# 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.

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**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

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arXiv:1202.1580v1 [cond-mat.mes-hall] 8 Feb 2012

Transport through side-coupled double quantum dots: from weak to strong interdot

coupling

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.

I. INTRODUCTION

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.

II.EXPERIMENT

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

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2

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

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3

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

this point.

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

FIG. 2.

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-

Page 4

4

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

conductance maps.

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. [39] and references therein)

H = HC+ Ht+ He+ HV + Hel. (1)

Here HCdescribes the electrostatic interactions

HC=

?

ℓ=d,D

+ UdD(ˆ ND− ND)(ˆ Nd− Nd)

Uℓ

2(ˆ Nℓ− Nℓ)2

(2)

where Nℓ

dot ℓ with its corresponding gate electrode, Uℓ is the

charging energy and UdD is given by the QDs mutual

capacitance.17,19–21

= CgℓVgℓ/Uℓ, Cgℓ is the capacitance of

Ht=

?

σ,α,β

tαβ

dD

?

d†

dασdDβσ+ h.c

?

, (3)

describes the tunneling coupling between the different

levels on the two dots with effective single electron energy

levels ˜ ǫℓα:

He=

?

ℓ=d,D

?

σ,α

˜ ǫℓαd†

ℓασdℓασ, (4)

and

HV =

?

α

?

ν=L,R

?

k,σ

Vkαν

?

c†

νkσddασ+ h.c.

?

, (5)

describes the coupling between QD d and the left (L)

and right (R) electrodes, which are modeled by two non-

interacting Fermi gases:

Hel=

?

ν,k,σ

ǫkc†

νkσcνkσ. (6)

We follow Refs. [18, 40, and 41] to calculate the con-

ductance through the system,

G =e2

?

?

dǫ

?

−∂f(ǫ)

∂ǫ

?

Tr

?

ΓRΓL

ΓR+ ΓLA(ǫ)

?

.(7)

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∗

L(R),ℓ(EF)VL(R),ℓ′(EF). (8)

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 =

ǫk.

To lowest order in Γ/kBT we replace in Eq. (7) the

exact spectral density A(ǫ) of the isolated DQD:

Aσ

n,m(ǫ) =1

Z

?

i,j

(e−βEi+ e−βEj)?Ψj|d†

nσ|Ψi?

× ?Ψi|dmσ|Ψj?δ[ǫ − (Ej− Ei)], (9)

where |Ψi? and Ei are the exact eigenfunctions and

eigenenergies of the DQD, and Z =?

G =e2

?

n,m

i,j

× ?Ψj|d†

ie−βEiis the par-

tition function. We get:

?

Γn,m

?

(Pi+ Pj)

?

−∂f(ω)

∂ω

????

Ei−Ej

?

nσ|Ψi??Ψi|dmσ|Ψj?

(10)

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:

Γn,m=

?Γδn,m if n ∈ d

0 if n ∈ D

(11)

We finally have:

G =e2

?

Γ

kBT

?

i,j

(Pi+ Pj)f(Ei− Ej)f(Ej− Ei)

×

?

n

|?Ψj|d†

dnσ|Ψi?|2

(12)

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-

trix elements?

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.

n|?Ψj|d†

dnσ|Ψi?|2are small and suppress

the conductance.

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5

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

coupling tαβ

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

and VgD:

tdD∝ Vgt+ αDVgD+ αdVgd

(13)

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

FIG. 3.

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

case.

Such changes are expected

VI.TUNNELING REGIMES

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

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6

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

C+?N

αεα, where the εαare

C=U

2(N −

Nd+ ND= N +1

2+ εN+1/U(14)

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

tdD. Then

dD=

Htb=

?˜ εd1T

T

˜ εD1

?

, (15)

where 1 is the identity matrix and

T =

tdD tdD ··· tdD

tdD ··· ··· tdD

······ ···

tdD ··· ··· tdD

···

.(16)

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+1

2+˜ εdNd+1

2+˜ εDND+1

Ud

+UdD

Ud

(ND− ND),

ND= ND+1

UD

+UdD

UD

(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

Page 7

7

FIG. 5.

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

electron valley).

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

electron.

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→ ∞

limits.

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-

Page 8

8

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-

served regimes.

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

(Vgd,VgD) plane.

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 ≫

tdD.

ACKNOWLEDGMENTS

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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42In the experiments, a crosstalk is observed between gates

Vgdand VgD. Assuming a linear crosstalk we have: NdUd=

CgdVgd+CD

relations can be easily inverted to obtain the experimental

axes Vgd and VgD. The result is a linear transformation

that may include scaling, shear and rotation. This type of

transformations need to be considered to obtain a more

quantitative agreement between theory and experiment.

H.Mebrahtu,

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gDVgd. These