequence,
different regions of the (Vgd,VgD) plane have associated
different intensities of tunneling coupling and it is pos-
sible to observe the above-mentioned regimes in a single
conductance map. The crossover between the different
regimes gives rise to a complex evolution of the CB peaks
with mergings and apparent crossings.
Finally, we analyzed the effect of the temperature on
the transport properties. For kBT > tdD, several levels
contribute to the conductance of each Coulomb blockade
peak, the pattern of conductance maxima is similar to
that of weakly coupled dots in the kBT ∼ tdDregime, and
resemble those of uncoupled dots (tdD→ 0) for kBT ≫
We thank P. Simon for driving our interest towards
such a double dot setup, and S. Florens for stimulating
discussions.P.S.C., G.U. and C.A.B. acknowledge fi-
nancial support from PIP 11220080101821 of CONICET
and PICT-Bicentenario 2010-1060 of the ANPCyT. T.
M. acknowledges partial funding from Marie Curie ERG
224786. D.M., C.B. and L.S. acknowledge financial sup-
port from ANR project “QuSpin”. D.F. acknowledges
support from PICS 5755 of CNRS.
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